MIPT/TH-26/25

ITEP/TH-35/25

1 Introduction↩︎

Yangian algebras were originally introduced by Drinfeld [1] for simple finite-dimensional Lie algebras \(\mathfrak{g}\). The Yangians were defined as a canonical deformation of the universal enveloping algebra U\((\mathfrak{g}[z])\) for the corresponding current algebra \(\mathfrak{g}[z]\). However, this definition turned out to be inconvenient for describing the representations of these algebras.

Later Drinfeld [2] addressed these limitations and introduced an alternative definition of these algebras that resembled the Chevalley bases of the corresponding Lie algebras. The new construction, also known as the Drinfeld second realization, highlighted the highest-weight structure of the finite-dimensional irreducible representations and completely classified them in terms of Drinfeld polynomials [2], [3].

Another fruitful perspective on Yangian algebras has emerged from the context of superstring theory. In particular, the study of BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds gives rise to algebraic structures known as BPS algebras [4]–[9]. These algebras can be classified by quivers that encode the effective gauge theories describing the low-energy dynamics of brane systems. In these settings, the quivers are constructed from toric diagrams via brane tiling [10], [11]. These Yangians are commonly referred to as quiver Yangian algebras in the literature.

One of the richest and most interesting directions of research is to study the representation theory of these algebras. Although the full theory is still under development, numerous important results have already been established. In the paper, we focus on the special class of Yangian representations — called crystal representations — whose states can be described by the statistical model of crystal melting [12]–[14]. The requirement that crystals transform into valid neighboring crystals under the action of the Yangian allows one to bootstrap the corresponding matrix elements of the generators in terms of meromorphic functions [11]. An alternative and more computationally convenient approach is to use equivariant localization techniques [15], [16], where the matrix elements are computed as integrals over quiver moduli spaces [17]. Duistermaat-Heckman integration formulae [18], [19] ensure that the calculations localize to contributions from the neighborhoods of fixed points on the quiver varieties.

There has been a particular interest in affine quiver Yangians, especially \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{n})\) and its supersymmetric counterparts \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{n|m})\) that also fall into the latter category. These affine algebras have been studied extensively, have wide-ranging applications, and have rich algebraic structures. The simplest example of these algebras is the Yangian \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{1})\) [20]–[22], which admits the well-known Fock representation. This representation is a crystal representation whose states are parameterized by Young diagrams, and it plays a central role in the theory of quantum integrable systems such as Calogero-Moser-Sutherland systems [23], [24] and WLZZ models [25]–[27]. The eigenfunctions of Calogero Hamiltonians are classical Schur/Jack polynomials [28]–[30], which are tightly related to superintegrability [31], [32]. The study of similar families of polynomials is itself a broad and active research area [33]. Notably, the Yangian structure allows natural generalizations of these families. For instance, the MacMahon representation of \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{1})\) yields 3-dimensional analogues of Young diagrams and Jack polynomials [34]. The lift to the \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{n})\) algebras gives rise to colored Young diagrams and Uglov polynomials [31], [35].

In the supersymmetric case, affine super Yangians \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{n|m})\) generalize these structures further. The most prominent example is the algebra \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{1|1})\) [28], [36], [37], which admits semi-Fock crystal representation with states enumerated by super-Young diagrams. In a similar way, the corresponding family of orthogonal polynomials emerges, known as super-Schur/Jack polynomials [36], [38].

One can extend the ideas above and consider the quiver Yangians associated with arbitrary affine Dynkin diagrams [39], [40]. While affine Yangians exhibit rich structure and wide applications, their analysis is often complicated by the presence of infinite-dimensional representations and intricate algebraic relations. In this paper, we focus instead on the Yangian algebras associated with finite Dynkin diagrams. These algebras admit finite-dimensional irreducible representations and possess simpler, more tractable structures, which makes them suitable for explicit construction.

More specifically, we explore the Yangian algebras \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) associated with Dynkin diagrams of A type. We use the quiver approach combined with equivariant integration techniques to explicitly describe their representations. The case of \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{2})\) was previously described using similar methods in [28]; see also [41], [42]. While the approach is robust and, in theory, can extend beyond Dynkin classification, it naturally describes only the highest-weight representations with a single non-zero highest weight. These representations are labeled by rectangular Young diagrams; therefore, we refer to them as rectangular. Notably, the states of these representations still have crystal structure. Moreover, they allow natural parametrization in terms of the Gelfand-Tsetlin bases [43], [44]. This correspondence we gradually introduce through the text.

The paper is organized as follows. Section 2 provides an overview of the construction of quiver Yangian algebras. We start with general data and gradually focus on the Dynkin quivers of A type. In section 3 we describe the algebras \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) and their representations in detail. We begin with a warm-up example, the Yangian \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\), in section 3.1, proceed to the representations of the \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\) algebra in section 3.2, and generalize the previous results to an arbitrary \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) algebra in section 3.3. Finally, in section 3.4 we show that the constructed Yangians are in fact isomorphic to Drinfeld Yangians [2] and give our comments on the construction.

2 Quiver Yangian algebras \(\mathsf{Y}({\cal Q})\)↩︎

2.1 Quiver Data↩︎

We start with a pair \(({\cal Q}, {\cal W})\) of a quiver diagram \({\cal Q}= (Q_{0}, Q_{1})\) where \(Q_{0}\) is the set of vertices of the quiver \({\cal Q}\), and \(Q_{1}\) is the set of the arrows between these vertices. The function \({\cal W}\) is a superpotential. We also introduce some useful notations:

• \(\{a\to b\}\) – a set of arrows flowing from node \(a\) to node \(b\).

• \(|a\to b|\) – a number of arrows flowing from node \(a\) to node \(b\).

If we are to specify the head and the tail of an arrow \(I\), we denote it as \(I\colon a \to b\). For a pair of nodes \(a, b\in Q_{0}\) we define their chirality by: \[\chi_{ab} = |a \to b| - |b \to a|\,.\] The quiver is called non-chiral when \(\chi_{ab} = 0\) for any pair of nodes; otherwise the quiver is called chiral. For some notes related to chiral quivers, see, for example, [45] or [46]. From now on we consider only non-chiral quivers.

The quiver formalism has previously been applied to describe BPS algebras arising from D-brane systems on Calabi–Yau three-folds [11], [16], [28]. In this work, we employ essentially the same framework, adapted to the study of Dynkin quivers, following the general ideas and motivations presented, for example, in [39], [40].

We now discuss the quiver data in more detail.

1. The quiver nodes can be divided into gauge nodes and framing nodes. We draw them using different shapes in the diagram: the gauge nodes are denoted as round nodes, whereas framing nodes are denoted as square nodes (see fig. 1). With each node \(a\in Q_{0}\) we associate a vector space \(V_{a} = \mathbb{C}^{d_{a}}\) and a gauge or a flavor group \(GL(d_{a}, \mathbb{C})\) with a dimension parameter \(d_{a}\geqslant 0\). We also assume that there is a field \(\Phi_{a} \in {\rm Hom}(V_{a}, V_{a})\) associated with the node.

2. We work with the quivers with only one type of arrows. To each arrow \(I\in \{a \to b\}\) we assign a chiral bifundamental field \(q_{I}\) that is charged by \(\overline{GL(d_{a})}\times GL(d_{b})\). If a node \(a\) or \(b\) is a framing node, \(q_{I}\) is just an (anti-)fundamental field. We suppose it to be a linear map from \(V_{a}\) to \(V_{b}\), therefore, \(q_{I:a\to b} \in {\rm Hom}(V_{a}, V_{b})\). We also assume that additional \(U(1)\) flavor symmetries are associated with framing nodes and assign (equivariant) weight \(h_{I} \in \mathbb{C}\) to each field. Moreover, we introduce R-charges \(R_{I}\) for each arrow of the quiver. There are \(|Q_{1}|\) weights, or R-charges, that are, in general, unconstrained.

3. The superpotential \({\cal W}\) is a holomorphic gauge invariant function of fields. We impose that the superpotential can be decomposed into a sum of monomials in \(q_{I}\). Gauge invariance requires the monomials to form closed loops \(\{L\}\) in the quiver \({\cal Q}\). The superpotential is flavor invariant as well. This leads us to the loop constraints that, together with vertex constraints, define the relations between equivariant weights: \[\label{Loop32and32Vertex32Constraints} \begin{align} &\text{Loop constraint: } \quad\sum_{I\in L} h_{I} = 0\,, \quad \forall L\in {\cal W}\\ &\text{Vertex constraint: } \quad \sum_{I \in a}\mathop{\mathrm{\mathop{sgn}}}_{a}(I)\,h_{I} = 0\,, \quad \forall a \in Q_{0}\,, \end{align}\tag{1}\] where \(\mathop{\mathrm{\mathop{sgn}}}_{a}(I)\) is equal to \(+1\) if the arrow flows towards the vertex \(a\), \(-1\) if the arrow flows outwards from the vertex \(a\), and \(0\) otherwise. The reasons behind the vertex constraints we address later in the text. The R-charges are also constrained by the superpotential due to the fact that its R-charge is fixed: \[\label{Loop32Constraints32R-charges} \sum_{I\in L} R_{I} = 2\,, \quad \forall L\in {\cal W}\,.\tag{2}\]

2.2 Yangian Algebra↩︎

Having defined the quiver data and superpotential, we now proceed to define the quadratic relations of the Yangian algebra following [11], [40].

The generators of the algebra are divided into \(|Q_{0}|\) triplets \((e^{(a)}_{n}, f^{(a)}_{n}, \psi^{(a)}_{n})\) where \(n\in \mathbb{Z}\), \(a\in Q_{0}\). They can be organized into generating functions for convenience²: \[\label{General32generating32functions} e^{(a)}(z)=\sum\limits_{n=0}^{\infty}\frac{e_n^{(a)}}{z^{n+1}},\quad f^{(a)}(z)=\sum\limits_{n=0}^{\infty}\frac{f_n^{(a)}}{z^{n+1}},\quad \psi^{(a)}(z)=1 + \sum\limits_{n=0}^{\infty}\frac{\psi_n^{(a)}}{z^{n+1}}\,.\tag{3}\] Generators \(e^{(a)}(z)\) and \(f^{(a)}(z)\) have parity: \[|a|=(|a\to a|+1)\;{\rm mod}\;2\,,\] that counts the number of loops at vertex \(a\). Generators \(\psi^{(a)}(z)\) always have parity 0.

For each pair of vertices, we introduce the bonding factors with the help of equivariant weights of arrows between these vertices: \[\label{bonding} \varphi_{a,b}(z)=\frac{\prod\limits_{I\in\{a\to b\}}(z+h_I)}{\prod\limits_{J\in\{b\to a\}}(z-h_J)}\,.\tag{4}\] The generating functions defined above help us to compactly write the quadratic relations of the algebra: \[\label{Yangian32Relations} \begin{align} &e^{(a)}(z)e^{(b)}(w)\simeq (-1)^{|a||b|}\varphi_{a,b}(z-w)\,e^{(b)}(w)e^{(a)}(z)\,,\\ &\psi^{(a)}(z)e^{(b)}(w)\simeq \varphi_{a,b}(z-w)e^{(b)}(w)\,\psi^{(a)}(z)\,,\\ &f^{(a)}(z)f^{(b)}(w)\simeq (-1)^{|a||b|}\varphi_{a,b}(z-w)^{-1}f^{(b)}(w)f^{(a)}(z)\,,\\ &\psi^{(a)}(z)f^{(b)}(w)\simeq \varphi_{a,b}(z-w)^{-1}f^{(b)}(w)\psi^{(a)}(z)\,,\\ &\psi^{(a)}(z)\psi^{(b)}(w)=\psi^{(b)}(w)\psi^{(a)}(z)\,,\\ &\left[e^{(a)}(z),f^{(b)}(w)\right\}\simeq-\delta_{ab}\frac{\psi^{(a)}(z)-\psi^{(b)}(w)}{z-w}\,, \end{align}\tag{5}\] where \([*,*\}\) denotes a super-commutator, and \(\simeq\) denotes an equivalence of series up to \(z^{n}w^{k\geq 0}\) and \(z^{n\geq 0}w^{k}\). One could unfold these relations in terms of modes. We refer the interested reader to a more detailed review [11].

In general, Yangian algebras have higher-order algebraic relations that are called Serre relations. However, cubic and higher-order relations require additional consideration and depend on the choice of the algebra. Some useful notes on Serre relations are given in [11]. A conjecture in the case of a general quiver is given in [49]. The affine cases \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{m|n})\) are covered in [50], [51]. The Serre relations for \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) algebras are mentioned later in the text.

2.3 Basic Properties↩︎

Let us now mention some basic properties of the algebra \(\mathsf{Y}({\cal Q})\) defined above. They are useful in understanding the structure of the relations 5 and address the motivation behind the vertex constraints 1 . All the properties that we discuss are tightly related to automorphisms of the algebra and were analyzed, for example, in [11], [20].

• The algebra admits \(\mathbb{Z}_{2}\) symmetry: \[\label{Z95232symmetry} e^{(a)}(z) \leftrightarrow f^{(a)}(z), \qquad \psi^{(a)}(z) \to \psi^{(a)}(z)^{-1}\,,\tag{6}\] that introduces a \(\mathbb{Z}_{2}\) grading of the generators.

• The relations also have rescaling symmetry³: \[\label{Z32symmetry} \begin{align} &h_{I} \to \sigma h_{I}\,, \qquad z \to \sigma z\,,\\ &e^{(a)}(z) \to \sigma^{-1}e^{(a)}(z)\,, \quad f^{(a)}(z) \to \sigma^{-1}f^{(a)}(z)\,, \quad \psi^{(a)}(z) \to \sigma^{-1}\psi^{(a)}(z)\,,\\ &e_{n}^{(a)} \to \sigma^{n}e_{n}^{(a)}\,, \quad f_{n}^{(a)} \to \sigma^{n}f_{n}^{(a)}\,, \quad \psi_{n}^{(a)} \to \sigma^{n}\psi_{n}^{(a)}\,. \end{align}\tag{7}\] The transformation rewritten in terms of modes shows that there is a natural \(\mathbb{Z}\) grading. This grading is also known as level grading, or spin filtration [11].

• In fact, there is another scaling symmetry: \[\label{normalization32symmetry} \begin{align} &e^{(a)}(z) \to \alpha e^{(a)}(z)\,, \quad f^{(a)}(z) \to \alpha^{-1}f^{(a)}(z)\,, \quad \psi^{(a)}(z) \to \psi^{(a)}(z)\,,\\ &e^{(a)}_{n} \to \alpha e_{n}^{(a)}\,, \quad f^{(a)}_{n} \to \alpha^{-1}f_{n}^{(a)}\,, \quad \psi_{n}^{(a)} \to \psi_{n}^{(a)}\,. \end{align}\tag{8}\] This symmetry has a simpler form and can be fixed by choosing a bilinear form with respect to which the generators are conjugate, in other words, a normalization.

• The last symmetry that we mention in this text has a more complicated structure. It changes the spectral parameter \(z\) of the algebra and therefore is known as spectral32shift. The symmetry is parameterized by a complex number \(s_{a} \in \mathbb{C}\) and takes the following form: \[\label{spectral32shift} \begin{align} &e^{(a)}(u) \to e^{(a)}(u - s_{a})\,, \qquad e^{(a)}_{j} \to \sum_{k = 0}^{j}\binom{j}{k}s_{a}^{j-k}e^{(a)}_{k}\,,\\ &f^{(a)}(u) \to f^{(a)}(u - s_{a})\,, \qquad f^{(a)}_{j} \to \sum_{k = 0}^{j}\binom{j}{k}s_{a}^{j-k}f^{(a)}_{k}\,,\\ &\psi^{(a)}(u) \to \psi^{(a)}(u - s_{a})\,, \qquad \psi^{(a)}_{j} \to \sum_{k = 0}^{j}\binom{j}{k}s_{a}^{j-k}\psi^{(a)}_{k}\,. \end{align}\tag{9}\] One could easily confirm that the relations 5 are invariant under this transformation in terms of generating functions. The binomial identities are useful to prove it in terms of modes [20].

We related the equivariant parameters \(h_{I}\) to the global symmetry of the theory. We, therefore, imposed the loop constraints 1 . One could notice that these constraints are invariant under the following shift of equivariant parameters: \[\label{gauge32shift} h_{I} \quad \to \quad h_{I} + \varepsilon_{a}\mathop{\mathrm{\mathop{sgn}}}_{a}(I) - \varepsilon_{b}\mathop{\mathrm{\mathop{sgn}}}_{b}(I)\,,\tag{10}\] where \(\varepsilon_{a}\) are parameters of the transformation and \(I\in \{a\to b\}\). This means, we have redundancy in the parametrization. From physical point of view, it represents the fact that some flavor symmetries are gauge [11]. Indeed, if we mix the global symmetry with a gauge symmetry associated with a vertex \(a\), we get 10 .

Now, let us modify the generators in each vertex \(a\in Q_{0}\): \[\widetilde{e}^{(a)}(z) = e^{(a)}\bigl(z + \varepsilon_{a}\bigr)\,, \quad \widetilde{f}^{(a)}(z) = f^{(a)}\bigl(z + \varepsilon_{a}\bigr)\,, \quad \widetilde{\psi}^{(a)}(z) = \psi^{(a)}\bigl(z + \varepsilon_{a}\bigr)\,.\] One could ask if these new generators still form a quiver Yangian algebra. In order to satisfy the relations 5 , we should also modify the bond factors 4 in the following way: \[\varphi_{a,b}(u)=\frac{\prod\limits_{I\in\{a\to b\}}(u+h_I)}{\prod\limits_{J\in\{b\to a\}}(u-h_J)} \qquad \to \qquad \widetilde{\varphi}_{a, b}(u) = \frac{\prod\limits_{I\in\{a\to b\}}\bigl(u+h_I + \varepsilon_{a} - \varepsilon_{b}\bigr)}{\prod\limits_{J\in\{b\to a\}}\bigl(u-h_J + \varepsilon_{a} - \varepsilon_{b}\bigr)}\,.\] Using appropriate spectral shifts 9 , one could compensate 10 . This procedure only shuffles the generators by linear combinations. This means, one could regard the spectral shift 9 as gauge symmetry.

One could deal with this gauge freedom directly; however, the more common choice [11], [45], [52] is to fix it by imposing the vertex constraint 1 .

2.4 States↩︎

Having defined the algebra relations, we now focus on its representations. The first step is to describe the states. Mathematically, the states are fixed points on the moduli space of quiver representations with respect to equivariant action originating from the toric geometry [16], [22], [53]. First, we review the effective description of the fixed points that considers the states as sets of paths or posets⁴ on the quiver. This approach is covered, for example, in reviews [40], [54], [55]. Notably, for the toric Calabi-Yau threefolds we end up with 3-dimensional molten crystals [11], [12], [14], [28]. This discussion applies (with new features) to toric CY fourfolds [56]–[58]. Even more general crystal models were constructed recently using the Jeffrey-Kirwan residue formulas [59] in [52].

Sets of Paths↩︎

To define the states directly in terms of quiver data, we briefly review some key notions of the path algebras of a quiver. The algebra \(\mathbb{C}{\cal Q}\) is generated by arrows in the quiver diagram, and the multiplication is defined by concatenation. A path \({\cal P}\) is a sequence of \(n\) arrows \(I_{s}\): \[\label{path32example} I_{1}\cdot I_{2}\cdot\ldots\cdot I_{n}\,,\tag{11}\] where each target of a previous arrow points at the source of the next arrow, which means that a path is connected.

In the presence of a superpotential, one should impose equivalence relations on the paths given by \(F\)-term relations: \[\label{general32quiver32F-terms} F_{I} = \partial_{I}{\cal W}= 0\,.\tag{12}\] For example, let us consider the superpotential: \[{\cal W}= {\rm Tr}\,(I_{1}I_{3} - I_{1}I_{4}I_{2})\,.\] The relation \(\partial_{I_{1}}{\cal W}= 0\) defines the path equivalence: \[I_{1} \simeq I_{4}I_{2}\,.\] This algebra is known as Jacobian algebra of a quiver: \[\label{Jacobian32algebra} {\cal J}_{(Q, W)} = \mathbb{C}{\cal Q}\,/\,F_{I}\,.\tag{13}\]

We have already mentioned the framing nodes in section 2.1. A choice of framing nodes and the arrows connecting them with the remaining quiver is called “framing". The states and therefore representations of the quiver depend on the choice of framings. We assume that we have a single framing node in the quiver and its framing dimension is \(d_{\mathfrak{f}} = 1\), and label the arrow connecting the framing node to a gauge node as \(R\). This choice is known as a canonical framing. For more consistent discussion of various framings, see, for example, [9], [40], [45], [60], [61].

Framing projects the Jacobian algebra to a subalgebra that we denote by \({\cal J}^{\sharp}\). The paths grow from the framing field \(R\): \[\label{Jacobian32algebra32framed} {\cal J}^{\sharp}_{(Q, W)} = {\cal J}_{(Q, W)}\cdot R\,.\tag{14}\] Analogously to the case of toric Calabi-Yau threefolds, we call each path an “atom", and a state of the representation should be viewed as a set of these atoms, up to the homotopic equivalence defined by eq. ¿eq:general32quiver32F-terms? , which is called”crystal"⁵. In these settings, one can think of a framing as a choice of a root atom.

There are two sets of parameters that are related to the arrows and therefore the paths on a quiver, namely, equivariant weights and R-charges. The constraints on equivariant weights can be resolved, and the parameters form a \(k\)-dimensional equivariant linear space: \[\label{general32arrow32weights} h_{I} = \sum_{j = 1}^{k}x^{j}_{I}\epsilon_{j}\,,\tag{15}\] where \(x^{j}_{I}\) is the \(j\)-th coordinate of the \(h_{I}\) in the equivariant space. For example, in the case of toric \({\boldsymbol{C}Y}_{3}\), there are two independent equivariant parameters [11].

Naturally, the equivariant weights and R-changes can also be defined for the paths \({\cal P}\) 11 : \[\label{general32path32weights} \begin{align} &x_{{\cal P}}^{j} = x^{j}_{I_{1}} + x^{j}_{I_{2}} + \ldots + x_{I_{n}}^{j}, \qquad R_{{\cal P}} = R_{I_{1}} + R_{I_{2}} + \ldots + R_{I_{n}}\,,\\ &h_{{\cal P}} = h_{I_{1}} + h_{I_{2}} + \ldots + h_{I_{n}}\,. \end{align}\tag{16}\] The starting point of a path is fixed by \(R\). Together with the homotopic equivalence, it means that the atoms are specified by their heads; therefore, they can be represented as points on a \(k + 1\)-dimensional space: \[{\cal P}\quad \leadsto \quad (\vec{x}_{{\cal P}}, R_{{\cal P}})\,.\] We will refer to this space as the R-equivariant space \(\Xi\). For later convenience, we denote a path with the ending node \(a\) as \({ \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\). The ending vertex is called a color of an atom. When two atoms \({ \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\) and \({ \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\) are connected by an arrow of the quiver \(I \in \{a \to b\}\), we connect these points in \(\Xi\) by an arrow. These arrows are also known as the chemical bonds. Therefore the crystals are graphs in this space.

We also impose an extra condition on the crystals known as the no-overlap condition [52]:

The condition means that the atoms in a crystal do not overlap in the space \(\Xi\). This holds for the known molten crystals for toric Calabi-Yau manifolds.

Earlier we introduced the vertex constraints 1 on the weights, and in section 2.3 we stated that they fix gauge redundancies. However, one should be careful while imposing them or any other gauge-fixing relations. There are examples [52] where one cannot satisfy the no-overlap condition if they impose the vertex constraints. As we will see later in the paper, this holds for our cases as well. Since we can work with the gauge freedom using the spectral shift 9 , the correct procedure is to impose the vertex constraints only after the computations.

Fixed Points on Quiver Varieties↩︎

Now, we proceed further and describe the states from a more mathematical point of view. We refer the reader to [15], [16], [22], [53], [62] for more concise reviews. Instead, we limit ourselves with technical details of the construction.

With each arrow \(I\in \{a \to b\}\), we have associated a matrix \(q_{I}\) that acts from \(V_{a}\) to \(V_{b}\). The F-relations 12 are translated into the matrix equations \[\label{general32F-terms} F_{I} = \partial_{q_{I}}{\cal W}= 0\,,\tag{17}\] and cut off a complex algebraic variety. Factorizing it modulo gauge transformations, we end up with the moduli space of quiver representations [17], [63], [64]: \[\mathscr{M}_{\vec{d}} = \left\{\begin{array}{c} \text{stable}\\ F_{I} = 0 \end{array}\right\}/\prod_{a \in Q_{0}}GL(d_{a}, \mathbb{C})\,.\] We do not address the definition of stability here. We emphasize, however, that the effective approach that was mentioned above allows us to describe only the so-called cyclic stability chamber of the moduli spaces⁶.

Now, we narrow this space further using the equivariant group generated by the vector fields: \[\label{equiv32vector32fields} v_{I:a\to b}={\rm Tr}\,(\Phi_b q_I-q_I\Phi_a-h_I q_I)\frac{\partial}{\partial q_I}\,.\tag{18}\] The states in this construction are associated with the fixed points of 18 . This procedure is analogous to assigning the equivariant weights to the paths. Notably, the crystals are represented as a set of matrices \(q_{I}\) with a given dimension vector \(\vec{d}\). The paths on the quiver 11 are naturally identified with the fixed points as follows: \[{\cal P}= I_{1}\cdot I_{2}\cdot\ldots\cdot I_{n} \quad \leadsto \quad q_{{\cal P}} = q_{I_{1}}\cdot q_{I_{2}}\cdot \ldots \cdot q_{I_{n}}\,.\] For homotopic paths we have \(q_{{\cal P}} = q_{{\cal P}'}\), which is ensured by the F-relations 17 . The atoms become vectors in the spaces \(V_{a}\), where \(a\in Q_{0}\): \[\label{cyc} V_a={\rm Span}\left\{q_{I_n}\cdot\ldots\cdot q_{I_2}\cdot q_{I_1}\cdot R \right\}\,.\tag{19}\]

One could directly solve the F-term relations, fix the gauge freedom, and find the corresponding fixed points for each possible choice of the dimension vector \(\vec{d}\). However, in our case, we can construct the fixed points by using the information from the Jacobian algebra \({\cal J}^{\sharp}\) of the quiver. This approach is also described in [28].

First, we fix the dimension vector and identify the crystal structure \(\Lambda\) of the state in the \(k + 1\)-dimensional space. We also choose the numeration of the vectors in \(V_{a}\): \[V_a=\bigoplus_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\in \Lambda}\mathbb{C}|{ \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\rangle, \quad a\in Q_0\,,\] where \({ \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\) specifies a color of an atom.

Second, we construct the vacuum expectation values of the matrices \(\Phi_{a}\): \[\label{gen32Phi} \Phi_a={\rm diag}\left\{\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_1},\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_2},\ldots,\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_{d_a}}\right\},\quad \phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_i}= \sum_{j = 1}^{k} x_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_i}^{j} \epsilon_{j}\,.\tag{20}\] The eigenvalues of these matrices encode the weights 16 of the corresponding atoms.

Finally, we construct explicitly the matrices of \(q_I\) in fixed points as follows (\(i=1,\ldots,d_b\), \(j=1,\ldots,d_a\)): \[\label{morphisms} \left(q_{I:a\to b}\right)_{ij}=\left\{\begin{array}{ll} 1, &if(\vec{x}_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_j} +\vec{x}_I,R_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_j}+R_I)=(\vec{x}_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_i},R_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }_i})\,;\\ 0, & otherwise\,. \end{array}\right.\tag{21}\]

2.5 Yangian Representations↩︎

Now we proceed to define the representation itself. Vectors in our modules are described in section 2.4. Given a state \(\Lambda\), one should define how the generators 3 act on it. We use the model where a raising operator \(e^{(a)}(z)\) adds a single atom of a color \(a\) to a crystal, and a lowering operator \(f^{(a)}(z)\) removes an atom from a crystal.

When we add/remove an atom, we can accidentally violate the molten crystal structure. It leads us to a natural constraint that a new set of atoms \(\Lambda \pm { \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\) is also a crystal. The sets \(\text{Add}(\Lambda)\) and \(\text{Rem}(\Lambda)\) consist of all possible atoms that can be added to (removed from) a given crystal \(\Lambda\) correspondingly.

Finally, we end up with the ansatz for Yangian action on crystal states: \[\label{General32Ansatz} \begin{align} &\psi^{(a)}(z)|\Lambda\rangle = {\boldsymbol{\Psi}}_{\Lambda}^{(a)}(z)|\Lambda\rangle\,, \\ &e^{(a)}(z)|\Lambda\rangle = \sum_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\in \text{Add}(\Lambda)}\dfrac{{\boldsymbol{E}}_{\Lambda, \Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}{z - \phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}|\Lambda + { \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\rangle\,,\\ &f^{(a)}(z)|\Lambda\rangle = \sum_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\in \text{Rem}(\Lambda)}\dfrac{{\boldsymbol{F}}_{\Lambda, \Lambda - { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}{z - \phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}|\Lambda - { \setbox 0=\Box \setbox 1=\raisebox{0.35ex}{\small a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\rangle\,. \end{align}\tag{22}\] The eigenvalues of operators \(\psi^{(a)}(z)\) are given by the formula: \[\label{general32eigenvalues} {\boldsymbol{\Psi}}^{(a)}_{\Lambda}(z)=\prod\limits_{I\in\{a\to a\}}\left(-\frac{1}{h_I}\right)\times\frac{\prod\limits_{K\in\{a\to\mathfrak{f}\}}(-z-h_K)}{\prod\limits_{J\in\{\mathfrak{f}\to a\}}(z-h_I)}\times \prod\limits_{b\in Q_0}\prod\limits_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }\in \Lambda}\varphi_{a,b}(z-\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }})\,.\tag{23}\] In fact, the poles of functions \({\boldsymbol{\Psi}}^{(a)}_{\Lambda}(z)\) coincide with the set \(\text{Add}(\Lambda) \cup \text{Rem}(\Lambda)\); we refer the interested reader to [40] for a detailed review. If a pole is equal to a weight of an atom in the crystal \(\Lambda\), it belongs to \(\text{Rem}(\Lambda)\) and to \(\text{Add}(\Lambda)\) otherwise.

One could derive that for 22 to be a representation of the Yangian 5 , matrix elements have to satisfy so-called hysteresis relations [11], [28]: \[\label{general32hysteresis} \begin{align} &{\boldsymbol{E}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}{\boldsymbol{F}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}=(-1)^{|a||b|}{\boldsymbol{F}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda}{\boldsymbol{E}}_{\Lambda,\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}\,,\\ &\frac{{\boldsymbol{E}}_{\Lambda,\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}{\boldsymbol{E}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}{{\boldsymbol{E}}_{\Lambda,\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}{\boldsymbol{E}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}\varphi_{a,b}(\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}-\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }})=(-1)^{|a||b|}\,,\\ &\frac{{\boldsymbol{F}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}{\boldsymbol{F}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda}}{{\boldsymbol{F}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}{\boldsymbol{F}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda}}\varphi_{a,b}(\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}-\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny b} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }})=(-1)^{|a||b|}\,,\\ &{\boldsymbol{E}}_{\Lambda,\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}{\boldsymbol{F}}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda}=\mathop{\rm res}\limits_{z=\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}\Psi^{(a)}_{\Lambda}(z)\,. \end{align}\tag{24}\]

2.6 Equivariant Matrix Coefficients↩︎

For 22 to be a representation, we need an explicit formula for the matrix elements \({\boldsymbol{E}}_{\Lambda, \Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}\) and \({\boldsymbol{F}}_{\Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }, \Lambda}\). The ansatz allows a slight ambiguity in the definition of the coefficients that was briefly addressed in [28]. This is caused by the algebra symmetries that we listed in section 2.3.

One of the options is a square-root representation [28]. We refer the interested reader to [11] for a more detailed review. In this case the coefficients are chosen as follows: \[\label{sq95rt} {\boldsymbol{E}}^{(root)}_{\Lambda,\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}={\boldsymbol{F}}^{(root)}_{\Lambda+{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} },\Lambda}\sim\sqrt{\mathop{\rm res}\limits_{z=\phi_{{ \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}\Psi^{(a)}_{\Lambda}(z)}\,.\tag{25}\] The states in this representation are normalized to unity. However, due to a lack of a canonical way to choose the branch of the square root function, we do not use 25 to calculate the matrix elements.

Instead, we exploit the geometric approach that relies on an equivariant integration over quiver representation moduli spaces [15], [62], [65]. For more physical motivations, see, for example [16], [28]. Here, we briefly review this construction.

One of the key ideas is that equivariant localization [19], [66], [67] allows us to extract all the information about the system properties in the vicinities of the fixed points. The equivariant fields 18 take diagonal form in a neighborhood of a fixed point and grade the corresponding tangent space \({\cal N}\): \[v=\sum\limits_i w_i \, z_i\frac{\partial}{\partial z_i}, \qquad {\cal N}= \bigoplus_{i}\mathbb{C}|w_{i}\rangle\,,\] where \(z_{i}\) are the local coordinates on \({\cal N}\). Note that some of the weights \(w_{i}\) can be zero.

In these settings, natural characteristics of a fixed point are the corresponding Euler classes. One should be careful, however, in directly applying the algorithm. The varieties \(\mathscr{M}\) that we work with can happen to be singular. It leads to jumps in the dimensions of the tangent spaces. For more details on the regularization in these cases, we refer to [16], [28]. After that, we end up with the modified version of the Euler classes: \[\label{reg95Eul} {\rm Eul}\;{\cal N}\,=\,(-1)^{\left\lfloor\frac{1}{2}\#\{i:\, w_i=0\}\right\rfloor}\prod\limits_{i:\,w_i\neq 0}w_i\,.\tag{26}\]

For each fixed point we define the corresponding Euler class: \[{\rm Eul}_{\Lambda}={\rm Eul}\; \mathsf{T}_{\Lambda}\mathscr{M}\,.\]

To proceed further and define the matrix elements \({\boldsymbol{E}}_{\Lambda,\Lambda'}\), \({\boldsymbol{F}}_{\Lambda',\Lambda}\) where \(\Lambda'=\Lambda+\Box\) we consider two corresponding quiver representations \(q_{I},~ q'_{I}\). The representations are homomorphic if there exists a set of maps \(\tau_a\), \(a\in Q_0\) making the following diagrams commutative: \[\tag{27} \begin{array}{c} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/akvecbxn.png}\tag{28}\end{figure} \end{array},\quad q_{I:a\to b}'\cdot \tau_a=\tau_b\cdot q_{I:a\to b},\quad \forall I\in Q_1\,.\]

We call an incidence locus \({{\cal I}}\) a surface in the Cartesian product of two representations, \(q_I\) and \(q_I'\), where this homomorphism exists. The tangent space to the incidence locus \(\mathsf{T}_{\Lambda,\Lambda'}{{\cal I}}\subset \mathsf{T}_{\Lambda}\mathscr{M}\oplus \mathsf{T}_{\Lambda'}\mathscr{M}\) is naturally an equivariantly weighted space, which means that we are able to define the corresponding Euler class: \[{\rm Eul}_{\Lambda,\Lambda'}={\rm Eul}\;\mathsf{T}_{\Lambda,\Lambda'}{{\cal I}}\,.\]

The matrix coefficients are constructed as Fourier-Mukai transforms [68] from \(\mathsf{T}_{\Lambda}\mathscr{M}\) to \(\mathsf{T}_{\Lambda'}\mathscr{M}\) and inverse with a kernel given by the structure sheaf of \({{\cal I}}\). The construction boils down to the canonical pullback-pushforward integration [15], [62], [65]. All the integrals are equivariant, and after applying the canonical Berline-Vergne-Atiyah-Bott localization formula [69]–[71] we acquire the result as a ratio of Euler classes: \[\label{equiv95matrix95coeffs} \begin{align} &{\boldsymbol{E}}^{(equiv)}_{\Lambda, \Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }} = \dfrac{\text{Eul}_{\Lambda}}{\text{Eul}_{\Lambda, \Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}\,, \\ &{\boldsymbol{F}}^{(equiv)}_{\Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }, \Lambda} = \dfrac{\text{Eul}_{\Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}{\text{Eul}_{\Lambda, \Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}}\,. \end{align}\tag{29}\] Equivariant integration gives us the normalization of the states [16]: \[\langle \Lambda |\Lambda \rangle = {\rm Eul}_{\Lambda}\,.\] One could expect that changing the norm to 1 returns the root representation. Although the statement is not proven in the general case, there are a few examples [28] where the representations coincide. This paper provides even more examples to the point. The coefficients are related using the formula 8 : \[\label{Rescaling32Equiv-to-Root} {\boldsymbol{E}}^{(root)}_{\Lambda, \Lambda + \Box} = {\boldsymbol{E}}^{(equiv)}_{\Lambda, \Lambda + \Box}\sqrt{\dfrac{{\rm Eul}_{\Lambda + \Box}}{{\rm Eul}_{\Lambda}}}, \quad {\boldsymbol{F}}^{(root)}_{\Lambda + \Box, \Lambda} = {\boldsymbol{F}}^{(equiv)}_{\Lambda + \Box, \Lambda}\sqrt{\dfrac{{\rm Eul}_{\Lambda}}{{\rm Eul}_{\Lambda + \Box}}}\,.\tag{30}\] For the rest of the paper, if not specified, the notation \({\boldsymbol{E}}_{\Lambda, \Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}\) stands for \({\boldsymbol{E}}^{(equiv)}_{\Lambda, \Lambda + { \setbox 0=\scriptstyle\Box \setbox 1=\raisebox{0.35ex}{\tiny a} \raisebox{-0.2ex}{\rlap{ to \wd 0{\hss{\box 1}\hss}}\box 0} }}\).

2.7 Dynkin Diagrams↩︎

In the preceding sections, we reviewed the general machinery for the construction of Yangian algebras and their representations using the quiver data. We now specialize to the class of quivers derived from Dynkin diagrams. This bridge between quivers and Dynkin diagrams is provided by the McKay correspondence [72]–[75]. The original correspondence [72] connects the smooth resolutions of orbifold singularities \(\mathbb{C}^{2}/\Gamma\), for finite groups \(\Gamma \subset SL(2, \mathbb{C})\), and Dynkin graphs of ADE type⁷. The generalization to the non-simply laced cases is given by the Slodowy correspondence [76], [77], see also [78], [79] for the geometric construction.

Here we present an algorithm to acquire the quivers using a given Dynkin diagram. We refer to [39], [40], [75] for more details.

A McKay quiver \({\cal Q}_{\Gamma}\), for a finite group \(\Gamma\), is defined as follows. Irreducible representations \(\rho_{a}\) of \(\Gamma\) label the set of vertices \(Q_{0}\) of the quiver⁸. One can determine the number of arrows \(n_{ab}\) between arbitrary nodes \(a, b\) using the decomposition: \[\label{McKay32arrows} {\cal R}\otimes \rho_{a} = \bigoplus_{b \in Q_{0}}n_{ab}\rho_{b}\,,\tag{31}\] where \({\cal R}\subset \mathbb{C}^{2}\) is the fundamental representation of \(\Gamma\) induced by inclusion \(\Gamma \subset SL(2, \mathbb{C})\). This results in “doubling" of the original Dynkin graph. We also extend this quiver further by adding a self-loop \(C_{a}\) to each vertex. These quivers are sometimes referred to as triple quivers [39], [40], [80]. We depict this construction, including non-simply laced cases⁹, and assign the corresponding superpotential in 32 : \[\tag{32} \begin{tblr}{c|c|c} Dynkin graph & Quiver & Superpotential\delta W_{ab} \\ \hline \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/etsuwpfh.png}\tag{33}\end{figure} & \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/hyiflnet.png}\tag{34}\end{figure} & {\rm Tr}\,(X_{ab}X_{ba}C_{a} - X_{ba}X_{ab}C_{b})\\ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ajkgclmo.png}\tag{35}\end{figure} & \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/jwtylgdh.png}\tag{36}\end{figure} & {\rm Tr}\,\Bigl(X_{ab}X_{ba}C_{a}^{|{\cal A}_{ab}|} - X_{ba}X_{ab}C_{b}^{|{\cal A}_{ba}|}\Bigr) \end{tblr}\] In 32 \({\cal A}_{ab}\) denotes the corresponding Cartan matrix. The full unframed superpotential is recovered by summing up all the possible pairs in the quiver: \({\cal W}= \sum\limits_{(a, b)}\delta W_{ab}\). By construction: \[|Q_{0}| = \mathop{\mathrm{\mathop{rk}}}(\mathfrak{g})\,,\] where \(\mathfrak{g}\) is the corresponding Lie algebra. Also, we would like to emphasize that although the quiver diagram in 32 seems to be the same for non-simply laced cases, the resulting theory is different. Indeed, the superpotentials have different forms, which change the weight assignment to the arrows of the quiver.

Although the procedure can be applied in theory for an arbitrary Dynkin diagram, we restrict ourselves to the A-type diagrams. When supplemented with suitable framings and superpotentials, the construction defines the Yangian algebras \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) and their representations. An arbitrary Dynkin diagram A\(_{n - 1}\) is given by the picture: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/uybomcrq.png}\label{dnjygzxu}\end{figure}\tag{37}\]

Applying the algorithm, we end up with the following family of quivers to describe rectangular representations of \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) algebras: \[\tag{38} {\cal Q}_{n,p,\lambda}=\left\{\begin{array}{c} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/jibokewg.png}\tag{39}\end{figure}\\ W={\rm Tr}\,\left[A_1C_1B_1+\sum\limits_{a=2}^{n-2}\left(A_aC_aB_a-B_{a-1}C_aA_{a-1}\right)-B_{n-2}C_{n-1}A_{n-2}+{\color{burgundy}C_p^{\lambda}R_{p}S_{p}}\right] \end{array}\right\}\,.\]

Note that for all \(a \in {\cal Q}_{n,p,\lambda}\), we have \(|a| = 0\). The fields \(R_{a}, S_{a}\) are indexed depending on what node they are attached to. The unframed quiver is independent of fields \(R_{a}\), \(S_{a}\) and of parameters \(p\), \(\lambda\). We emphasize that technically, we should consider the perturbations of all the framing nodes in equivariant calculations: \[R_{a} = \overline{R}_{a} + \delta R_{a}\,, \quad S_{a} = \overline{S}_{a} + \delta S_{a}\,,\] including the fields where \(a \ne p\). This means that we added more than 1 framing with the dimensions \(\mathfrak{f}_{a} = 1\). This could affect the construction of the states described in section 2.4. However, we will see later that \(\overline{R}_{a} = 0,~ \overline{S}_{a} = 0\) for \(a \ne p\). The latter ensures that there are no additional branches starting at \(R_{a},~ a\ne p\).

The \(F\)-terms cut off toric varieties in this case. In fact, this is no longer true for any other Dynkin diagram [39], [52], [61] and complicates the analysis in the general case.

Our framing choice, depending explicitly on \(p\) and \(\lambda\), cuts out a specific representation of \(\mathfrak{s}\mathfrak{l}_n\), classified by the following Young diagram: \[\tag{40} \Upsilon_{p,\lambda}=\begin{array}{c} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/zyilnbxh.png}\tag{41}\end{figure} \end{array}\,,\] The quivers have an additional \(\mathbb{Z}_{2}\)-symmetry: \[\label{sl40n4132quiver32symmetry32Z952} \mathcal{Q}_{n, p, \lambda} \to \mathcal{Q}_{n, n - p, \lambda}\,,\tag{42}\] that corresponds to complex conjugation of the representations \(\Upsilon_{p,\lambda} \to \Upsilon_{n - p,\lambda}\). Due to this fact, we can omit the description of the representations with \(p > \left\lceil \frac{n-1}{2} \right\rceil\). That leaves us with the representations: \[\label{gen32rectangular32reps32after32cutoff} \Upsilon_{p,\lambda}\,, \quad \text{where } \quad p \leqslant \left\lceil \frac{n-1}{2} \right\rceil\,.\tag{43}\]

Counting the number of equivariant parameters↩︎

Since we have specified the quivers, we need to check the independence of their equivariant parameters. The dependencies are linear and defined by the system of equations 1 . Let us denote the matrix corresponding to this system as \({\cal M}\). Basic linear algebra tells us that the number of independent variables can be calculated as follows: \[\label{Number32of32equivariant32parameters} \#(\epsilon) = |Q_{1}| - \mathop{\mathrm{\mathop{rk}}}({\cal M})\,.\tag{44}\] For the given quivers we have: \[|Q_{1}| = (n - 1) + 2(n-2) = 3n - 5\,.\] The number of all constraints is: \[\#(\text{Constraints}) = \underset{\text{Vertex}}{\underbrace{n - 1}} + \underset{\text{Loop}}{\underbrace{2(n - 2)}} = 3n - 5\,.\] The next step is to determine the rank. One can notice that the loop constraints and the vertex constraints are separable in our case 38 . This is because the loop constraints involve the weights \(h_{C_{a}}\) whereas the vertex constraints do not. We can check that the vertex constraints are not independent: \[\sum_{a = 1}^{n - 1} \text{Vertex constraint}_{a} = 0\,.\] For example, in the case of \(n = 4\): \[\begin{align} &\text{Vertex constraint}_{1} = h_{B_{1}} - h_{A_{1}}\,,\\ &\text{Vertex constraint}_{2} = h_{A_{1}} - h_{B_{1}} + h_{B_{2}} - h_{A_{2}}\,,\\ &\text{Vertex constraint}_{3} = h_{A_{2}} - h_{B_{2}}\,,\\ &\sum_{a = 1}^{3} \text{Vertex constraint}_{a} = h_{B_{1}} - h_{A_{1}} + h_{A_{1}} - h_{B_{1}} + h_{B_{2}} - h_{A_{2}} + h_{A_{2}} - h_{B_{2}} = 0\,. \end{align}\] The matrix of the loop constraints has a full rank. It means that the rank of the whole matrix \({\cal M}\) reads: \[\mathop{\mathrm{\mathop{rk}}}({\cal M}) = n-2 + 2(n - 2) = 3n - 6\,.\] Finally, we end with the result: \[\fbox{\#(\epsilon) = 3n - 5 - (3n -6) = 1}\,.\]

Analysis of the result↩︎

Let us examine the weight assignment in the presence of the constraints 1 more closely. First, one can solve the loop constraints as follows: \[\begin{align} &h_{C_{1}} = \dots = h_{C_{a}} = \dots = h_{C_{n-1}} = \epsilon\,,\\ &h_{A_{a}} = h_{a} - \dfrac{\epsilon}{2}\,, \quad h_{B_{a}} = - h_{a} - \frac{\epsilon}{2}\,, \end{align}\] where \(\epsilon\) and \(h_{a}\) are \(n - 1\) parameters. For example, for the fields \(C_{a}\) and \(C_{a + 1}\) we have: \[\begin{align} &h_{A_{a}} + h_{C_{a + 1}} + h_{B_{a}} = 0,\,\\ &h_{A_{a}} + h_{C_{a}} + h_{B_{a}} = 0\,. \end{align}\] Subtracting the one from the other we get \(h_{C_{a}} = h_{C_{a + 1}}\).

We now examine the vertex constraints. There are two types of these constraints. The constraints that correspond to the vertices from \(2\) to \(n - 2\), and the other constraints that correspond to the vertices labeled by \(1\) and \(n - 1\). There are \(n - 3\) constraints of the first type, but all of them have the same structure. For a vertex \(a\), the constraint takes the following form: \[\begin{align} &h_{A_{a - 1}} - h_{B_{a - 1}} + h_{B_{a}} - h_{A_{a}} = 0\,\\ &h_{a - 1} - \frac{\epsilon}{2} + h_{a - 1} + \frac{\epsilon}{2} - h_{a} - \frac{\epsilon}{2} - h_{a} + \frac{\epsilon}{2} = 0\,\\ &h_{a - 1} = h_{a}\,. \end{align}\] This means that all the parameters \(h_{a}\) are equal to each other. We denote them as \(h\).

The “edge" conditions resolve as: \[\label{h3261320} h = -h \qquad \rightarrow \qquad h = 0\,,\tag{45}\] which, as expected, mirrors the result of our calculations above that we have a single independent parameter \(\epsilon\).

In other words, we demonstrated that our R-equivariant space \(\Xi\) happens to be two-dimensional. However, as we will see, for example, in section 3.2 atoms in the crystals overlap if we impose the vertex constraints. In section 2.4 we mentioned that, in this case, one should impose the loop constraints, construct a representation, and only after that impose the vertex constraints, ensuring that the theory is still gauge invariant.

We will use the suggested procedure, although with a slight modification. In our work we do not need to keep track of all the \(n - 1\) parameters \((h_{a}, \epsilon)\) that are left after the loop constraints. Therefore, we just loose the vertex constraints and impose only the constraints of the first type. That leaves us with only two equivariant parameters \(h, \epsilon\). Together with R-charge, the space \(\Xi\) becomes three-dimensional. In the final formulas we substitute \(h = 0\).

3 Yangian Algebras \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\)↩︎

In this section, we consider a few examples and applications of the construction introduced in section 2 in detail.

3.1 \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\) example↩︎

3.1.1 The Algebra↩︎

In this section we discuss one of the simplest examples, the \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\) algebra, and its rectangular representations that are related to \((\lambda, 0)\) or \((0, \lambda)\) representations¹⁰ of \(\mathfrak{s}\mathfrak{l}_{3}\) algebra. These representations are complex conjugates of each other; see 43 . Therefore, it is sufficient to focus on \((\lambda, 0)\) representations. The quiver depicted in fig. 2 can be used to describe these representations.

The dimensions of the vector spaces that correspond to the nodes are \(n_{1}\) and \(n_{2}\). We also add the following superpotential: \[\label{sl4034132superpotential} \mathcal{W} = {\rm Tr}\,(C_{1}BA - C_{2}AB + C_{1}^{\lambda}R_{1}S_{1} + R_{2}S_{2})\,.\tag{46}\] The weights and R-charges of fields read: \[\label{sl4034132weights} \begin{tblr}{c|c|c|c|c|c|c} Fields & C_{1} & C_{2} & A & B & R_{1} & S_{1}\\ \hline Weights & \epsilon & \epsilon & -\frac{\epsilon}{2} + h & -\frac{\epsilon}{2} - h & 0 & -\lambda \epsilon \\ R-charges & 0 & 0 & 1 & 1 & 0 & 2 \end{tblr}\,.\tag{47}\]

Yangian algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\) contains two families of generators \((e^{(a)}_{k}, f^{(a)}_{k}, \psi^{(a)}_{k})\), \(a\in \{1, 2\}\), which can be assembled in generating functions: \[\label{sl4034132generating32functions} e^{(a)}(z) = \sum_{n = 0}^{\infty}\dfrac{e^{(a)}_{n}}{z^{n + 1}}, \quad f^{(a)}(z) = \sum_{n = 0}^{\infty}\dfrac{f^{(a)}_{n}}{z^{n + 1}}, \quad \psi^{(a)}(z) = 1 + \sum_{n = 0}^{\infty}\dfrac{\psi^{(a)}_{n}}{z^{n + 1}}\,.\tag{48}\] Using quiver 2 we construct the bonding factors using the formula 4 : \[\label{sl4034132bond32factors} \begin{align} &\varphi_{1, 1}(z) = \dfrac{z + \epsilon}{z - \epsilon}\,, \quad \varphi_{1, 2}(z) = \dfrac{z - \frac{\epsilon}{2} + h}{z + \frac{\epsilon}{2} + h}\,,\\ &\varphi_{2, 1}(z) = \dfrac{z - \frac{\epsilon}{2} - h}{z + \frac{\epsilon}{2} - h}\,, \quad \varphi_{2, 2}(z) = \dfrac{z + \epsilon}{z - \epsilon}\,, \end{align}\tag{49}\] The generating functions satisfy the equations 5 : \[\label{sl4034132relations} \begin{align} &e^{(a)}(z)e^{(b)}(w) \simeq \varphi_{a, b}(z- w)e^{(b)}(w)e^{(a)}(z)\,,\\ &\psi^{(a)}(z)e^{(b)}(w) \simeq \varphi_{a, b}(z - w)e^{(b)}(w)\psi^{(a)}(z)\,,\\ &f^{(a)}(z)f^{(b)}(w) \simeq \varphi_{a, b}(z - w)^{-1} f^{(b)}(w)f^{(a)}(z)\,,\\ &\psi^{(a)}(z)f^{(b)}(w) \simeq \varphi_{a, b}(z - w)^{-1} f^{(b)}(w)\psi^{(a)}(z)\,,\\ &\psi^{(a)}(z)\psi^{(b)}(w) = \psi^{(b)}(w)\psi^{(a)}(z)\,,\\ &\Bigl[e^{(a)}(z), f^{(b)}(w)\Bigr] \simeq -\delta_{ab}\dfrac{\psi^{(a)}(z) - \psi^{(b)}(w)}{z - w}\,, \end{align}\tag{50}\] together with the Serre relations [81]: \[\label{sl4034132Serre32relations} \begin{align} &\sum_{\sigma \in \mathfrak{S}_{2}}\Bigl[e^{(a)}\bigl(u_{\sigma(1)}\bigr)\,, \Bigl[e^{(a)}\bigl(u_{\sigma(2)}\bigr), e^{(b)}(v)\Bigr]\Bigr] = 0\,,\\ &\sum_{\sigma \in \mathfrak{S}_{2}}\Bigl[f^{(a)}\bigl(u_{\sigma(1)}\bigr)\,, \Bigl[f^{(a)}\bigl(u_{\sigma(2)}\bigr), f^{(b)}(v)\Bigr]\Bigr] = 0\,, \end{align}\tag{51}\] where \(a \ne b\), and \(\mathfrak{S}_{2}\) is the symmetric group on two letters. These relations can be translated into the mode relations. From the relations with \(\varphi_{1, 1}\) we get: \[\label{sl4034132phi951132relations} \begin{align} &[e^{(1)}_{n + 1}, e^{(1)}_{m}] - [e^{(1)}_{n}, e^{(1)}_{m + 1}] = \epsilon \{e^{(1)}_{n}, e^{(1)}_{m}\}\,,\\ &[\psi^{(1)}_{n + 1}, e^{(1)}_{m}] - [\psi^{(1)}_{n}, e^{(1)}_{m + 1}] = \epsilon \{\psi^{(1)}_{n}, e^{(1)}_{m}\}\,,\\ &[f^{(1)}_{n + 1}, f^{(1)}_{m}] - [f^{(1)}_{n}, f^{(1)}_{m + 1}] = -\epsilon \{f^{(1)}_{n}, f^{(1)}_{m}\}\,,\\ &[\psi^{(1)}_{n + 1}, f^{(1)}_{m}] - [\psi^{(1)}_{n}, f^{(1)}_{m + 1}] = -\epsilon \{\psi^{(1)}_{n}, f^{(1)}_{m}\}\,; \end{align}\tag{52}\] From \(\varphi_{1, 2}\) we get: \[\label{sl4034132phi951232relations} \begin{align} &[e^{(1)}_{n + 1}, e^{(2)}_{m}] - [e^{(1)}_{n}, e^{(2)}_{m + 1}] = -\frac{\epsilon}{2} \{e^{(1)}_{n}, e^{(2)}_{m}\} - h[e^{(1)}_{n}, e^{(2)}_{m}]\,,\\ &[\psi^{(1)}_{n + 1}, e^{(2)}_{m}] - [\psi^{(1)}_{n}, e^{(2)}_{m + 1}] = -\frac{\epsilon}{2} \{\psi^{(1)}_{n}, e^{(2)}_{m}\} - h[\psi^{(1)}_{n}, e^{(2)}_{m}]\,,\\ &[f^{(1)}_{n + 1}, f^{(2)}_{m}] - [f^{(1)}_{n}, f^{(2)}_{m + 1}] = \frac{\epsilon}{2} \{f^{(1)}_{n}, f^{(2)}_{m}\} + h[f^{(1)}_{n}, f^{(2)}_{m}]\,,\\ &[\psi^{(1)}_{n + 1}, f^{(2)}_{m}] - [\psi^{(1)}_{n}, f^{(2)}_{m + 1}] = \frac{\epsilon}{2} \{\psi^{(1)}_{n}, f^{(2)}_{m}\} + h[\psi^{(1)}_{n}, f^{(2)}_{m}]\,; \end{align}\tag{53}\] And from the relations with \(\varphi_{2, 1},~ \varphi_{2, 2}\) we get the mode relations similar to 53 and 52 correspondingly. This fact is even more transparent when we set \(h = 0\), because after that \(\varphi_{1, 1}(z) = \varphi_{2, 2}(z)\) and \(\varphi_{1, 2}(z) = \varphi_{2, 1}(z)\).

The latter two lines of 50 are translated into: \[\begin{align} &[\psi_{n}^{(a)}, \psi_{m}^{(b)}] = 0\,,\\ &[e^{(a)}_{n}, f^{(b)}_{m}] = \delta_{ab}\psi_{n + m}\,, \end{align}\] In the discussion above we skipped so-called “boundary conditions" that are also derived from 50 : \[\label{sl4034132boundary32conditions} \begin{align} &[\psi_{0}^{(1)}, e_{m}^{(1)}] = 2 e_{m}^{(1)},\quad &[\psi_{0}^{(1)}, f_{m}^{(1)}] = -2 f_{m}^{(1)}\,, \\ &[\psi^{(2)}_{0}, e_{m}^{(1)}] = -e_{m}^{(1)},\quad &[\psi_{0}^{(2)}, f_{m}^{(1)}] = f_{m}^{(1)}\,, \\ &[\psi_{0}^{(1)}, e^{(2)}_{m}] = -e_{m}^{(2)},\quad &[\psi_{0}^{(1)}, f^{(2)}_{m}] = f_{m}^{(2)}\,,\\ &[\psi_{0}^{(2)}, e_{m}^{(2)}] = 2 e_{m}^{(2)},\quad &[\psi_{0}^{(2)}, f_{m}^{(2)}] = -2 f_{m}^{(2)}\,, \\ \end{align}\tag{54}\] Finally, for \(a \ne b\) the Serre relations 51 take the form [2]: \[\label{sl4034132Serre32relations32in32modes} \begin{align} &\sum_{\sigma \in \mathfrak{S}_{2}}\big[e_{n_{\sigma(1)}}^{(a)}\big[e_{n_{\sigma(2)}}^{(a)}, e^{(b)}_{m}\big]\big] = 0\,,\\ &\sum_{\sigma \in \mathfrak{S}_{2}}\big[f_{n_{\sigma(1)}}^{(a)}\big[f_{n_{\sigma(2)}}^{(a)}, f^{(b)}_{m}\big]\big] = 0\,. \end{align}\tag{55}\] In this section we provided the calculations with the parameter \(h\). However, we will not focus on it later in the text and will automatically assume \(h = 0\) in any calculations except the crystal structures.

3.1.2 The States↩︎

From the superpotential 46 , we can derive the corresponding F-term equations: \[\label{sl4034132F-term32relations} \begin{align} &\partial_{A}{\cal W}= C_{1}B - BC_{2} = 0\,,\quad &\partial_{B}{\cal W}= AC_{1} - C_{2}A = 0\,, \\ &\partial_{C_{1}}\mathcal{W} = BA = 0\,, &\partial_{C_{2}}\mathcal{W} = AB = 0\,, \\ &\partial_{S_{1}}\mathcal{W} = C_{1}^{\lambda}R_{1} = 0\,, &\partial_{S_{2}}{\cal W}= R_{2} = 0\,. \end{align}\tag{56}\] Fixed points on this variety are defined by the following equations: \[\label{sl4034132fixed32points} \begin{align} &[\Phi_{1}, C_{1}] = \epsilon~C_{1}\,, &[\Phi_{2}, C_{2}] = \epsilon~C_{2}\,,\\ &\Phi_{2}A - A\Phi_{1} = \Bigl(-\frac{\epsilon}{2} + h\Bigr)~A \,, &\Phi_{1}B - B\Phi_{2} = \Bigl(-\frac{\epsilon}{2} - h\Bigr)~B \,, \\ &\Phi_{1} R_{1} = 0\,, & S_{1}\Phi_{1} = -\lambda\epsilon~S_{1}\,. \end{align}\tag{57}\] It is helpful to depict fixed points in terms of sets of paths on the quiver 2. Using the equivalence relations given by the F-terms, one can prove that: \[C_{1}^{a}B = BC_{2}^{a}\,, \quad AC_{1}^{b} = C_{2}^{b}A\,.\] It can be shown by exploiting the first two relations from 56 . For example: \[C_{1}^{2}B = C_{1}(C_{1}B) \overset{F}{=} (C_{1}B)C_{2} \overset{F}{=} BC_{2}C_{2} = BC_{2}^{2}\,.\] Suppose that the field \(B\) appears in a path. Then: \[BC_{2}^{a}AC_{1}^{b}R_{1} = BC_{2}^{a}C_{2}^{b}AR_{1} = BC_{2}^{a + b}AR_{1} = C_{1}^{a + b}BAR_{1} = C_{1}^{a + b}(BA)R_{1} = 0\,.\] Therefore all the paths have the following form: \[C_{1}^{a}R_{1}\,, \qquad AC_{1}^{a}R_{1} = C_{2}^{a}AR_{1}\,.\] Note that the equivariant weights and the R-charges of the paths are: \[\begin{align} &h(C_{1}^{a}R_{1}) = a\epsilon\,, \qquad h(AC_{1}^{a}R_{1}) = -\frac{\epsilon}{2} + h + a\epsilon\,,\\ &R(C_{1}^{a}R_{1}) = 0\,, \qquad R(AC_{1}^{a}R_{1}) = 1\,. \end{align}\] Having these two building blocks and denoting them as points, we depict the random fixed point of the quiver in the space of weights and R-charges \(\Xi\): \[\tag{58} \begin{array}{c} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/frcpeovz.png}\tag{59}\end{figure} \end{array}\] The figure requires the fulfillment of the F-terms 56 . Otherwise the point \(AC_{1}R_{1}\) would be split into two different points: \(AC_{1}R_{1}\) and \(C_{2}AR_{1}\), which would mean that we could not form a square. It also applies to any rectangle that we can see on the figure. The F-relation \(C_{1}^{\lambda}R_{1} = 0\) serves as a cut-off.

Note also that the picture 58 shows that it is necessary that \(n_{1} \geqslant n_{2}\). A state grows from the field \(R_{1}\). We now assume that there is a path where: \[C_{2}^{m}AR_{1} \ne 0, \quad \text{and } \quad C_{1}^{m}R_{1} = 0\,.\] Then we can apply the equivalence relation above and derive: \[C_{2}^{m}AR_{1} = AC_{1}^{m}R_{1} = 0\,,\] which was not true due to our suggestion.

State Counting↩︎

The next step is to check the dimension of the representation \((\lambda, 0)\). The dimension of the finite-dimensional representations of Yangian algebras \(\mathfrak{s}\mathfrak{l}_{n}\) coincide with the dimensions of highest-weight representations of \(\mathfrak{s}\mathfrak{l}_{n}\), as was shown in [2]. Therefore, the dimension of our representations on the fixed points should match with: \[\dim_{\mathfrak{s}\mathfrak{l}_{3}}|(\lambda, 0)| = \frac{1}{2}(\lambda + 1)(\lambda + 2)\,.\]

To count the number of fixed points in \((\lambda, 0)\) representations, we reimagine the states. First, we introduce an empty box (a planar rectangular graph in this case): \[\tag{60} \begin{array}{c} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/gvlrkmfj.png}\tag{61}\end{figure} \end{array}\] The graph is embedded in the R-equivariant space \(\Xi\) and represents the allowed positions of atoms in the crystals. This is also our vacuum state where \(n_{1} = 0,~n_{2} = 0\). When we increase the dimensions \(n_{1}\) and \(n_{2}\), “put atoms into the box", we obtain the remaining states. The atoms, the filled spaces, we will denote as bigger nodes in contrast to the empty nodes.

The states can be uniquely parameterized by \(n_{1}\) and \(n_{2}\) since in each node there are only vectors of one type: \[\begin{align} &\text{First node: } \quad &C_{1}^{a}R_{1}\,,\\ &\text{Second node: } \quad &AC_{1}^{a}R_{1}\,. \end{align}\]

It leads us to the picture: \[\tag{62} \begin{array}{c} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/qhafbcpj.png}\tag{63}\end{figure} \end{array}\] Starting from the vacuum 60 , we can add the paths into the first row without restrictions. It means \(n_{1} \in [0, \lambda]\). Then, when we are to add the paths into the second line, we must keep in mind that \(n_{2} \leqslant n_{1}\). Therefore \(n_{2} \in [0, n_{1}]\), and the number of states reads: \[\dim_{f.p.}|(\lambda, 0)| = \sum_{n_{2} = 0}^{\lambda}\sum_{n_{1} = n_{2}}^{\lambda}1 = \sum_{n_{1} = 0}^{\lambda}\sum_{n_{2} = 0}^{n_{1}}1 = \sum_{n_{1} = 0}^{\lambda}(n_{1} + 1) = \sum_{k = 1}^{\lambda + 1}k = \dfrac{1}{2}(\lambda + 1)(\lambda + 2) = \dim_{\mathfrak{s}\mathfrak{l}_{3}}|(\lambda, 0)|\,.\]

Gelfand-Tsetlin Bases↩︎

In the 62 we mentioned the Gelfand-Tsetlin bases [43], [44]. In this case they parametrize the states equivalently as \((n_{1}, n_{2})\). However, this description will be useful within the general approach. We address this correspondence later in sections 3.2 and 3.3.

3.1.3 The Representations \(\Upsilon_{1,\lambda}\)↩︎

Having defined the states, we can proceed and construct the Yangian representations. First, we adapt the ansatz 22 to our case. It takes the following form: \[\label{Y40sl403414132anzatz} \begin{align} &e^{(1)}_{[\lambda, 0]}(z)|n_{1}, n_{2}\rangle = \dfrac{{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1} + 1, n_{2})}}{z - n_{1}\epsilon}|n_{1} + 1, n_{2}\rangle\,, \\ &e^{(2)}_{[\lambda, 0]}(z)|n_{1}, n_{2}\rangle = \dfrac{{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1}, n_{2} + 1)}}{z + \frac{\epsilon}{2} - n_{2}\epsilon}|n_{1}, n_{2} + 1\rangle\,,\\ &f^{(1)}_{[\lambda, 0]}(z)|n_{1}, n_{2}\rangle = \dfrac{{\boldsymbol{F}}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1} - 1, n_{2})}}{z - (n_{1} - 1)\epsilon}|n_{1} - 1, n_{2}\rangle\,, \\ &f^{(2)}_{[\lambda, 0]}(z)|n_{1}, n_{2}\rangle = \dfrac{{\boldsymbol{F}}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1}, n_{2} - 1)}}{z + \frac{\epsilon}{2} - (n_{2} - 1)\epsilon}|n_{1}, n_{2} - 1\rangle\,,\\ &\psi^{(s)[\lambda, 0]}(z)|n_{1}, n_{2}\rangle = \Psi^{(s)[\lambda, 0]}_{(n_{1}, n_{2})}|n_{1}, n_{2}\rangle\,. \end{align}\tag{64}\] The eigenvalues of \(\psi^{(s)[\lambda, 0]}(z)\) read as follows: \[\label{sl4034132eigenfunctions} \begin{align} \Psi^{(1)[\lambda, 0]}_{(n_{1}, n_{2})}(z) &= -\frac{1}{\epsilon}\dfrac{(z - \lambda\epsilon)}{z}\prod_{k = 1}^{n_{1}}\varphi_{1, 1}\bigl(z - (k-1)\epsilon\bigr)\prod_{k = 1}^{n_{2}}\varphi_{1, 2}\Bigl(z + \frac{\epsilon}{2} - (k-1)\epsilon\Bigr) =\\ &= -\frac{1}{\epsilon}\dfrac{(z - \lambda\epsilon)\bigl(z - (n_{2} - 1)\epsilon\bigr)}{(z - n_{1}\epsilon)\bigl(z - (n_{1} - 1)\epsilon\bigr)}\,, \\ \Psi^{(2)[\lambda, 0]}_{(n_{1}, n_{2})}(z) &= -\frac{1}{\epsilon}\prod_{k = 1}^{n_{1}}\varphi_{2, 1}\bigl(z - (k - 1)\epsilon\bigr)\prod_{k = 1}^{n_{2}}\varphi_{2, 2}\Bigl(z + \frac{\epsilon}{2} - (k - 1)\epsilon\Bigr) = \\ &= -\frac{1}{\epsilon}\frac{\Bigl(z + \frac{3\epsilon}{2}\Bigr)\bigl(z + \frac{\epsilon}{2} - n_{1}\epsilon\bigr)}{\Bigl(z + \frac{\epsilon}{2} - n_{2}\epsilon\Bigr)\Bigl(z + \frac{\epsilon}{2} - (n_{2} - 1)\epsilon\Bigr)}\,. \end{align}\tag{65}\] Again, we have set \(h = 0\) in these formulas.

Amplitudes↩︎

For later convenience we introduce useful “ICO" matrices: \[\tag{66} \begin{align} \text{If } n \leqslant m: \quad &I(n, m) := \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fqeuzows.png}\tag{67}\end{figure}\,,\quad n > m\,: \quad &I(n, m) := \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/hypbvdxu.png}\tag{68}\end{figure}\,, \\ &C(n) = \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/qpniudoa.png}\tag{69}\end{figure}\,, &O(n, m) = \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/xbcinrmt.png}\tag{70}\end{figure}\,, \end{align}\] where \(I(n, m)\) is a generalization of an identity matrix and \(O(n, m)\) is just a null matrix. These matrices will be building blocks of the matrices that correspond to the fixed points.

The vacuum expectation values of the fields take the form: \[\label{sl4034132vacuum32matrices} \begin{align} &\overline{C}_{1} = C(n_{1})\,, \quad \overline{C}_{2} = C(n_{2})\,, \quad \overline{A} = I(n_{2}, n_{1})\,, \quad \overline{B} = O(n_{1}, n_{2})\,,\\ &\overline{R}_{1} = I(n_{1}, 1)\,, \quad \overline{S}_{1} = O(1, n_{1})\,, \quad \overline{R}_{2} = O(n_{2}, 1)\,, \quad \overline{S}_{2} = O(1, n_{2})\,. \end{align}\tag{71}\] In the next examples we do not highlight the fields that have zero expectation values.

Implementing the algorithm described in section 2.6, we can evaluate the corresponding Euler classes: \[\label{sl4034132Euler32classes} \begin{align} &\text{Eul}^{[\lambda, 0]}_{(n_{1}, n_{2})} = (\epsilon)^{2n_{1}}\frac{\lambda!}{(\lambda - n_{1})!}(n_{1} - n_{2})!n_{2}!\prod_{k = 1}^{n_{2}}\Bigl(\frac{1}{2}(2k-3)\epsilon\Bigr)^{2}, \\ &\text{Eul}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1}+1, n_{2})} = (-\epsilon)(\epsilon)^{2n_{1}}\frac{\lambda!}{(\lambda - n_{1})!}(n_{1} - n_{2})!n_{2}!\prod_{k = 1}^{n_{2}}\Bigl(\frac{1}{2}(2k-3)\epsilon\Bigr)^{2}\,, \\ &\text{Eul}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1}, n_{2} + 1)} = \frac{(2n_{2} - 1)}{2}\epsilon~(\epsilon)^{2n_{1}}\dfrac{\lambda!}{(\lambda - n_{1})!}(n_{1} - n_{2} - 1)!n_{2}!\prod_{k = 1}^{n_{2}}\Bigl(\frac{1}{2}(2k-3)\epsilon\Bigr)^{2}\,. \end{align}\tag{72}\] Next we derive the corresponding matrix coefficients using the formulas 29 : \[\label{sl4034132yangian32coefficients} \begin{align} &{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1} + 1, n_{2})} = -\dfrac{1}{\epsilon}, \\ &{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2}) \to (n_{1}, n_{2} + 1)} = \dfrac{(n_{1} - n_{2})}{n_{2}\epsilon - \frac{\epsilon}{2}}\,, \\ &{\boldsymbol{F}}^{[\lambda, 0]}_{(n_{1}, n_{2})\to (n_{1} - 1, n_{2})} = -\epsilon(n_{1} - n_{2})(\lambda - n_{1} + 1)\,, \\ &{\boldsymbol{F}}^{[\lambda, 0]}_{(n_{1}, n_{2}) \to (n_{1}, n_{2} - 1)} = n_{2}\Bigl(n_{2} - \frac{3}{2}\Bigr)\epsilon\,. \end{align}\tag{73}\]

As in the case of \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{2})\) [28], we expect that the lowest operators of the algebra form the representation of the \(\mathfrak{s}\mathfrak{l}_{3}\) algebra itself. In order to see that, one should rescale the states to have unit norm. The corresponding rescaling of the coefficients is given in the formula 30 .

The coefficients then take the following form: \[\label{sl4034132normalized32coefficients} \begin{align} &e^{(1)}_{0}|n_{1}, n_{2}\rangle = \sqrt{(\lambda - n_{1})(n_{1} - n_{2} + 1)}|n_{1} + 1, n_{2}\rangle\,,\\ &e^{(2)}_{0}|n_{1}, n_{2}\rangle = \sqrt{(n_{1} - n_{2})(n_{2} + 1)}|n_{1}, n_{2} + 1\rangle\,,\\ &f^{(1)}_{0}|n_{1}, n_{2}\rangle = \sqrt{(n_{1} - n_{2})(\lambda - n_{1} + 1)}|n_{1} - 1, n_{2}\rangle\,,\\ &f^{(2)}_{0}|n_{1}, n_{2}\rangle = \sqrt{n_{2}(n_{1} - n_{2} + 1)}|n_{1}, n_{2} - 1\rangle\,. \end{align}\tag{74}\] These expressions take the same form as the coefficients of the representations \((\lambda, 0)\) of \(\mathfrak{s}\mathfrak{l}_{3}\) Lie algebra that was given in [44].

Finally, we check the hysteresis relations 24 to ensure that the algebra is self-consistent. For our Yangian \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\), these relations are satisfied and take the form: \[\label{sl4034132hysteresis32relations} \begin{align} &{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2}) \to (n_{1} + 1, n_{2})}{\boldsymbol{F}}^{[\lambda, 0]}_{(n_{1} +1, n_{2})\to (n_{1}, n_{2})} = (n_{1} - n_{2} + 1)(\lambda - n_{1}) = \mathop{\rm res}\limits_{z= n_{1}\epsilon}\Psi^{(1)}_{\Lambda}(z)\,,\\ &{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2}) \to (n_{1}, n_{2} + 1)}{\boldsymbol{F}}^{[\lambda, 0]}_{(n_{1}, n_{2} + 1)\to (n_{1}, n_{2})} = (n_{1} - n_{2})(n_{2} + 1) = \mathop{\rm res}\limits_{z= -\frac{\epsilon}{2} + n_{2}\epsilon}\Psi^{(2)}_{\Lambda}(z)\,,\\ &\frac{{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2}) \to (n_{1} + 1, n_{2})}{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1} +1, n_{2})\to (n_{1} + 1, n_{2} + 1)}}{{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2}) \to (n_{1}, n_{2} + 1)}{\boldsymbol{E}}^{[\lambda, 0]}_{(n_{1}, n_{2} + 1)\to (n_{1}+1, n_{2}+1)}} = \dfrac{n_{1} + 1 - n_{2}}{n_{1} - n_{2}} = \varphi_{2, 1}\Bigl(n_{2}\epsilon - \frac{\epsilon}{2} - n_{1}\epsilon\Bigr)\,. \end{align}\tag{75}\]

3.2 \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\) example↩︎

Our next example is the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\). In this case we have two different types of finite-dimensional representations. The first are symmetric representations \((\lambda, 0, 0)\); their description is similar to the previous case. The second are the \((0, \lambda, 0)\) representations. Their description requires a more involved approach. These representations serve as a crucial stepping point in the later generalization of our results to an arbitrary \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) algebra.

3.2.1 The Algebra↩︎

The quivers that we use to describe the \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\) algebra and its representations take the forms presented in fig. 3.

The dimensions of the vector spaces associated with the nodes are \(n_{1},~n_{2},\) and \(n_{3}\) correspondingly, and the superpotential takes the form: \[\label{sl4044132superpotential} {\cal W}= {\rm Tr}\,\bigl(A_{1}C_{1}B_{1} + A_{2}C_{2}B_{2} - B_{1}C_{2}A_{1} - B_{2}C_{3}A_{2} + C_{1}^{\lambda_{1}}R_{1}S_{1} + C_{2}^{\lambda_{2}}R_{2}S_{2} + C_{3}^{\lambda_{3}}R_{3}S_{3}\bigr)\,,\tag{76}\] where \((\lambda_{1}, \lambda_{2}, \lambda_{3})\) is equal to \((\lambda, 0, 0)\) or \((0, \lambda, 0)\) correspondingly.

We can determine the weights and the R-charges of the fields: \[\label{sl4044132weights} \begin{tblr}{c|c|c|c|c|c|c|c|c|c|c|c|c|c} Fields & C_{1} & C_{2} & C_{3} & A_{1} & B_{1} & A_{2} & B_{2} & R_{1} & S_{1} & R_{2} & S_{2} & R_{3} & S_{3} \\ \hline Weights & \epsilon & \epsilon & \epsilon & -\frac{\epsilon}{2} + h & -\frac{\epsilon}{2} - h & -\frac{\epsilon}{2} + h & -\frac{\epsilon}{2} - h & 0 & -\lambda_{1} \epsilon & 0 & -\lambda_{2}\epsilon & 0 & -\lambda_{3}\epsilon\\ R-charges & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 2 & 0 & 2 & 0 & 2 \end{tblr}\,.\tag{77}\]

The Yangian algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\) involves three families of generators \((e^{(a)}_{k}, f^{(a)}_{k}, \psi^{(a)}_{k})\), \(a\in \{1, 2, 3\}\), which we assemble in generating functions: \[\label{sl4044132generating32functions} e^{(a)}(z) = \sum_{n = 0}^{\infty}\dfrac{e^{(a)}_{n}}{z^{n + 1}}\,, \quad f^{(a)}(z) = \sum_{n = 0}^{\infty}\dfrac{f^{(a)}_{n}}{z^{n + 1}}\,, \quad \psi^{(a)}(z) = 1 + \sum_{n = 0}^{\infty}\dfrac{\psi^{(a)}_{n}}{z^{n + 1}}\,.\tag{78}\] Using the unframed quiver, we construct the bonding factors: \[\label{sl4044132bonding32factors} \begin{align} &\varphi_{1, 1}(z) = \varphi_{2, 2}(z) = \varphi_{3, 3}(z) = \frac{z + \epsilon}{z - \epsilon}\,,\\ &\varphi_{1, 2}(z) = \varphi_{2, 1}(z) = \varphi_{3, 2}(z) = \varphi_{2, 3}(z) = \dfrac{z - \frac{\epsilon}{2}}{z + \frac{\epsilon}{2}}\,. \end{align}\tag{79}\] The generating functions satisfy the equation identical to 50 . We also add the Serre relations: \[\label{sl4044132Serre32relations} \begin{align} &\text{If } |a - b| = 1\colon \quad \sum_{\sigma \in \mathfrak{S}_{2}}\Big[e^{(a)}\bigl(u_{\sigma(1)}\bigr),\Bigl[e^{(a)}\bigl(u_{\sigma(2)}\bigr), e^{(b)}(v)\Bigr]\Big] = 0\,,\\ &\text{If } |a - b| = 2\colon \quad [e^{(a)}(u), e^{(b)}(v)] = 0\,;\\ &\text{If } |a - b| = 1\colon \quad \sum_{\sigma \in \mathfrak{S}_{2}}\Big[f^{(a)}\bigl(u_{\sigma(1)}\bigr),\Bigl[f^{(a)}\bigl(u_{\sigma(2)}\bigr), f^{(b)}(v)\Bigr]\Big] = 0\,,\\ &\text{If } |a - b| = 2\colon \quad [f^{(a)}(u), f^{(b)}(v)] = 0\,. \end{align}\tag{80}\] We do not unfold the relations in terms of modes here, leaving it to the general case 134 , 135 , 136 .

Having the superpotential 76 , we find the corresponding F-term relations: \[\label{sl4044132F-terms} \begin{align} &C_{1}B_{1} - B_{1}C_{2} = 0\,, &A_{1}C_{1} - C_{2}A_{1} = 0\,,\\ &C_{2}B_{2} - B_{2}C_{3} = 0\,, &A_{2}C_{2} - C_{3}A_{2} = 0\,, \\ &B_{1}A_{1} = A_{2}B_{2} = 0\,, &B_{2}A_{2} - A_{1}B_{1} = 0\,,\\ &C_{1}^{\lambda_{1}}R_{1} = 0\,, \qquad C_{2}^{\lambda_{2}}R_{2} = 0\,, \qquad &C_{3}^{\lambda_{3}}R_{3} = 0\,. \end{align}\tag{81}\]

The fixed points on the variety above are defined by: \[\label{sl4044132fixed32points} \begin{align} &[\Phi_{1}, C_{1}] = \epsilon C_{1}\,, \quad [\Phi_{2}, C_{2}] = \epsilon C_{2}\,, \quad [\Phi_{3}, C_{3}] = \epsilon C_{3}\,,\\ &\Phi_{2}A_{1} - A_{1}\Phi_{1} = \Bigl(-\frac{\epsilon}{2} + h\Bigr)A_{1}\,, \quad \Phi_{1}B_{1} - B_{1}\Phi_{2} = \Bigl(-\frac{\epsilon}{2} - h\Bigr)B_{1}\,,\\ &\Phi_{3}A_{2} - A_{2}\Phi_{2} = \Bigl(-\frac{\epsilon}{2} + h\Bigr)A_{2}\,, \quad \Phi_{2}B_{2} - B_{2}\Phi_{3} = \Bigl(-\frac{\epsilon}{2} - h\Bigr)B_{2}\,,\\ &\Phi_{1}R_{1} = 0\,, \quad \Phi_{2}R_{2} = 0\,, \quad \Phi_{3}R_{3} = 0\,, \\ &S_{1}\Phi_{1} = -\lambda_{1}\epsilon S_{1}\,, \quad \Phi_{2}S_{2} = -\lambda_{2}\epsilon S_{2}\,, \quad S_{3}\Phi_{3} = -\lambda_{3}\epsilon S_{3}\,. \end{align}\tag{82}\]

3.2.2 The Representations \(\Upsilon_{1,\lambda}\)↩︎

In this section we use the quiver [quiver32sl404413240l443204432041] to describe the representations \(\Upsilon_{1, \lambda} = (\lambda, 0, 0)\) of the Yangian \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\). The analysis of these representations is quite similar to the previous example described in section 3.1, so we skip some details of the construction.

The States↩︎

The paths correspond to the words of the following form \(\mathcal{P}_{1} = m(C_{1}, C_{2}, C_{3}, A_{1}, A_{2}, B_{1}, B_{2})\cdot R_{1}\), where \(m\) is a monomial. As before, using F-terms, we can convert the C-fields \(C_{3} \to C_{2}, C_{2} \to C_{1}\): \[\begin{align} &B_{2}C_{3}^{a} = C_{2}^{a}B_{2}\,, &B_{1}C_{2}^{a} = C_{1}^{a}B_{1}\,, \\ &C_{3}^{a}A_{2} = A_{2}C_{2}^{a}\,, &C_{2}^{a}A_{1} = A_{1}C_{1}^{a}\,, \end{align}\] so we can consider only the paths that contain the field \(C_{1}\).

• If the path \(\mathcal{P}_{1}\) doesn’t contain the field \(A_{1}\), it takes the only possible form: \[C_{1}^{a}R_{1}\,.\]

• If \(A_{1} \in \mathcal{P}_{1}\), but \(A_{2}, B_{1}, B_{2} \notin \mathcal{P}_{1}\), the path takes form: \[A_{1}C_{1}^{a}R_{1}\,.\]

• If \(A_{1}, A_{2} \in \mathcal{P}_{1}\), the path takes form: \[A_{2}A_{1}C_{1}^{a}R_{1}\,.\]

• However, if \(B_{1}\in \mathcal{P}_{1}\): \[B_{1}A_{1}C_{1}^{a}R_{1} = 0\,,\] so these monomials do not acquire expectation values in the vacuum, and thus \(B_{1} \notin \mathcal{P}_{1}\).

• The last case when \(B_{2} \in \mathcal{P}_{1}\): \[B_{2}A_{2}A_{1}C_{1}^{a}R_{1} = A_{1}B_{1}A_{1}C_{1}^{a}R_{1} = A_{1}(B_{1}A_{1})C_{1}^{a}R_{1} = 0\,,\] so \(B_{2} \notin \mathcal{P}_{1}\).

Summarizing, we observe that a set of possible paths contains elements of three types: \[\label{sl4044132possible32paths3240l443204432041} C_{1}^{a}R_{1}\,, \quad A_{1}C_{1}^{a}R_{1}\,, \quad A_{2}A_{1}C_{1}^{a}R_{1}\,,\tag{83}\] that corresponds to three vertices of the quiver.

The weights and the R-charges of these paths read: \[\begin{align} &h(C_{1}^{a}R_{1}) = a\epsilon\,, \quad h(A_{1}C_{1}^{a}R_{1}) = -\frac{\epsilon}{2} + h + a\epsilon\,, \quad h(A_{2}A_{1}C_{1}^{a}R_{1}) = -\epsilon + 2h + a\epsilon\,, \\ &R(C_{1}^{a}R_{1}) = 0\,, \quad R(A_{1}C_{1}^{a}R_{1}) = 1\,, \quad R(A_{2}A_{1}C_{1}^{a}R_{1}) = 2\,. \end{align}\] The general state of the representation can be depicted in the space of equivariant weights and R-charges similar to 58 . The relation \(C_{1}^{\lambda}R_{1} = 0\) serves as a cut-off. As before, we note that \(n_{1} \geqslant n_{2} \geqslant n_{3}\); this is a direct result of the imposed \(F\)-terms.

State counting↩︎

Here we want to count the fixed points of the representation. Due to the arguments similar to that given in the previous example, the dimension should equal the expression from the algebra \(\mathfrak{s}\mathfrak{l}_{4}\): \[\dim_{\mathfrak{s}\mathfrak{l}_{4}}|(\lambda, 0, 0)| = \dfrac{(\lambda + 1)(\lambda + 2)(\lambda + 3)}{3!}\,.\]

The empty box is still a planar rectangle and takes the form: \[\tag{84} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/vkqdiotw.png}\tag{85}\end{figure} \qquad \Rightarrow \qquad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/letrmjhv.png}\tag{86}\end{figure}\] Next we obtain the other states by increasing the dimensions \(n_{1}, n_{2}\), and \(n_{3}\). We can add the nodes into the first row before \(n_{1} = \lambda\), so \(n_{1} \in [0, \lambda]\). Similarly, the dimension \(n_{2}\) is restricted by \(n_{1}\) as follows: \(n_{2} \in [0, n_{1}]\). Finally, the last dimension \(n_{3}\in [0, n_{2}]\) and the number of the fixed points reads: \[\dim_{f.p.}|(\lambda, 0, 0)| = \sum_{n_{1} = n_{2}}^{\lambda}\sum_{n_{2} = n_{3}}^{\lambda}\sum_{n_{3} = 0}^{\lambda}1 = \sum_{n_{1} = 0}^{\lambda}\sum_{n_{2} = 0}^{n_{1}}\sum_{n_{3} = 0}^{n_{2}}1 = \sum_{n_{1} = 0}^{\lambda}\sum_{n_{2} = 0}^{n_{1}}(n_{2} + 1) = \dfrac{(\lambda + 1)(\lambda + 2)(\lambda + 3)}{3!} = \dim_{\mathfrak{s}\mathfrak{l}_{4}}|(\lambda, 0, 0)|\,.\]

Gelfand-Tsetlin Bases↩︎

In this representation each node of the quiver has the paths of only one type. Therefore the states can be uniquely parameterized by the dimensions \(n_{1}, n_{2}, n_{3}\). The Gelfand-Tsetlin bases take the following form: \[\tag{87} |n_{1}, n_{2}, n_{3}\rangle = \quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/aiwzuytf.png}\tag{88}\end{figure}\,.\] We use the notation \(|n_{1}, n_{2}, n_{3}\rangle\) for simplicity, as the descriptions are equivalent in this case.

The Representation↩︎

The next step is to adopt the ansatz 22 . It reads as follows: \[\label{sl404413240l44320443204132ansatz} \begin{align} &e^{(1)}_{[\lambda, 0, 0]}(z)|n_{1}, n_{2}, n_{3}\rangle = \dfrac{{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})\to (n_{1} + 1, n_{2}, n_{3})}}{z - n_{1}\epsilon}|n_{1} + 1, n_{2}, n_{3}\rangle\,, \\ &e^{(2)}_{[\lambda, 0, 0]}(z)|n_{1}, n_{2}, n_{3}\rangle = \dfrac{{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})\to (n_{1}, n_{2} + 1, n_{3})}}{z + \frac{\epsilon}{2} - n_{2}\epsilon}|n_{1}, n_{2} + 1, n_{3}\rangle\,,\\ &e^{(3)}_{[\lambda, 0, 0]}(z)|n_{1}, n_{2}, n_{3}\rangle = \dfrac{{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})\to (n_{1}, n_{2}, n_{3} + 1)}}{z + \epsilon - n_{3}\epsilon}|n_{1}, n_{2}, n_{3} + 1\rangle\,,\\ &f^{(1)}_{[\lambda, 0, 0]}(z)|n_{1}, n_{2}, n_{3}\rangle = \dfrac{{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})\to (n_{1} - 1, n_{2}, n_{3})}}{z - (n_{1} - 1)\epsilon}|n_{1} - 1, n_{2}, n_{3}\rangle\,, \\ &f^{(2)}_{[\lambda, 0, 0]}(z)|n_{1}, n_{2}, n_{3}\rangle = \dfrac{{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})\to (n_{1}, n_{2} - 1, n_{3})}}{z + \frac{\epsilon}{2} - (n_{2} - 1)\epsilon}|n_{1}, n_{2} - 1, n_{3}\rangle\,,\\ &f^{(3)}_{[\lambda, 0, 0]}(z)|n_{1}, n_{2}, n_{3}\rangle = \dfrac{{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})\to (n_{1}, n_{2}, n_{3} - 1)}}{z + \epsilon - (n_{3} - 1)\epsilon}|n_{1}, n_{2}, n_{3} - 1\rangle\,,\\ &\psi^{(s)[\lambda, 0, 0]}(z)|n_{1}, n_{2}, n_{3}\rangle = \Psi^{(s)[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})}(z)|n_{1}, n_{2}, n_{3}\rangle\,. \end{align}\tag{89}\] The eigenfunctions take the form: \[\label{sl404413240l44320443204132eigenfunctions} \begin{align} \Psi^{(1)[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})}(z) &= -\frac{1}{\epsilon}\dfrac{(z - \lambda\epsilon)}{z}\prod_{k = 1}^{n_{1}}\varphi_{1, 1}\bigl(z - (k-1)\epsilon\bigr)\prod_{k = 1}^{n_{2}}\varphi_{1, 2}\Bigl(z + \frac{\epsilon}{2} - (k-1)\epsilon\Bigr) =\\ &= -\frac{1}{\epsilon}\dfrac{(z - \lambda\epsilon)\bigl(z - (n_{2} - 1)\epsilon\bigr)}{(z - n_{1}\epsilon)\bigl(z - (n_{1} - 1)\epsilon\bigr)}\,, \\ \Psi^{(2)[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})}(z) &= -\frac{1}{\epsilon}\prod_{k = 1}^{n_{1}}\varphi_{2, 1}\bigl(z - (k - 1)\epsilon\bigr)\prod_{k = 1}^{n_{2}}\varphi_{2, 2}\Bigl(z + \frac{\epsilon}{2} - (k - 1)\epsilon\Bigr)\prod_{k = 1}^{n_{3}}\varphi_{2, 3}\Bigl(z + \epsilon - (k-1)\epsilon\Bigr) = \\ &= -\frac{1}{\epsilon}\frac{\Bigl(z + \frac{\epsilon}{2} - n_{1}\epsilon\Bigr)\Bigl(z + \frac{3\epsilon}{2} - n_{3}\epsilon\Bigr)}{\Bigl(z + \frac{\epsilon}{2} - n_{2}\epsilon\Bigr)\Bigl(z + \frac{\epsilon}{2} - (n_{2} - 1)\epsilon\Bigr)}\,, \\ \Psi^{(3)[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3})}(z) &= \frac{1}{\epsilon}\prod_{k = 1}^{n_{2}}\varphi_{3, 2}\Bigl(z + \frac{\epsilon}{2} - (k-1)\epsilon\Bigr)\prod_{k = 1}^{n_{3}}\varphi_{3, 3}\Bigl(z + \epsilon - (k - 1)\epsilon\Bigr) = \\ &= -\frac{1}{\epsilon}\frac{(z + 2\epsilon)(z + \epsilon - n_{2}\epsilon)}{\bigl(z + \epsilon - n_{3}\epsilon\bigr)\bigl(z + \epsilon - (n_{3} - 1)\epsilon\bigr)}\,. \end{align}\tag{90}\]

Amplitudes↩︎

The vacuum expectation values of the fields take the form: \[\label{sl404413240l44320443204132vacuum32fixed32points} \begin{align} &\overline{C}_{1} = C(n_{1}), \quad \overline{C}_{2} = C(n_{2})\,, \quad \overline{C}_{3} = C(n_{3})\,,\\ &\overline{A}_{1} = I(n_{2}, n_{1})\,, \quad \overline{A}_{2} = I(n_{3}, n_{2})\,, \quad \overline{R}_{1} = I(n_{1}, 1)\,, \end{align}\tag{91}\] whereas the remaining fields \(B_{1}, B_{2}, S_{1}, R_{2}, S_{2}, R_{3},\) and \(S_{3}\) acquire zero expectation values.

Applying the equivariant methods, we get the matrix elements of the representation: \[\label{sl404413240l44320443204132equiv32representation} \begin{align} &{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}+1, n_{2}, n_{3})} = -\dfrac{1}{\epsilon}\,,\\ &{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2}+1, n_{3})} = \dfrac{n_{1} - n_{2}}{n_{2}\epsilon - \frac{\epsilon}{2}}\,,\\ &{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2}, n_{3} + 1)} = \dfrac{n_{2}-n_{3}}{n_{3}\epsilon - \epsilon}\,,\\ &{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}-1, n_{2}, n_{3})} = -\epsilon(n_{1}-n_{2})(\lambda-n_{1}+1)\,,\\ &{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2}-1, n_{3})} = (n_{2}-n_{3})\Bigl(n_{2} - \frac{3}{2}\Bigr)\epsilon\,,\\ &{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2}, n_{3}-1)} = n_{3}(n_{3}-2)\epsilon\,. \end{align}\tag{92}\] The formulas for the Euler classes are quite bulky, so we do not write them explicitly here.

Next, we normalize the coefficients using 30 . The result reads as follows: \[\label{sl4044132Lie32algebra32coefficients} \begin{align} &e^{(1)}_{0}|n_{1}, n_{2}, n_{3}\rangle = \sqrt{(\lambda - n_{1})(n_{1} - n_{2} + 1)}|n_{1} + 1, n_{2}, n_{3}\rangle\,,\\ &e^{(2)}_{0}|n_{1}, n_{2}, n_{3}\rangle = \sqrt{(n_{1} - n_{2})(n_{2} - n_{3} + 1)}|n_{1}, n_{2} + 1, n_{3}\rangle\,,\\ &e^{(3)}_{0}|n_{1}, n_{2}, n_{3}\rangle = \sqrt{(n_{2} - n_{3})(n_{3} + 1)}|n_{1}, n_{2}, n_{3} + 1\rangle\,;\\ &f^{(1)}_{0}|n_{1}, n_{2}, n_{3}\rangle = \sqrt{(n_{1} - n_{2})(\lambda - n_{1} + 1)}|n_{1} - 1, n_{2}, n_{3}\rangle\,,\\ &f^{(2)}_{0}|n_{1}, n_{2}, n_{3}\rangle = \sqrt{(n_{2} - n_{3})(n_{1} - n_{2} + 1)}|n_{1}, n_{2} - 1, n_{3}\rangle\,,\\ &f^{(3)}_{0}|n_{1}, n_{2}, n_{3}\rangle = \sqrt{n_{3}(n_{2} - n_{3} + 1)}|n_{1}, n_{2}, n_{3}-1\rangle\,. \end{align}\tag{93}\] As expected, these expressions coincide with the formulas introduced by Gelfand in [44].

Finally, we check the hysteresis relations 24 . For the representation \(\Upsilon_{1, \lambda}\) of the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\), the relations are satisfied and take the form: \[\label{sl404413240l44320443204132hysteresis} \begin{align} &{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1} + 1, n_{2}, n_{3})}{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1} +1, n_{2}, n_{3})\to (n_{1}, n_{2}, n_{3})} = (n_{1} - n_{2} + 1)(\lambda - n_{1}) = \mathop{\rm res}\limits_{z= n_{1}\epsilon}\Psi^{(1)}_{\Lambda}(z)\,,\\ &{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2} + 1, n_{3})}{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2} + 1, n_{3})\to (n_{1}, n_{2}, n_{3})} = (n_{1} - n_{2})(n_{2} - n_{3} + 1) = \mathop{\rm res}\limits_{z= -\frac{\epsilon}{2} + n_{2}\epsilon}\Psi^{(2)}_{\Lambda}(z)\,,\\ &{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2}, n_{3} + 1)}{\boldsymbol{F}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3} - 1)\to (n_{1}, n_{2}, n_{3})} = (n_{2} - n_{3})(n_{3} + 1) = \mathop{\rm res}\limits_{z= -\epsilon + n_{3}\epsilon}\Psi^{(2)}_{\Lambda}(z)\,;\\ &\frac{{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1} + 1, n_{2}, n_{3})}{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1} +1, n_{2}, n_{3})\to (n_{1} + 1, n_{2} + 1, n_{3})}}{{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2} + 1, n_{3})}{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2} + 1, n_{3})\to (n_{1}+1, n_{2}+1, n_{3})}} = \dfrac{n_{1} + 1 - n_{2}}{n_{1} - n_{2}} = \varphi_{2, 1}\Bigl(n_{2}\epsilon - \frac{\epsilon}{2} - n_{1}\epsilon\Bigr)\,,\\ &\frac{{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2} + 1, n_{3})}{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2} + 1, n_{3})\to (n_{1}, n_{2} + 1, n_{3} + 1)}}{{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3}) \to (n_{1}, n_{2}, n_{3} + 1)}{\boldsymbol{E}}^{[\lambda, 0, 0]}_{(n_{1}, n_{2}, n_{3} + 1)\to (n_{1}, n_{2}+1, n_{3} + 1)}} = \dfrac{n_{2} + 1 - n_{3}}{n_{2} - n_{3}} = \varphi_{3, 2}\Bigl(n_{3} - \epsilon - n_{2}\epsilon + \frac{\epsilon}{2}\Bigr)\,. \end{align}\tag{94}\]

3.2.3 The Representations \(\Upsilon_{2,\lambda}\)↩︎

The \(F\)-terms that serve as a cut-off for crystal growth in this case read: \[R_{1} = 0\,, \quad C_{2}^{\lambda}R_{2} = 0\,, \quad R_{3} = 0\,.\]

The paths correspond to the words of the form \(\mathcal{P}_{2} = m(C_{1}, C_{2}, C_{3}, A_{1}, A_{2}, B_{1}, B_{2})\cdot R_{2}\). Here we use the field \(C_{2}\) instead of \(C_{1}, C_{3}\) because this is more convenient.

• We have the paths \[C_{2}^{a}R_{2}, \quad A_{2}C_{2}^{a}R_{2}\,,\] similarly to the case above.

• If \(B_{1} \in \mathcal{P}_{2}\) and \(A_{1}, A_{2}, B_{2} \notin \mathcal{P}_{2}\) we have the paths: \[B_{1}C_{2}^{a}R_{2}\,.\] This was not true for the case described above.

• We can proceed and get new paths using \(A_{1}\) or \(B_{2}\): \[A_{1}\underline{B_{1}C_{2}^{a}R_{2}} = B_{2}\underline{A_{2}C_{2}^{a}R_{2}}\,,\] where we have applied the relation \(A_{1}B_{1} = B_{2}A_{2}\). We emphasize that the vacuum expectation values of these monomials do not vanish. This is the main difference from the case described above.

• However, when we have more than two fields \(A_{1}, A_{2}, B_{1}, B_{2}\) in a monomial \(m\), the paths equal zero: \[\begin{align} &A_{2}A_{1}B_{1}C_{2}^{a}R_{2} = (A_{2}B_{2})A_{2}C_{2}^{a}R_{2} = 0\,, \\ &B_{1}B_{2}A_{2}C_{2}^{a}R_{2} = (B_{1}A_{1})B_{1}C_{2}^{a}R_{2} = 0\,. \end{align}\]

Therefore, the set of the paths \(\mathcal{P}_{2}\) can contain only the elements of four types: \[\label{sl4044132possible32paths324004432l4432041} C_{2}^{a}R_{2}, \quad A_{2}C_{2}^{a}R_{2}, \quad B_{1}C_{2}^{a}R_{2}, \quad A_{1}B_{1}C_{2}^{a}R_{2} = B_{2}A_{2}C_{2}^{a}R_{2}\,.\tag{95}\]

The equivariant weights and the R-charges of these atoms take the form: \[\label{sl40441324004432l443204132paths32weights32and32r-charges} \begin{align} &h(C_{2}^{a}R_{2}) = a\epsilon\,, \quad h(A_{2}C_{2}^{a}R_{2}) = -\frac{\epsilon}{2} + h + a\epsilon\,, \quad h(B_{1}C_{2}^{a}R_{2}) = -\frac{\epsilon}{2} - h + a\epsilon\,, \quad h(A_{1}B_{1}C_{2}^{a}R_{2}) = -\epsilon + a\epsilon\,,\\ &R(C_{2}^{a}R_{2}) = 0\,, \quad R(A_{2}C_{2}^{a}R_{2}) = 1\,, \quad R(B_{1}C_{2}^{a}R_{2}) = 1\,, \quad R(A_{1}B_{1}C_{2}^{a}R_{2}) = 2\,. \end{align}\tag{96}\] We emphasize that the additional weight \(h\) helps us to distinguish the paths \(A_{2}C_{2}^{a}R_{2}\) and \(B_{1}C_{2}^{a}R_{2}\), which would overlap if we assumed \(h = 0\) from the beginning. One could also note that the paths \(C_{2}^{a}R_{2}\) and \(A_{1}B_{1}C_{2}^{a}R_{2}\) live in the same vector space \(V_{2}\) yet have different R-charges, therefore, they also do not overlap in the space \(\Xi\).

We can highlight these facts on the R-equivariant space and draw the corresponding empty box 97 : \[\tag{97} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/aihzlebp.png}\tag{98}\end{figure} \qquad \Rightarrow \qquad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/czlautir.png}\tag{99}\end{figure}\] The picture explicitly shows that the crystals of the representation \(\Upsilon_{2,\lambda}\) naturally form a 3-dimensional graph. It justifies our assumption about the existence of an additional parameter \(h\).

We want to emphasize some key differences between this case and the previous cases that corresponded to symmetric representations. Although the previous crystals also lived in the 3-dimensional space, they were effectively 2-dimensional. All the nodes consisted of the atoms of only one type. Here, we have a visible 3-dimensional structure. The vector space \(V_{2}\) include the atoms of two types: \[\label{sl40441324004432l443204132structure32of32V952} V_{2} = {\rm Span}\left\{C_{2}^{a}\cdot R_{2}\,,~~ A_{1} B_{1} C_{2}^{a}\cdot R_{2} \right\} = \nu_{1, 2}\oplus\nu_{2, 2}\,.\tag{100}\] That means that we do not have a unique way to add an atom to the second node. In order to solve this problem, we utilize the Gelfand-Tsetlin bases, which, as we demonstrate, naturally arise in our description.

Example: Representation \((0,1,0)\)↩︎

We now demonstrate the statement above in the simplest example, the representation \((0, 1, 0)\) where \(\lambda = 1\). We need to distinguish the vectors in the space \(V_{2}\). The number \(m_{1}\) will count the atoms of the type \(C_{2}^{a}R_{2}\), whereas the number \(m_{2}\) will count the others: \[\label{sl40441324004432l443204132the32numeration} \begin{align} \dim\nu_{1, 2} = \#(C_{2}^{a}R_{2}) = m_{1}\,,\\ \dim\nu_{2, 2} = \#(A_{1}B_{1}C_{2}^{a}R_{2}) = m_{2}\,. \end{align}\tag{101}\] The formula 100 gives the connection between the dimension \(n_{2}\) of \(V_{2}\) and the parameters \(m_{1},~m_{2}\): \[n_{2} = m_{1} + m_{2}\,.\] The cut-off takes the form \(C_{2}R_{2} = 0\) in this case; therefore, our empty box is of the form: \[ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/jnefspwb.png}\label{pzteaifu}\end{figure}\tag{102}\] Next, we construct the crystals in this representation by gradually adding atoms into the box.

We start with the atom \({R_{2}}\). It corresponds to the action of the generator \(e^{(2)}(z)\) of the Yangian:

The path \(R_{2}\) belongs to the first type of vectors in 100 ; therefore, the parameter \(m_{1}\) equals \(1\) now.

Then we add a second atom. We can only add an atom to the first vector space \(V_{1}\) or to the third one \(V_{3}\) using the operators \(e^{(1)}(z)\) or \(e^{(3)}(z)\) correspondingly. This gives us two new states. These crystals take the form:

We can proceed further and get the remaining states. The whole structure of the states of this representation is depicted in fig. 6. We emphasize that the dimension of the vector space \(V_{2}\) 100 is shifted twice. The first shift corresponds to the parameter \(m_{1}\), whereas the second shift corresponds to the parameter \(m_{2}\). Nonetheless, both shifts are performed by the generator \(e^{(2)}(z)\).

Multiplicities and Gelfand-Tsetlin bases↩︎

In this section we parametrize the states of the representations \(\Upsilon_{2,\lambda}\) and emphasize the importance of Gelfand-Tsetlin bases in our description. In all previous examples¹¹, we were able to assign the dimension vector \(\vec{n}\) to a particular state of the representation. However, this is no longer true for the representations \(\Upsilon_{2,\lambda}\), which means that a crystal cannot be parameterized just by three numbers \(n_{1}, n_{2}, n_{3}\).

This problem is not new in the theory of quiver Yangian algebras. For example, the quiver for \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{1})\) has a single node with the dimension parameter \(n\). It is a well-known fact [9], [16], [60] that there are \(p(n)\) corresponding states in the Fock representation, where \(p(n)\) is the number of Young diagrams of size \(n\).

Notably, this is also tightly related to a classical problem in the representation theory of Lie algebras. Having a highest weight state \(|\vec{\lambda}\rangle\), all the states can be acquired by the action of the raising operators of an algebra: \[\text{States: }\quad |\vec{\mu}\rangle = \prod_{i}e_{i}|\vec{\lambda}\rangle \quad \leftrightarrow \quad \vec{\mu} = \vec{\lambda} - \sum_{i}n_{i}\vec{\alpha}_{i}\,,\] where \(\vec{\alpha}_{i}\) are simple roots. The numbers \(n_{i}\) resemble the dimension vector \(\vec{n}\) of the quiver by construction. The weights \(\vec{\mu}\) form so-called weight system. Although the approach allows to systematically construct all the weights in the representation, it does not track multiplicities. This means that we can have different states \(|\vec{\mu}_{1}\rangle \ne |\vec{\mu}_{2}\rangle\) with \(\vec{\mu}_{1} = \vec{\mu}_{2}\). Depending on the needs of a researcher, there are various ways to overcome this problem. The one we are interested in is introducing the Gelfand-Tsetlin bases [44] as the parametrization of the states.

In order to connect this to the crystal states, we start with the simplest example, where we encounter non-trivial multiplicities in our construction. Namely, we consider the particular states in the representation \((0, 2, 0)\).

The empty box is fixed by the relation \(C_{2}^{2}R_{2} = 0\). We do not unwrap the whole set of states in the representation due to its high dimension. Instead, let us focus on the state \(e^{(3)}(z_{3})e^{(1)}(z_{1})e^{(2)}(z_{2})|\varnothing\rangle\). Its dimension vector is \(\vec{n} = (1, 1, 1)\), and the crystal diagram takes the following form: \[\tag{103} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/xufgyqzt.png}\tag{104}\end{figure}\] We are interested in expanding this crystal to get new states. In fact, there are two possibilities that we highlight in red for a moment: \[\tag{105} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ifxsylcj.png}\tag{106}\end{figure}\] Adding the atoms of the colors of the first and the third nodes is prohibited due to the F-terms. The direct calculation shows that \(C_{1}B_{1}R_{2}\) indeed acquires zero expectation: \[\begin{align} &C_{1}B_{1}R_{2} = B_{1}C_{2}R_{2} = B_{1}(C_{2}R_{2}) = 0\,, \end{align}\] where we used the fact that \(C_{2}R_{2} = 0\) because we do not have this atom in the crystal 103 .

Now, we focus on the two states that we get from 103 . Both crystals are obtained by the action of the generator \(e^{(2)}(z)\). Moreover, the dimension vector of both states equals \(\vec{n} = (1, 2, 1)\).

1. We get the first one by shifting the parameter \(m_{1}\) from \(1\) to \(2\). The crystal diagram takes the form:

   The corresponding vacuum expectation values of the fields read as follows: \[\begin{align} &C_{1} = C_{3} = (0)\,, \\ &C_{2} = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\,,\\ &A_{1} = (0, 0)^{T}, ~ A_{2} = (1, 0)\,,\\ &B_{1} = (1, 0)\,, ~ B_{2} = (0, 0)^{T}\,,\\ &R_{2} = (1, 0)^{T}\,, ~ S_{2} = (0, 0)\,. \end{align}\]

2. By shifting the parameter \(m_{2}\) from \(0\) to \(1\) we acquire the second state. Its diagram takes the form:

   The fields acquire the following vacuum expectation values: \[\begin{align} &C_{1} = C_{3} = (0)\,, \\ &C_{2} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\,,\\ &A_{1} = (0, 0)^{T}\,, ~ A_{2} = (0, 0)\,,\\ &B_{1} = (1, 0)\,, ~ B_{2} = (0, 0)^{T}\,,\\ &R_{2} = (1, 0)^{T}\,, ~ S_{2} = (0)\,. \end{align}\]

We just provided the example of the representation, where one cannot parametrize the states by the dimension vector \(\vec{n}\). We now lift the construction to an arbitrary \(\Upsilon_{2,\lambda}\) representation. Effectively, we have four vector spaces: \[\{V_{1},~V_{2},~V_{3}\} = \{V_{1}, \nu_{1, 2}, \nu_{2, 2}, V_{3}\}\,.\] We suppose that their corresponding dimensions \(n_{1},~m_{1},~m_{2},~n_{3}\) parametrize the crystal in the representation. We encode this in the following form: \[\tag{107} \mu =\quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/vioayxqz.png}\tag{108}\end{figure}\,,\] where \(n_{2} = m_{1} + m_{2}\). This resembles the Gelfand-Tsetlin bases for the representation \((0, \lambda, 0)\). The generators \(e^{(1)}(z)\) and \(e^{(3)}(z)\), or their lowering analogues, shift the dimensions \(n_{1},~n_{3}\). We denote the corresponding patterns as follows: \[\tag{109} \mu_{3, 3 \pm 1}^{2} =\quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/pqwyvxtu.png}\tag{110}\end{figure}\,, \qquad \mu_{1, 1 \pm 1}^{1} \quad = \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/gjexqsyd.png}\tag{111}\end{figure}\,.\] As for the second node, the operator \(e^{(2)}(z)\), or \(f^{(2)}(z)\), can shift the parameters \(m_{1}\) and \(m_{2}\). These options we encode in the following form: \[\tag{112} \mu_{2, 2 \pm 1}^{1} =\quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/klmjcfxn.png}\tag{113}\end{figure}\,, \qquad \mu_{2, 2 \pm 1}^{2} \quad= \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/sowapunj.png}\tag{114}\end{figure}\,.\] One might find the notation of the patterns 109 and 112 confusing at this moment. The symbol \(\mu_{k, k \pm 1}^{j}\) encodes the fact that we shift the dimension of the \(j\)-th element of the \(k\)-th row. For our needs we sometimes stack this notation to encode the fact that we shift two dimensions. For example: \[\bigl(\mu_{2, 2 + 1}^{1}\bigr)_{1, 1 + 1}^{1} = \bigl(\mu_{1, 1 + 1}^{1}\bigr)_{2, 2 + 1}^{1} = \quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fvwxoera.png}\label{lykezohq}\end{figure}\,.\tag{115}\]

State Counting↩︎

Strictly speaking, the introduced patterns 107 have not proven to be the Gelfand-Tsetlin bases yet. In order to associate them with each other, one should verify the triangular inequalities: \[\label{sl40441324004432l443204132triangular32inequalities} \begin{align} &0 \leqslant m_{2} \leqslant n_{3}\,, \quad n_{3} \leqslant m_{1} \leqslant \lambda\,, &m_{2} \leqslant n_{1} \leqslant m_{1}\,. \end{align}\tag{116}\] Note that by construction of the crystal box, all the dimensions are limited by \(\lambda\): \[\begin{align} &0 \leqslant n_{1} \leqslant \lambda\,,\quad 0 \leqslant n_{3} \leqslant \lambda\,, &0 \leqslant m_{1} \leqslant \lambda\,, \quad 0 \leqslant m_{2} \leqslant \lambda\,, \end{align}\] which means that there are only four inequalities in 116 that we need to prove. They can be organized as follows: \[m_{2} \leqslant n_{1} \leqslant m_{2}\,, \quad m_{2} \leqslant n_{3} \leqslant m_{2}\,.\] In fact, when we prove the first sequence, we automatically prove the second due to the quiver symmetry.

We start with the inequality \(n_{1} \leqslant m_{1}\). Let us assume the contrary. It means there is an atom for which: \[C_{1}^{m}B_{1}R_{2} \ne 0\,, \quad C_{2}^{m}R_{2} = 0\,.\] However, we have equivalence relations 81 . Using them, we get: \[C_{1}^{m}B_{1}R_{2} = B_{1}C_{2}^{m}R_{2} = B_{1}(C_{2}^{m}R_{2}) = 0\,,\] which was not true in our suggestion. Therefore, \(n_{1} \leqslant m_{1}\).

The second inequality can be proven in a similar way. The key calculation reads as follows: \[C_{2}^{m}A_{1}B_{1}R_{2} = A_{1}C_{1}^{m}B_{1}R_{2} = A_{1}(C_{1}^{m}B_{1}R_{2}) = 0\,.\] and shows that \(m_{2} \leqslant n_{1}\). It concludes the proof of both sequences of inequalities and verifies that the patterns 107 are indeed the Gelfand-Tsetlin bases.

Next, we count the dimension of our crystal representation. The inequalities 116 limit our dimensions. Without loss of generality, we assume also that \(n_{1} \geqslant n_{3}\) since the quiver [quiver32sl40441324004432l4432041] has \(\mathbb{Z}_{2}\) symmetry 42 . This leads us to the sum: \[\dim_{f.p.}|(0, \lambda, 0)| = \sum_{m_{2} = 0}^{\lambda}\sum_{n_{3} = m_{2}}^{\lambda}\sum_{n_{1} = m_{2}}^{\lambda}\sum_{m_{1} = n_{1}}^{\lambda} 1 = \sum_{m_{2} = 0}^{\lambda}\sum_{n_{3} = m_{2}}^{\lambda}\sum_{n_{1} = m_{2}}^{\lambda}(\lambda - n_{1} + 1) = \dots\,.\] After the calculation, we end up with the result: \[\label{sl40441324004432l443204132dimension} \dim_{f.p.}|(0, \lambda, 0)| = \dfrac{(\lambda + 1)(\lambda + 2)(\lambda + 2)(\lambda + 3)}{3!2!} =\dim_{\mathfrak{s}\mathfrak{l}_{4}}|(0, \lambda, 0)|\,,\tag{117}\] which is exactly the dimension of the \((0, \lambda, 0)\) representation of the algebra \(\mathfrak{s}\mathfrak{l}_{4}\).

The representation↩︎

We need to adjust the ansatz 22 to the representations \(\Upsilon_{2,\lambda}\). Our states are now denoted as Gelfand-Tsetlin bases 107 . The operators corresponding to the first node and the third node transform them into the bases 109 . As for the operators \(e^{(2)}(z)\) and \(f^{(2)}(z)\), their action is now split in two and gives the patterns 112 . Therefore, we end up with the ansatz of the following form: \[\label{sl40441324004432l443204132ansatz} \begin{align} &e^{(1)}_{\Upsilon_{2,\lambda}}(z)|\mu\rangle = \dfrac{{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{1, 1 + 1}^{1}]}{z + \frac{\epsilon}{2} - n_{1}\epsilon}|\mu_{1, 1 + 1}^{1}\rangle\,, \\ &e^{(2)}_{\Upsilon_{2,\lambda}}(z)|\mu\rangle = \dfrac{{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu\to \mu_{2, 2 + 1}^{1}]}{z - m_{1}\epsilon}|\mu_{2, 2 + 1}^{1}\rangle + \dfrac{{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu\to \mu_{2, 2 + 1}^{2}]}{z + \epsilon - m_{2}\epsilon}|\mu_{2, 2 + 1}^{2}\rangle\,,\\ &e^{(3)}_{\Upsilon_{2,\lambda}}(z)|\mu\rangle = \dfrac{{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{3, 3 + 1}^{2}]}{z + \frac{\epsilon}{2} - n_{3}\epsilon}|\mu_{3, 3 + 1}^{2}\rangle\,,\\ &f^{(1)}_{\Upsilon_{2,\lambda}}(z)|\mu\rangle = \dfrac{{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{1, 1 - 1}^{1}]}{z + \frac{\epsilon}{2} - (n_{1} - 1)\epsilon}|\mu_{1, 1 - 1}^{1}\rangle\,, \\ &f^{(2)}_{\Upsilon_{2,\lambda}}(z)|\mu\rangle = \dfrac{{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2, -1}^{1}]}{z - (m_{1} - 1)\epsilon}|\mu_{2, 2-1}^{1}\rangle + \dfrac{{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2 -1}^{2}]}{z + \epsilon - (m_{2} - 1)\epsilon}|\mu_{2, 2-1}^{2}\rangle\,,\\ &f^{(3)}_{\Upsilon_{2,\lambda}}(z)|\mu\rangle = \dfrac{{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{3, 3, -1}^{2}]}{z + \frac{\epsilon}{2} - (n_{3} - 1)\epsilon}|\mu_{3, 3 - 1}^{2}\rangle,\\ &\psi^{(s)}_{\Upsilon_{2,\lambda}}(z)|\mu\rangle = \Psi^{(s)}_{\mu, \Upsilon_{2,\lambda}}(z)|\mu\rangle\,. \end{align}\tag{118}\] The eigenfunctions of the operators \(\psi^{(s)}_{\Upsilon_{2,\lambda}}(z)\) read: \[\label{sl40441324004432l443204132eigenfunctions} \begin{align} \Psi^{(1)}_{\mu, \Upsilon_{2,\lambda}}(z) &= -\frac{1}{\epsilon}\prod_{k = 1}^{n_{1}}\varphi_{1, 1}\Bigl(z + \frac{\epsilon}{2} - (k-1)\epsilon\Bigr)\prod_{k = 1}^{m_{1}}\varphi_{1, 2}\Bigl(z - (k-1)\epsilon\Bigr)\prod_{k = 1}^{m_{2}}\Bigl(z + \epsilon - (k-1)\epsilon\Bigr) =\\ &= -\dfrac{1}{\epsilon}\dfrac{\Bigl(z + \dfrac{\epsilon}{2} - m_{1}\epsilon\Bigr)\Bigl(z + \frac{\epsilon}{2} - (m_{2} - 1)\epsilon\Bigr)}{\Bigl(z + \dfrac{\epsilon}{2}- n_{1}\epsilon\Bigr)\Bigl(z + \frac{ \epsilon}{2}- (n_{1} - 1)\epsilon\Bigr)}\,, \\ \Psi^{(2)}_{\mu, \Upsilon_{2,\lambda}}(z) &= -\frac{1}{\epsilon}\dfrac{(z - \lambda\epsilon)}{z}\prod_{k = 1}^{n_{1}}\varphi_{2, 1}\Bigl(z + \frac{\epsilon}{2} - (k - 1)\epsilon\Bigr)\prod_{k = 1}^{m_{1}}\varphi_{2, 2}\Bigl(z - (k - 1)\epsilon\Bigr)\prod_{k = 1}^{m_{2}}\varphi_{2, 2}\Bigl(z + \epsilon - (k-1)\epsilon\Bigr)\cdot \\ &\cdot\prod_{k = 1}^{n_{3}}\varphi_{2, 3}\Bigl(z + \dfrac{\epsilon}{2} - (k-1)\epsilon\Bigr) = -\frac{1}{\epsilon}\frac{(z - \lambda \epsilon)(z + 2\epsilon)(z + \epsilon - n_{1}\epsilon)(z + \epsilon - n_{3}\epsilon)}{(z - m_{1}\epsilon)(z - (m_{1} - 1)\epsilon)(z + \epsilon - m_{2}\epsilon)(z + \epsilon - (m_{2} - 1)\epsilon)}\,, \\ \Psi^{(3)}_{\mu, \Upsilon_{2,\lambda}}(z) &= \frac{1}{\epsilon}\prod_{k = 1}^{n_{2}}\varphi_{3, 2}\Bigl(z + \frac{\epsilon}{2} - (k-1)\epsilon\Bigr)\prod_{k = 1}^{n_{3}}\varphi_{3, 3}\Bigl(z + \epsilon - (k - 1)\epsilon\Bigr) = \\ &= -\frac{1}{\epsilon}\frac{\Bigl(z + \dfrac{\epsilon}{2} - m_{1}\epsilon\Bigr)\Bigl(z + \dfrac{\epsilon}{2} - (m_{2} - 1)\epsilon\Bigr)}{\Bigl(z + \dfrac{\epsilon}{2} - n_{3}\epsilon\Bigr)\Bigl(z + \dfrac{\epsilon}{2} - (n_{3} - 1)\epsilon\Bigr)}\,. \end{align}\tag{119}\] The function \(\Psi_{\mu, \Upsilon_{2,\lambda}}^{(2)}(z)\) now has four poles, which reflects the fact that there are more possibilities to add or remove an atom from the crystal \(|\mu\rangle\).

Amplitudes↩︎

In order to determine the coefficients in the ansatz 118 , we need to construct the vacuum expectation values of the fields that correspond to a fixed point in 81 . As we have mentioned, the space \(V_{2}\) consists of two parts, \(\nu_{1, 2}\) and \(\nu_{2, 2}\). We remind the reader that the matrices \(A_{1}, B_{1}, A_{2}, B_{2}, C_{2}\) are linear maps of the form: \[\label{sl40441324004432l443204132linear32maps32involving32V2} \begin{align} &A_{1} \colon V_{1} \to V_{2}\,, \qquad B_{1} \colon V_{2} \to V_{1}\,,\\ &A_{2} \colon V_{2} \to V_{3}\,, \qquad B_{2} \colon V_{3} \to V_{2}\,,\\ &C_{2} \colon V_{2} \to V_{2}\,, \end{align}\tag{120}\] and all of them involve the vector space \(V_{2}\). The structure of this space 100 is a direct sum of two subspaces. This results in the fact that the fields act on these subspaces independently, which means the matrices have a block structure.

To separate the two subspaces \(\nu_{1, 2},~\nu_{2, 2}\), we fix the ordering of the vectors in \(V_{2}\). First, we enumerate all the vectors of the form \(C_{2}^{a}R_{2}\), then the remaining ones. As an example, let us consider another state of the representation \((0, 2, 0)\), where we have four vectors in \(V_{2}\): \[\tag{121} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/oehtlbgz.png}\tag{122}\end{figure} \qquad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fivutbce.png}\tag{123}\end{figure} \quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/iwxsflma.png}\tag{124}\end{figure}\] In 121 we present the structure of the vector spaces, the enumeration of the atoms of the state, and the crystal itself.

We can now define the vacuum expectation values of the fields for an arbitrary GT base 107 using the ICO matrices introduced earlier 66 : \[\label{sl40441324004432l443204132vacuum32fixed32points} \begin{align} &\overline{C}_{1} = C(n_{1})\,, \quad \overline{C}_{2} = \left( \begin{array}{c|c} C(m_{1}) & O(m_{1}, m_{2}) \\ \hline O(m_{2}, m_{1}) & C(m_{2}) \end{array} \right)\,, \quad \overline{C}_{3} = C(n_{3})\,, \\ &\overline{A}_{1} = \left(\begin{array}{c} O(m_{1}, n_{1}) \\ \hline I(m_{2}, n_{1}) \end{array}\right)\,, \quad \overline{A}_{2} = \left(\begin{array}{c|c} I(n_{3}, m_{1}) & O(n_{3}, m_{2}) \end{array}\right)\,, \\ &\overline{B}_{1} = \left(\begin{array}{c|c} I(n_{1}, m_{1}) & O(n_{1}, m_{2}) \end{array}\right)\,, \quad \overline{B}_{2} = \left(\begin{array}{c} O(m_{1}, n_{3}) \\ \hline I(m_{2}, n_{3}) \end{array}\right)\,,\\ &\overline{R}_{1} = O(n_{1}, 1)\,, \quad \overline{R}_{2} = I(m_{1} + m_{2}, 1)\,, \quad \overline{R}_{3} = O(n_{3}, 1)\,. \end{align}\tag{125}\]

Substituting the fixed point into the algorithm discussed in section 2.6, we end up with the matrix elements of the following form: \[\label{sl40441324004432l443204132equivariant32coefficients} \begin{align} &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{1, 1 + 1}^{1}] = \frac{(m_{1} - n_{1})}{n_{1}\epsilon - \frac{\epsilon}{2}}\,,\\ &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2 + 1}^{1}] = -\frac{1}{\epsilon}\,,\\ &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2 + 1}^{2}] = -\frac{(n_{1} - m_{2})(n_{3} - m_{2})}{(m_{1} - m_{2})(m_{1} - m_{2} + 1)}\frac{1}{\epsilon}\,,\\ &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{3, 3 + 1}^{2}] = \frac{m_{1} - n_{3}}{n_{3}\epsilon - \frac{\epsilon}{2}}\,,\\ &{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{1, 1 - 1}^{1}] = (n_{1} - m_{2})\Bigl(n_{1} - \frac{3}{2}\Bigr)\epsilon\,,\\ &{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2 - 1}^{1}] = -\frac{(m_{1} + 1)(\lambda - m_{1} + 1)(m_{1} - n_{1})(m_{1} - n_{3})}{(m_{1} - m_{2} + 1)(m_{1} - m_{2})}\epsilon\,,\\ &{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2 - 1}^{2}] = - m_{2}(\lambda - m_{2} + 2)\epsilon\,,\\ &{\boldsymbol{F}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{3, 3 - 1}^{2}] = (n_{3} - m_{2})\Bigl(n_{3} - \frac{3}{2}\Bigr)\epsilon\,. \end{align}\tag{126}\] Using the normalization 30 , we verify that at the zero level the construction gives us representations \((0, \lambda, 0)\) of the algebra \(\mathfrak{s}\mathfrak{l}_{4}\): \[\label{sl40441324004432l443204132Lie32algebra32coefficients} \begin{align} e^{(1)}_{0}|\mu\rangle &= \sqrt{(m_{1} - n_{1})(n_{1} - m_{2} + 1)}|\mu_{1, 1 + 1}^{1}\rangle\\ e_{0}^{(2)}|\mu\rangle &= \sqrt{\frac{(m_{1} + 2)(\lambda - m_{1})(m_{1} - n_{1} + 1)(m_{1} - n_{3} + 1)}{(m_{1} - m_{2} + 2)(m_{1} - m_{2} + 1)}}|\mu_{2, 2 + 1}^{1}\rangle +\\ &+ \sqrt{\frac{(m_{2} + 1)(\lambda - m_{2} + 1)(n_{1} - m_{2})(n_{3} - m_{2})}{(m_{1} - m_{2})(m_{1} - m_{2} + 1)}}|\mu_{2, 2 + 1}^{2}\rangle\\ e^{(3)}_{0}|\mu\rangle &= \sqrt{(m_{1} - n_{3})(n_{3} - m_{2} + 1)}|\mu_{3, 3 + 1}^{2}\rangle,\\ f^{(1)}_{0}|\mu\rangle &= \sqrt{(m_{1} - n_{1} + 1)(n_{1} - m_{2})}|\mu_{1, 1 - 1}^{1}\rangle\\ f_{0}^{(2)}|\mu\rangle &= \sqrt{\frac{(m_{1} + 1)(\lambda - m_{1} + 1)(m_{1} - n_{1})(m_{1} - n_{3})}{(m_{1} - m_{2} + 1)(m_{1} - m_{2})}}|\mu_{2, 2 - 1}^{1}\rangle +\\ &+ \sqrt{\frac{m_{2}(\lambda - m_{2} + 2)(n_{1} - m_{2} + 1)(n_{3} - m_{2} + 1)}{(m_{1} - m_{2} + 1)(m_{1} - m_{2} + 2)}}|\mu_{2, 2 - 1}^{2}\rangle\\ f^{(3)}_{0}|\mu\rangle &= \sqrt{(m_{1} - n_{3} + 1)(n_{3} - m_{2})}|\mu_{3, 3 - 1}^{2}\rangle, \end{align}\tag{127}\]

Lastly, we need to check the hysteresis relations of the Yangian 24 . They are also satisfied in this case and take the following form: \[\label{sl40441324004432l443204132hysteresis} \begin{align} &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{1, 1 + 1}^{1}]{\boldsymbol{F}}_{\Upsilon_{2, \lambda}}[\mu_{1, 1 + 1}^{1} \to \mu] = (m_{1} - n_{1})(n_{1} - m_{2} + 1) = \mathop{\rm res}_{z = n_{1}\epsilon - \frac{\epsilon}{2}}\Psi_{\mu, \Upsilon_{2,\lambda}}^{(1)}(z)\,,\\ &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2 + 1}^{1}]{\boldsymbol{F}}_{\Upsilon_{2, \lambda}}[\mu_{2, 2 + 1}^{1} \to \mu] = \frac{(m_{1} + 2)(\lambda - m_{1})(m_{1} - n_{1} + 1)(m_{1} - n_{3} + 1)}{(m_{1} - m_{2} + 2)(m_{1} - m_{2} + 1)} = \mathop{\rm res}_{z = m_{1}\epsilon}\Psi_{\mu, \Upsilon_{2,\lambda}}^{(1)}(z)\,,\\ &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{2, 2 + 1}^{2}]{\boldsymbol{F}}_{\Upsilon_{2, \lambda}}[\mu_{2, 2 + 1}^{2} \to \mu] = \frac{(m_{2} + 1)(\lambda - m_{2} + 1)(n_{1} - m_{2})(n_{3} - m_{2})}{(m_{1} - m_{2})(m_{1} - m_{2} + 1)} = \mathop{\rm res}_{z = m_{2}\epsilon - \epsilon}\Psi_{\mu, \Upsilon_{2,\lambda}}^{(1)}(z)\,,\\ &{\boldsymbol{E}}_{\Upsilon_{2,\lambda}}[\mu \to \mu_{3, 3 + 1}^{2}]{\boldsymbol{F}}_{\Upsilon_{2, \lambda}}[\mu_{3, 3 + 1}^{2} \to \mu] = (m_{1} - n_{3})(n_{3} - m_{2} + 1) = \mathop{\rm res}_{z = n_{3}\epsilon - \frac{\epsilon}{2}}\Psi_{\mu, \Upsilon_{2,\lambda}}^{(3)}(z)\,,\\ &\frac{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{2, 2 + 1}^{1}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{2, 2 + 1}^{1} \to \bigl(\mu_{2, 2 + 1}^{1}\bigr)_{1, 1 + 1}^{1}\Bigr]}{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{1, 1 + 1}^{1}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{1, 1 + 1}^{1} \to \bigl(\mu_{1, 1 + 1}^{1}\bigr)_{2, 2 + 1}^{1}\Bigr]} = \frac{m_{1} + 1 - n_{1}}{m_{1} - n_{1}} = \varphi_{1, 2}\Bigl(n_{1}\epsilon - \frac{\epsilon}{2} - m_{1}\epsilon\Bigr)\,,\\ &\frac{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{1, 1 + 1}^{1}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{1, 1 + 1}^{1} \to \bigl(\mu_{1, 1 + 1}^{1}\bigr)_{2, 2 + 1}^{2}\Bigr]}{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{2, 2 + 1}^{2}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{2, 2 + 1}^{2} \to \bigl(\mu_{2, 2 + 1}^{2}\bigr)_{1, 1 + 1}^{1}\Bigr]} = \frac{n_{1} + 1 - m_{2}}{n_{1} - m_{2}} = \varphi_{2, 1}\Bigl(m_{2}\epsilon - \epsilon - n_{1}\epsilon + \frac{\epsilon}{2}\Bigr)\,,\\ &\frac{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{2, 2 + 1}^{1}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{2, 2 + 1}^{1} \to \bigl(\mu_{2, 2 + 1}^{1}\bigr)_{3, 3 + 1}^{2}\Bigr]}{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{3, 3 + 1}^{2}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{3, 3 + 1}^{2} \to \bigl(\mu_{3, 3 + 1}^{2}\bigr)_{2, 2 + 1}^{1}\Bigr]} = \frac{m_{1} + 1 - n_{3}}{m_{1} - n_{3}} = \varphi_{3, 2}\Bigl(n_{3}\epsilon - \frac{\epsilon}{2} - m_{1}\epsilon\Bigr)\,,\\ &\frac{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{3, 3 + 1}^{2}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{3, 3 + 1}^{2} \to \bigl(\mu_{3, 3 + 1}^{2}\bigr)_{2, 2 + 1}^{2}\Bigr]}{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{2, 2 + 1}^{2}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{2, 2 + 1}^{2} \to \bigl(\mu_{2, 2 + 1}^{2}\bigr)_{3, 3 + 1}^{2}\Bigr]} = \frac{n_{3} + 1 - m_{2}}{n_{3} - m_{2}} = \varphi_{2, 3}\Bigl(m_{2}\epsilon - \epsilon - n_{3}\epsilon + \frac{\epsilon}{2}\Bigr)\,,\\ &\frac{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{2, 2 + 1}^{1}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{2, 2 + 1}^{1} \to \bigl(\mu_{2, 2 + 1}^{1}\bigr)_{2, 2 + 1}^{2}\Bigr]}{{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}[\mu \to \mu_{2, 2 + 1}^{2}]{\boldsymbol{E}}_{\Upsilon_{2, \lambda}}\Bigl[\mu_{2, 2 + 1}^{2} \to \bigl(\mu_{2, 2 + 1}^{2}\bigr)_{2, 2 + 1}^{1}\Bigr]} = \frac{m_{1} - m_{2}}{m_{1} - m_{2} + 2} = \varphi_{2, 2}\bigl(m_{2}\epsilon - \epsilon - m_{1}\epsilon\bigr)\,. \end{align}\tag{128}\]

3.3 \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) details↩︎

We are now ready to lift the construction and describe the key elements of the \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) algebras representations for an arbitrary \(n\).

3.3.1 The Algebra↩︎

We work with the quiver \({\cal Q}_{n,p,\lambda}\) 38 . The dimensions of the nodes are denoted as \(n_{1}, \dots, n_{r}\) where \(r = n - 1\). These parameters, as usual, can be assembled into the dimension vector \(\vec{n}\). We double the superpotential here for convenience: \[\label{sl40n4132superpotential} {\cal W}={\rm Tr}\,\left[A_1C_1B_1+\sum\limits_{a=2}^{n-2}\left(A_aC_aB_a-B_{a-1}C_aA_{a-1}\right)-B_{n-2}C_{n-1}A_{n-2}+{\color{burgundy}C_p^{\lambda}R_{p}S_{p}}\right]\,.\tag{129}\]

The equivariant weights and R-charges that are assigned to the fields take the following form: \[\label{sl40n4132weights} \begin{tblr}{c|c|c|c|c|c} Fields & C_{i} & A_{i} & B_{i} & R_{p} & S_{p}\\ \hline Weights & \epsilon & -\frac{\epsilon}{2} + h & -\frac{\epsilon}{2} - h & 0 & -\lambda \epsilon \\ R-charges & 0 & 1 & 1 & 0 & 2 \end{tblr}\,.\tag{130}\]

The generating functions of the Yangian read: \[\label{sl40n4132generating32functions} e^{(a)}(z) = \sum_{n = 0}^{\infty}\dfrac{e^{(a)}_{n}}{z^{n + 1}}\,, \quad f^{(a)}(z) = \sum_{n = 0}^{\infty}\dfrac{f^{(a)}_{n}}{z^{n + 1}}\,, \quad \psi^{(a)}(z) = 1 + \sum_{n = 0}^{\infty}\dfrac{\psi_{n}^{(a)}}{z^{n + 1}}\,.\tag{131}\] These functions satisfy the relations 5 , where the bonding factors for \(|a - b |\leqslant 1\) are given by: \[\label{sl40n4132bonding32factors} \varphi_{a, b}(z) = \delta_{b, a + 1}\dfrac{2z - \epsilon}{2z + \epsilon} + \delta_{b, a}\dfrac{z + \epsilon}{z - \epsilon} + \delta_{b, a - 1}\dfrac{2z - \epsilon}{2z + \epsilon}\,.\tag{132}\] When \(|a - b| > 1\), we set \(\varphi_{a, b}(z) \equiv 1\). Again, we substitute \(h = 0\) in the calculations related to the algebra.

We also impose the Serre relations [2], [81]: \[\label{sl40n4132serre32relations} \begin{align} &\sum_{\sigma \in \mathfrak{S}_{m}}\Bigl[e^{(a)}\bigl(u_{\sigma(1)}\bigr), \Bigl[e^{(a)}\bigl(u_{\sigma(2)}\bigr), \dots, \Bigl[e^{(a)}\bigl(u_{\sigma(m)}\bigr), e^{(b)}(v)\Bigl]\dots \Bigr]\Bigr] = 0,\\ &\sum_{\sigma \in \mathfrak{S}_{m}}\Bigl[f^{(a)}\bigl(u_{\sigma(1)}\bigr), \Bigl[f^{(a)}\bigl(u_{\sigma(2)}\bigr),\dots \Bigl[f^{(a)}(u_{\sigma(m)}), f^{(b)}(v)\Bigr]\dots \Bigr]\Bigr] = 0, \end{align}\tag{133}\] where \({\cal A}_{ab}\) is the Cartan matrix of \(\mathfrak{s}\mathfrak{l}_{n}\), \(m = 1 - {\cal A}_{ab}\), and \(a\ne b\).

Unfolding the relations 5 in modes for \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) we get: \[\label{sl40n4132Yangian32relations32in32modes} \begin{align} &[e_{n + 1}^{(a)}, e^{(a)}_{k}] - [e^{(a)}_{n}, e^{(a)}_{k + 1}] = \epsilon\{e_{n}^{(a)}, e_{k}^{(a)}\}\,, \\ &[e_{n + 1}^{(a)}, e_{k}^{(a + 1)}] - [e^{(a)}_{n}, e^{(a + 1)}_{k + 1}] = -\frac{\epsilon}{2}\{e^{(a)}_{n}, e^{(a + 1)}_{k}\}\,, \\ &[e_{n + 1}^{(a + 1)}, e^{(a)}_{k}] - [e^{(a + 1)}_{n}, e_{k + 1}^{(a)}] = -\frac{\epsilon}{2}\{e^{(a + 1)}_{n}, e^{(a)}_{k}\}\,, \\ &[\psi_{n + 1}^{(a)}, e_{k}^{(a)}] - [\psi^{(a)}_{n}, e^{(a)}_{k + 1}] = \epsilon\{\psi_{n}^{(a)}, e_{k}^{(a)}\}\,,\\ &[\psi_{n + 1}^{(a)}, e_{k}^{(a + 1)}] - [\psi^{(a)}_{n}, e_{k + 1}^{(a + 1)}] = -\frac{\epsilon}{2}\{\psi_{n}^{(a)}, e_{k}^{(a + 1)}\}\,,\\ &[\psi^{(a + 1)}_{n + 1}, e_{k}^{(a)}] - [\psi_{n}^{(a + 1)}, e_{k + 1}^{(a)}] = -\frac{\epsilon}{2}\{\psi_{n}^{(a + 1)}, e_{k}^{(a)}\}\,,\\ &[f_{n + 1}^{(a)}, f^{(a)}_{k}] - [f^{(a)}_{n}, f^{(a)}_{k + 1}] = -\epsilon\{f_{n}^{(a)}, f_{k}^{(a)}\}\,, \\ &[f_{n + 1}^{(a)}, f_{k}^{(a + 1)}] - [f^{(a)}_{n}, f^{(a + 1)}_{k + 1}] = \frac{\epsilon}{2}\{f^{(a)}_{n}, f^{(a + 1)}_{k}\}\,, \\ &[f_{n + 1}^{(a + 1)}, f^{(a)}_{k}] - [f^{(a + 1)}_{n}, f_{k + 1}^{(a)}] = \frac{\epsilon}{2}\{f^{(a + 1)}_{n}, f^{(a)}_{k}\}\,, \\ &[\psi_{n + 1}^{(a)}, f_{k}^{(a)}] - [\psi^{(a)}_{n}, f^{(a)}_{k + 1}] = -\epsilon\{\psi_{n}^{(a)}, f_{k}^{(a)}\}\,,\\ &[\psi_{n + 1}^{(a)}, f_{k}^{(a + 1)}] - [\psi^{(a)}_{n}, f_{k + 1}^{(a + 1)}] = \frac{\epsilon}{2}\{\psi_{n}^{(a)}, f_{k}^{(a + 1)}\}\,,\\ &[\psi^{(a + 1)}_{n + 1}, f_{k}^{(a)}] - [\psi_{n}^{(a + 1)}, f_{k + 1}^{(a)}] = \frac{\epsilon}{2}\{\psi_{n}^{(a + 1)}, f_{k}^{(a)}\}\,,\\ &[\psi_{n}^{(a)}, \psi_{k}^{(b)}] = 0\,,\\ &[e_{n}^{(a)}, f_{k}^{(b)}] = \delta_{ab}\psi^{(a)}_{n + k}\,, \end{align}\tag{134}\] with “boundary conditions": \[\label{sl40n4132boundary32relations} \begin{align} &[\psi_{0}^{(a)}, e_{k}^{(a)}] = 2 e_{k}^{(a)}\,,\quad &[\psi_{0}^{(a)}, f_{k}^{(a)}] = -2 f_{k}^{(a)}\,, \\ &[\psi^{(a + 1)}_{0}, e_{k}^{(a)}] = -e_{k}^{(a)}\,,\quad &[\psi_{0}^{(a + 1)}, f_{k}^{(a)}] = f_{k}^{(a)}\,, \\ &[\psi_{0}^{(a)}, e^{(a + 1)}_{k}] = -e_{k}^{(a + 1)}\,,\quad &[\psi_{0}^{(a)}, f^{(a + 1)}_{k}] = f_{k}^{(a + 1)}\,. \end{align}\tag{135}\] The Serre relations 133 in modes take the form introduced in [2]: \[\label{sl40n4132serre32relations32in32modes} \begin{align} &\sum_{\sigma \in \mathfrak{S}_{m}}\Bigl[e^{(a)}_{n_{\sigma(1)}}, \Bigl[e^{(a)}_{n_{\sigma(2)}}, \dots, \Bigl[e^{(a)}_{n_{\sigma(m)}}, e^{(b)}_{m}\Bigl]\dots \Bigr]\Bigr] = 0\,,\\ &\sum_{\sigma \in \mathfrak{S}_{m}}\Bigl[f^{(a)}_{n_{\sigma(1)}}, \Bigl[f^{(a)}_{n_{\sigma(2)}}, \dots, \Bigl[f^{(a)}_{n_{\sigma(m)}}, f^{(b)}_{m}\Bigl]\dots \Bigr]\Bigr] = 0\,. \end{align}\tag{136}\] Strictly speaking, we have not included the relations between the generators for which \(|a - b| \geqslant 2\) in 134 , 135 . They are trivial, and we can include them by rewriting the relations more compactly using the Cartan matrix: \[\label{sl40n4132relations32in32modes32via32Cartan32matrix} \begin{align} &[e_{n + 1}^{(a)}, e^{(b)}_{k}] - [e^{(a)}_{n}, e^{(b)}_{k + 1}] = \frac{\epsilon}{2}{\cal A}_{ab}\{e_{n}^{(a)}, e_{k}^{(b)}\}\,, \\ &[\psi_{n + 1}^{(a)}, e_{k}^{(b)}] - [\psi^{(a)}_{n}, e^{(b)}_{k + 1}] = \frac{\epsilon}{2}{\cal A}_{ab}\{\psi_{n}^{(a)}, e_{k}^{(b)}\}\,,\\ &[f_{n + 1}^{(a)}, f^{(b)}_{k}] - [f^{(a)}_{n}, f^{(b)}_{k + 1}] = -\frac{\epsilon}{2}{\cal A}_{ab}\{f_{n}^{(a)}, f_{k}^{(b)}\}\,, \\ &[\psi_{n + 1}^{(a)}, f_{k}^{(b)}] - [\psi^{(a)}_{n}, f^{(b)}_{k + 1}] = -\frac{\epsilon}{2}{\cal A}_{ab}\{\psi_{n}^{(a)}, f_{k}^{(b)}\}\,,\\ \hline &[\psi_{0}^{(a)}, e_{k}^{(b)}] = {\cal A}_{ab} e_{k}^{(b)}\,,\qquad [\psi_{0}^{(a)}, f_{k}^{(b)}] = -{\cal A}_{ab} f_{k}^{(b)}\,, \end{align}\tag{137}\] where the form of the remaining relations is not changed.

3.3.2 The Representations \(\Upsilon_{p,\lambda}\)↩︎

Having defined the algebraic relations of \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\), we proceed to its representation. We fix an arbitrary representation \(\Upsilon_{p, \lambda}\) 40 , where the parameter \(p\) is restricted by 43 , and \(\lambda \geqslant 0\).

The States↩︎

Now, we discuss the crystal structure of the states of this representation. The superpotential 129 defines the F-term relations as follows: \[\label{sl40n4132F-terms} \begin{align} &\partial_{A_{i}}{\cal W}= C_{i}B_{i} - B_{i}C_{i + 1} = 0\,, & i \in \overline{1, n - 2}\,,\\ &\partial_{B_{i}}{\cal W}= A_{i}C_{i} - C_{i + 1}A_{i} = 0\,, & i \in \overline{1, n - 2}\,,\\ &\partial_{C_{i}}{\cal W}= B_{i}A_{i} - A_{i - 1}B_{i - 1} = 0\,, & i \in \overline{2, n - 2}\,,\\ &\partial_{C_{1}}{\cal W}= B_{1}A_{1} = 0\,, &\partial_{C_{n - 1}}{\cal W}= A_{n - 2}B_{n - 2} = 0\,,\\ &\partial_{S_{p}}{\cal W}= C_{p}^{\lambda}R_{p} = 0\,. \end{align}\tag{138}\] The fixed points can be defined according to 130 : \[\label{sl40n4132fixed32points} \begin{align} &[\Phi_{i}, C_{i}] = \epsilon C_{i}\,,\quad \Phi_{i + 1}A_{i} - A_{i}\Phi_{i} = \Bigl(-\frac{\epsilon}{2} + h\Bigr)A_{i}\,, \quad \Phi_{i}B_{i} - B_{i}\Phi_{i + 1} = \Bigl(-\frac{\epsilon}{2} - h\Bigr)B_{i}\,,\\ &\Phi_{i}R_{i} = 0\,, \qquad S_{p}\Phi_{p} = -\lambda \epsilon S_{p}\,. \end{align}\tag{139}\] The first two relations in 138 allow us to interchange a field \(C_{i}\) and \(C_{j}\) for any \(i,~j\in Q_{0}\), just as in the previous examples. Therefore, we use the field \(C_{p}\). Again, the term \(C_{p}^{\lambda}R_{p} = 0\) serves as a cut-off.

The relations involving the fields \(A_{i}\) and \(B_{i}\) impose more interesting constraints on a crystal. A state grows from the field \(R_{p}\). Acting by the fields \(A_{i}, B_{j}\) (and \(C_{p}\)) on the field \(R_{p}\) we get some graph whose 2-dimensional slice \((h, R)\) in the space \(\Xi\) we depict in 140 . \[\tag{140} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ngimfodh.png}\tag{141}\end{figure}\]

As we have mentioned, the \(F\)-terms impose homotopic equivalence on paths of the R-equivariant space \(\Xi\). An example of the equivalence is presented in 140 .

Our aim is to construct the empty box, the set of all the admissible paths on \(\Xi\), for the representation \(\Upsilon_{p, \lambda}\). One could check that we always can form a rectangular structure 142 that, by construction, is limited by the edges of the quiver 38 . \[\tag{142} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/xnjtzydi.png}\tag{143}\end{figure}\]

In general, however, we could end up with one of the two cases depicted¹² in fig. 9 as well.

We claim that none of them is allowed in our case.

• In the first case, we have an extra structure near the edge of the quiver. Let us look closer at the graph. \[\tag{144} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/rqcaiotg.png}\tag{145}\end{figure}\] We note that the green path in 144 is of the form: \[B_{1}A_{1}\cdot\underbrace{\text{combination of the fields }A,\,B}\cdot\, C_{p}^{a}R_{p}\,.\] Using the relation: \[B_{1}A_{1} = 0\,,\] we verify that this type of growth is indeed prohibited.

• In the second case, we have an extra structure “inside" the quiver. The key fact that we use here is that the path equivalence holds in the whole space \(\Xi\), not only the graph we cut off. Indeed, let us consider the red path depicted in 146 . \[\tag{146} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fbyviocz.png}\tag{147}\end{figure}\] The path is equivalent to the green path by the relations \(B_{i}A_{i} = A_{i-1}B_{i-1}\); however, it goes through the node that was not included inside the box. Moreover, we can homotopically deform the path to go through the edge nodes and conclude that the vacuum expectation value of this path is zero, as in the previous case.

Therefore, we have demonstrated that the \(F\)-terms limit the growth of a crystal drastically. We solidify our discussion above by providing a concrete example, \({\cal Q}_{5,2,\lambda}\).

In this case, the height on the \(\epsilon\)-axis is simply restricted by \(C_{2}^{\lambda}R_{2} = 0\). Therefore, we are interested in the two-dimensional slice of \(\Xi\) that takes the following form: \[\tag{148} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/mxfvrosd.png}\tag{149}\end{figure}\] The numbers label the nodes of the quiver \({\cal Q}_{5,2,\lambda}\). We focus on a single path depicted in 150 . \[\tag{150} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/qvnsruba.png}\tag{151}\end{figure}\] This green path is a path of the form: \[B_{2}B_{3}A_{3}A_{2}\,C_{2}^{a}R_{2}\,.\] Now we use the relations 138 and get the result: \[B_{2}B_{3}A_{3}A_{2}\,C_{2}^{a}R_{2} = (B_{2}A_{2})(B_{2}A_{2})\,C_{2}^{a}R_{2} = A_{1}(B_{1}A_{1})B_{1}\,C_{2}^{a}R_{2} = 0\,,\] which proves that the path is not admissible.

We can now proceed further and define the empty box for \(\Upsilon_{p,\lambda}\). The two-dimensional projection reads as follows: \[\tag{152} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ktrsnhxv.png}\tag{153}\end{figure}\] As we can see in 152 , the crystals that correspond to the quiver \({\cal Q}_{n,p,\lambda}\) contain \(p(n - p)\) different types of paths. As in 97 , we can lift the picture to the whole R-equivariant space, where we can see the fields \(C_{i}\). The parameter \(\lambda\) becomes the maximal allowed height.

Gelfand-Tsetlin Bases↩︎

In this paragraph we lift the correspondence between the crystals and the Gelfand-Tsetlin bases to the representation \(\Upsilon_{p,\lambda}\).

We highlight the main ideas of the construction starting with the example \({\cal Q}_{5,2,\lambda}\). We have established earlier that the empty box for the corresponding representation takes the form 154 . \[\tag{154} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/yhlfpwtn.png}\tag{155}\end{figure} \qquad \Rightarrow \qquad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/szmiatuy.png}\tag{156}\end{figure}\] The vector spaces \(V_{1}, \dots, V_{4}\) in this case can be represented as: \[\label{sl4054132vector32spaces} \begin{align} &V_{1} = \nu_{1, 1} = {\rm Span}\left\{B_{1}C_{2}^{a}\cdot R_{2}\right\}\,,\\ &V_{2} = \nu_{1, 2} \oplus \nu_{2, 2} = {\rm Span}\left\{C_{2}^{a}\cdot R_{2}\,,~~ A_{1} B_{1} C_{2}^{a}\cdot R_{2} \right\}\,,\\ &V_{3} = \nu_{2, 3} \oplus \nu_{3, 3} = {\rm Span}\left\{A_{2}C_{2}^{a}\cdot R_{2}\,,~~ B_{3}A_{3}A_{2}C_{2}^{a}\cdot R_{2}\right\}\,,\\ &V_{4} = \nu_{3, 4} = {\rm Span}\left\{A_{3}A_{2}C_{2}^{a}\cdot R_{2}\right\}\,, \end{align}\tag{157}\] where the numeration of the subspaces is chosen to match the corresponding elements of the Gelfand-Tsetlin patterns. For example, the second number labels a node: \[V_{\color{blue} a} = \bigoplus \nu_{\bullet , \color{blue} a}\,.\] The first number runs through the different types of vectors that live in the corresponding vector space. In order to clarify the range of this parameter, we construct the GT bases next. The result is presented in 158 . \[\tag{158} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/xcazorlg.png}\tag{159}\end{figure}\] At the top we place the partition that corresponds to the Young diagram that labels the representation \(\Upsilon_{2, \lambda}\): \[\Upsilon_{2, \lambda} \quad \leftrightarrow \quad (0, \lambda, 0, 0) \quad \leftrightarrow \quad [\lambda, \lambda, \lambda, 0, 0]\,.\] The parameters \(m_{i, k}\) are the dimensions of the vector spaces 157 : \[m_{i, k} = \dim \nu_{i, k}\,.\] The pattern is organized to satisfy the triangular inequalities: \[\label{sl4054132triangular32inequalities} \begin{align} &\lambda \geqslant m_{3, 4} \geqslant 0\,, \quad m_{1, 2} \geqslant m_{1, 1} \geqslant m_{2,2}\,,\\ &\lambda \geqslant m_{1, 2} \geqslant m_{2, 3}\,, \quad m_{2, 3} \geqslant m_{2, 2} \geqslant m_{3, 3}\,,\\ &\lambda \geqslant m_{2, 3} \geqslant m_{3, 4}\,, \quad m_{3, 4} \geqslant m_{3, 3} \geqslant 0\,. \end{align}\tag{160}\] These inequalities define the structure of the GT bases, and from the crystal point of view, they are a direct consequence of the F-relations 138 , which we have seen in the example before in section 3.2.

Let us prove it in the case \({\cal Q}_{5,2,\lambda}\) as well. In order to get a visualization of a GT base, one could rotate the crystal: \[\tag{161} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/mftkcsba.png}\tag{162}\end{figure} \qquad \leftrightarrow \qquad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/vhkazsmg.png}\tag{163}\end{figure}\] The picture 161 is in agreement with the decomposition of the vector spaces 157 . We additionally highlight the dimensions of these vector spaces¹³.

We claim that there are only two types of inequalities we need to prove, namely \(m_{1, 2} \geqslant m_{2, 3}\) and \(m_{1, 2}\geqslant m_{1, 1}\). Indeed, every other inequality is of one of two types by construction.

Let us start with the first one, \(m_{1, 2} \geqslant m_{2, 3}\). We assume the contrary, which means that there is a natural number \(k\), such that: \[C_{2}^{k}\cdot R_{2} = 0\,, \qquad C_{3}^{k}A_{2}\cdot R_{2}\ne 0\,.\] However, applying the F-terms 138 , we get: \[0 \ne C_{3}^{k}A_{2}\cdot R_{2} = A_{2}C_{2}^{k}\cdot R_{2} = 0\,,\] which contradicts our initial suggestion. Therefore, \(m_{1, 2} \geqslant m_{2, 3}\).

Next, we prove the second inequality, \(m_{1, 2}\geqslant m_{1, 1}\). In a similar way, we assume the contrary. Therefore, there is a number \(k\) such that: \[C_{2}^{k}\cdot R_{2} = 0\,, \qquad C_{1}^{k}B_{1}\cdot R_{2}\ne 0\,.\] Again, we apply the F-terms and end up with the contradiction: \[0 \ne C_{1}^{k}B_{1}\cdot R_{2} = B_{1}C_{2}^{k}\cdot R_{2} = 0\,,\] which verifies that \(m_{1, 2}\geqslant m_{1, 1}\), and also concludes the proof that the patterns 158 are indeed the GT bases.

Now, we are ready to present the general construction of the GT bases for \({\cal Q}_{n,p,\lambda}\).

• We start with the partition: \[(0, \dots, 0, \underset{p}{\lambda}, 0, \dots, 0) \quad \leftrightarrow \quad [m_{1, n}, \dots, m_{i, n-p}, \dots, m_{n, n}]\,,\] where \(m_{i,n} = \lambda\) if \(i \leqslant n - p\) and \(m_{i, n} = 0\) otherwise. The partition is fixed by the representation.

• We decompose the vector spaces of the quiver nodes into direct sums of linear subspaces: \[\label{sl40n4132vector32space32decomposition} V_{k} = \bigoplus_{i}\nu_{i, k}\,,\tag{164}\] and assign to the subspaces their corresponding dimensions: \[m_{i, k} = \dim \nu_{i, k}\,.\]

• We form a triangular pattern using these dimensions that reads: \[\tag{165} \mu =\quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/daqywboi.png}\tag{166}\end{figure}\,.\] We impose the triangular inequalities on the fixed numbers \(\lambda,\,0\) in advance. Therefore, the partition cuts off a rectangle of free parameters similar to the case 161 . The shape resembles 152 .

• The numbers \(m_{i, j}\) are dimension parameters. By construction, they satisfy one of two types of inequalities: \[\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/jhxuargi.png}\tag{167}\end{figure} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/cvqdkxur.png}\tag{168}\end{figure}\] These inequalities are, in fact, the triangular inequalities: \[\label{sl40n4132triangual32inequalities} m_{i, j + 1} \geqslant m_{i, j} \geqslant m_{i + 1, j + 1}\,,\tag{169}\] which are a well-known property of the GT bases.

• The decompositions 164 can be rewritten in terms of the parameters \(m_{i, j}\) as follows: \[\dim V_{k} = n_{k} = \sum_{i = a[k]}^{b[k]}m_{i, k}\,,\] where the numbers \(a[k],\, b[k]\) can be determined by the structure of 165 . In the representation \(\Upsilon_{p, \lambda}\) they take the form: \[a[k] = \max\{1, k - p + 1\}\,, \quad b[k] = \min\{n-p, k\}\,.\]

The Representation↩︎

We proceed now to describe the representation \(\Upsilon_{p, \lambda}\) of the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\). Again, we adjust the ansatz 22 . The states of the representation are parameterized by the GT bases. As in the example presented in section 3.2, we denote by \(\mu_{k, k + 1}^{i}\) the pattern obtained from \(\mu\) by replacing \(m_{i, k}\) with \(m_{i, k} + 1\). And by \(\mu_{k, k - 1}^{i}\) we denote the pattern where \(m_{i, k}\) is replaced with \(m_{i, k} - 1\).

Implementing the facts above, we end up with the ansatz of the following form: \[\label{sl40n4132ansatz} \begin{align} &e^{(k)}_{\Upsilon_{p,\lambda}}(z)|\mu\rangle = \sum_{i = a[k]}^{b[k]}\dfrac{{\boldsymbol{E}}_{\Upsilon_{p, \lambda}}[\mu \to \mu_{k, k + 1}^{i}]}{z - m_{i, k}\epsilon + (i - a[k])\epsilon + |k - p|\frac{\epsilon}{2}}|\mu_{k, k + 1}^{i}\rangle\,,\\ &f^{(k)}_{\Upsilon_{p, \lambda}}(z)|\mu\rangle = \sum_{i = a[k]}^{b[k]}\dfrac{{\boldsymbol{F}}_{\Upsilon_{p, \lambda}}[\mu \to \mu_{k, k - 1}^{i}]}{z - (m_{i, k} - 1)\epsilon + (i - a[k])\epsilon + |k - p|\frac{\epsilon}{2}}|\mu_{k, k - 1}^{i}\rangle\,,\\ &\psi^{(k)}_{\Upsilon_{p, \lambda}}(z)|\mu\rangle = \Psi^{(k)}_{\mu, \Upsilon_{p, \lambda}}(z)|\mu\rangle\,, \end{align}\tag{170}\] where the eigenfunctions are defined by 23 . The explicit formulas for the \(\Upsilon_{p,\lambda}\) are presented below. The functions on the edges read as follows: \[\label{sl40n4132eigenfunctions32on32the32edge} \begin{align} \Psi^{(1)}_{\mu, \Upsilon_{p, \lambda}}(z) = &-\dfrac{1}{\epsilon}\prod_{l = 1}^{m_{1, 1}}\varphi_{1, 1}\Bigl(z - (l - 1)\epsilon + (i - a[1])\epsilon + |1 - p|\frac{\epsilon}{2}\Bigr)\,\times\\ &\times\,\prod_{i = a[2]}^{b[2]}\prod_{l = 1}^{m_{i, 2}}\varphi_{1, 2}\Bigl(z - (l - 1)\epsilon + (i - a[2])\epsilon + |2 - p|\frac{\epsilon}{2}\Bigr) =\\ &= -\frac{1}{\epsilon}\frac{\Bigl(z - m_{1, 2}\epsilon + |1 - p|\frac{\epsilon}{2}\Bigr)\Bigl(z - (m_{2, 2} - 1)\epsilon + |1 - p|\frac{\epsilon}{2}\Bigr)}{\Bigl(z - m_{1, 1}\epsilon + |1 - p|\frac{\epsilon}{2}\Bigr)\Bigl(z - (m_{1, 1} - 1)\epsilon + |1 - p|\frac{\epsilon}{2}\Bigr)}\,,\\ \Psi_{\mu, \Upsilon_{p,\lambda}}^{(n - 1)} = &-\frac{1}{\epsilon}\prod_{i = a[n - 1]}^{b[n - 1]}\prod_{l = 1}^{m_{i, n - 1}}\varphi_{n - 1, n - 1}\Bigl(z - (l - 1)\epsilon + (i - a[n - 1])\epsilon + |n - 1 - p|\frac{\epsilon}{2}\Bigr)\,\times\\ &\times\, \prod_{i = a[n - 2]}^{b[n - 2]}\prod_{l = 1}^{m_{i, n - 2}}\varphi_{n - 1, n - 2}\Bigl(z - (l - 1)\epsilon + (i - a[n - 2])\epsilon + |n - 2 - p|\frac{\epsilon}{2}\Bigr) = \\ &= -\frac{1}{\epsilon}\frac{\Bigl(z - m_{n - p - 1, n - 2}\epsilon + |n - 1 - p|\frac{\epsilon}{2}\Bigr)\Bigl(z - (m_{n - p, n - 2} - 1)\epsilon + |n - 1 - p|\frac{\epsilon}{2}\Bigr)}{\Bigl(z - m_{n - p, n - 1}\epsilon + |n - 1 - p|\frac{\epsilon}{2}\Bigr)\Bigl(z - m_{n - p, n - 1}\epsilon + |n - 1 - p|\frac{\epsilon}{2}\Bigr)}\,, \end{align}\tag{171}\] where we assumed that \(p \ne 1\). If \(p = 1\), the eigenfunction \(\Psi_{\mu, \Upsilon_{p,\lambda}}^{(1)}(z)\) includes the additional factor: \[\label{sl40n4132first32eigenfunction32when32p3261321} \begin{align} &\Psi^{(1)}_{\mu, \Upsilon_{1, \lambda}}(z) \to \frac{z - \lambda \epsilon}{z}\cdot\Psi^{(1)}_{\mu, \Upsilon_{p, \lambda}}(z)\,,\\ &\Psi^{(1)}_{\mu, \Upsilon_{1, \lambda}}(z) = -\frac{1}{\epsilon}\frac{\bigl(z - \lambda\epsilon\bigr)\bigl(z - (m_{2, 2} - 1)\epsilon\bigr)}{\bigl(z - m_{1, 1}\epsilon\bigr)\bigl(z - (m_{1, 1} - 1)\epsilon\bigr)}\,, \end{align}\tag{172}\] which in practice substitutes \(z\) by \(z - \lambda\epsilon\).

The remaining eigenfunctions for \(k \ne p\,, n-p\) read as follows: \[\label{sl40n4132eigenfunctions} \begin{align} \Psi_{\mu, \Upsilon_{p,\lambda}}^{(k)} = &-\frac{1}{\epsilon}\prod_{i = a[k]}^{b[k]}\prod_{l = 1}^{m_{i, k}}\varphi_{k,k}\Bigl(z - (l - 1)\epsilon + (i - a[k])\epsilon + |k - p|\frac{\epsilon}{2}\Bigr)\,\times\\ &\times\, \prod_{i = a[k + 1]}^{b[k + 1]}\prod_{l = 1}^{m_{i, k + 1}}\varphi_{k, k + 1}\Bigl(z - (l - 1)\epsilon + (i - a[k + 1])\epsilon + |k + 1 - p|\frac{\epsilon}{2}\Bigr)\,\times\\ &\prod_{i = a[k - 1]}^{b[k - 1]}\prod_{l = 1}^{m_{i, k - 1}}\varphi_{k, k - 1}\Bigl(z - (l - 1)\epsilon + (i - a[k - 1])\epsilon + |k - 1 - p|\frac{\epsilon}{2}\Bigr) =\\ &= -\dfrac{1}{\epsilon}\prod_{i = a[k]}^{b[k]}\frac{1}{\Bigl(z - m_{i, k}\epsilon + (i - a[k])\epsilon + |k - p|\frac{\epsilon}{2}\Bigr)\Bigl(z - (m_{i,k} - 1)\epsilon + (i - a[k])\epsilon + |k - p|\frac{\epsilon}{2}\Bigr)}\,\times\\ &\times\prod_{i = a[k - 1]}^{b[k - 1]}\Bigl(z - m_{i, k - 1}\epsilon + (i - a[k - 1])\epsilon + |k - p|\frac{\epsilon}{2}\Bigr)\prod_{i = a[k + 1]}^{b[k + 1]}\Bigl(z - (m_{i, k + 1} - 1)\epsilon + (i - a[k + 1])\epsilon + |k - p|\frac{\epsilon}{2}\Bigr)\,. \end{align}\tag{173}\]

Finally, we present the explicit form of eigenfunctions for \(k = p\) and \(k = n - p\): \[\label{sl40n4132eigenfunctions32k326132p} \begin{align} \Psi_{\mu, \Upsilon_{p,\lambda}}^{(p)} = &-\frac{1}{\epsilon}(z - \lambda\epsilon)\prod_{i = 1}^{b[p]}\frac{1}{\Bigl(z - m_{i, p}\epsilon + (i - 1)\epsilon\Bigr)\Bigl(z - (m_{i,p} - 1)\epsilon + (i - 1)\epsilon\Bigr)}\,\times\\ &\times\prod_{i = a[p - 1]}^{b[p - 1]}\Bigl(z - m_{i, p - 1}\epsilon + (i - a[p - 1])\epsilon\Bigr)\prod_{i = a[p + 1]}^{b[p + 1]}\Bigl(z - (m_{i, p + 1} - 1)\epsilon + (i - a[p + 1])\epsilon\Bigr)\,,\\ \Psi_{\mu, \Upsilon_{p,\lambda}}^{(n - p)} = &-\frac{1}{\epsilon}\Bigl(z - n\frac{\epsilon}{2}\Bigr)\prod_{i = 1}^{b[p]}\frac{1}{\Bigl(z - m_{i, p}\epsilon + (i - 1)\epsilon\Bigr)\Bigl(z - (m_{i,p} - 1)\epsilon + (i - 1)\epsilon\Bigr)}\,\times\\ &\times\prod_{i = a[p - 1]}^{b[p - 1]}\Bigl(z - m_{i, p - 1}\epsilon + (i - a[p - 1])\epsilon\Bigr)\prod_{i = a[p + 1]}^{b[p + 1]}\Bigl(z - (m_{i, p + 1} - 1)\epsilon + (i - a[p + 1])\epsilon\Bigr)\,. \end{align}\tag{174}\] If we have the case when \(p = n - p\), then: \[\begin{align} \Psi_{\mu, \Upsilon_{p,\lambda}}^{(p)} = &-\frac{1}{\epsilon}\bigl(z - \lambda\epsilon\bigr)\Bigl(z - n\frac{\epsilon}{2}\Bigr)\prod_{i = 1}^{b[p]}\frac{1}{\Bigl(z - m_{i, p}\epsilon + (i - 1)\epsilon\Bigr)\Bigl(z - (m_{i,p} - 1)\epsilon + (i - 1)\epsilon\Bigr)}\,\times\\ &\times\prod_{i = a[p - 1]}^{b[p - 1]}\Bigl(z - m_{i, p - 1}\epsilon + (i - a[p - 1])\epsilon\Bigr)\prod_{i = a[p + 1]}^{b[p + 1]}\Bigl(z - (m_{i, p + 1} - 1)\epsilon + (i - a[p + 1])\epsilon\Bigr)\,. \end{align}\]

Amplitudes↩︎

Our next step is to apply equivariant techniques and get the matrix coefficients of the raising and lowering operators in the ansatz 170 . In order to evaluate the corresponding Euler classes, we need to construct the vacuum expectation values of the fields in a fixed point \(\mu\).

We start with the fields \(C_{k}\). These matrices act from \(V_{k}\) to \(V_{k}\). As we have mentioned, the vector space \(V_{k}\) can be decomposed as follows: \[\label{sl40n4132V95k32decomposition} V_{k} = \bigoplus_{i = a[k]}^{b[k]}\nu_{i, k}\,.\tag{175}\] This allows us to determine the general structure of \(C_{k}\) as block matrices. The corresponding vacuum expectation values read: \[\begin{align} &\overline{C}_{1} = C(m_{1, 1}) = C(n_{1})\,, \\ &\overline{C}_{k} = \left(\begin{array}{c|c|c|c|c} \mathbf{C(m_{a[k], k})} & O(m_{a[k], k}, m_{a[k] + 1, k}) & \cdots & O(m_{a[k], k}, m_{b[k] - 1, k}) & O(m_{a[k], k}, m_{b[k], k})\\ \hline O(m_{a[k] + 1, k}, m_{a[k], k}) & \mathbf{C(m_{a[k] + 1, k})} & \cdots & O(m_{a[k] + 1, k}, m_{b[k] - 1, k}) & O(m_{a[k] + 1, k}, m_{b[k], k}) \\ \hline \vdots & \vdots & \ddots & \vdots & \vdots \\ \hline O(m_{b[k] - 1, k}, m_{a[k], k}) & O(m_{b[k]- 1, k}, m_{a[k] + 1, k}) & \cdots & \mathbf{C(m_{b[k] - 1, k})} & O(m_{b[k] - 1, k}, m_{b[k], k})\\ \hline O(m_{b[k], k}, m_{a[k], k}) & O(m_{b[k], k}, m_{a[k] + 1, k}) & \cdots & O(m_{b[k] - 1, k}, m_{b[k], k}) & \mathbf{C(m_{b[k], k})} \end{array}\right)\,. \end{align}\]

In order to construct the matrices \(\overline{A}_{k}\) and \(\overline{B}_{k}\), we need to split the region \(k \in [1, n-1]\) into three pieces. They correspond to the three different regions in 152 .

1. The First Case \(k \leqslant p - 1\): This area corresponds to the left triangle in 152 . The ranges read: \[\begin{align} &a[k] = 1\,, \quad b[k] = k\,,\\ &a[k + 1] = 1 \quad b[k + 1] = k + 1\,. \end{align}\] In this case the fields acquire the following vacuum expectation values: \[\begin{align} &\overline{A}_{k} = \left(\begin{array}{c|c|c|c|c} O(m_{1, k + 1}, m_{1, k}) & O(m_{1, k+1}, m_{2, k}) & \cdots & O(m_{1, k + 1}, m_{k - 1, k}) & O(m_{1, k + 1}, m_{k, k}) \\ \hline \mathbf{I(m_{2, k + 1}, m_{1, k})} & O(m_{2, k + 1}, m_{2, k}) & \cdots & O(m_{2, k + 1}, m_{k - 1, k}) & O(m_{2, k + 1}, m_{k, k}) \\ \hline \vdots & \ddots & \ddots & \vdots & \vdots \\ \hline O(m_{k -1, k + 1}, m_{1, k}) & O(m_{k - 1, k + 1}, m_{2, k}) & \cdots & O(m_{k - 1, k + 1}, m_{k - 1, k}) & O(m_{k - 1, k + 1}, m_{k, k}) \\ \hline O(m_{k, k + 1}, m_{1, k}) & O(m_{k, k + 1}, m_{2, k}) & \cdots & \mathbf{I(m_{k, k + 1}, m_{k - 1, k})} & O(m_{k, k + 1}, m_{k, k})\\ \hline O(m_{k + 1, k + 1}, m_{1, k}) & O(m_{k + 1, k + 1}, m_{2, k}) & \cdots & O(m_{k + 1, k + 1}, m_{k-1, k}) & \mathbf{I(m_{k + 1, k + 1}, m_{k, k})} \end{array}\right)\,,\\ \\ &\overline{B}_{k} = \left(\begin{array}{c|c|c|c|c} \mathbf{I(m_{1, k}, m_{1, k + 1})} & O(m_{1, k}, m_{2, k + 1}) & \cdots & O(m_{1, k}, m_{k, k + 1}) & O(m_{1, k}, m_{k + 1, k + 1})\\ \hline O(m_{2, k}, m_{1, k + 1}) & \mathbf{I(m_{2, k}, m_{2, k + 1})} & \cdots & O(m_{2, k}, m_{k, k + 1}) & O(m_{2, k}, m_{k + 1, k + 1})\\ \hline \vdots & \ddots & \ddots & \vdots & \vdots \\ \hline O(m_{k, k}, m_{1, k + 1}) & O(m_{k, k}, m_{2, k + 1}) & \cdots & \mathbf{I(m_{k, k}, m_{k, k + 1})} & O(m_{k, k}, m_{k + 1, k + 1}) \end{array}\right)\,. \end{align}\]

2. The Second Case \(p \leqslant k \leqslant n - p - 1\): This area corresponds to the parallelogram in the center. The ranges read: \[\begin{align} &a[k] = k - p + 1\,, \quad b[k] = k\,,\\ &a[k + 1] = k - p + 2\,, \quad b[k + 1] = k + 1\,. \end{align}\] We can notice that the number of the vertical blocks is equal to the number of the horizontal ones: \[b[k] - a[k] + 1 = b[k + 1] - a[k + 1] + 1 = p\] It means that the blocks of the matrices \(\overline{A}_{k}\) and \(\overline{B}_{k}\) form a “square" matrix themselves. Therefore, the vacuum expectation values of the fields read as follows: \[{!}{ \begin{align} &\overline{A}_{k} = \left(\begin{array}{c|c|c|c} \mathbf{I(m_{a[k + 1], k + 1}, m_{a[k], k})} & O(m_{a[k + 1], k + 1}, m_{a[k] + 1, k}) & \cdots & O(m_{a[k + 1], k + 1}, m_{k, k})\\ \hline O(m_{a[k + 1] + 1, k + 1}, m_{a[k], k}) & \mathbf{I(m_{a[k + 1] + 1, k + 1}, m_{a[k] + 1, k})} & \cdots & O(m_{a[k + 1] + 1, k + 1}, m_{k, k}) \\ \hline \vdots & \vdots & \ddots & \vdots \\ \hline O(m_{k + 1, k + 1}, m_{a[k], k}) & O(m_{k + 1, k + 1}, m_{a[k] + 1, k}) & \cdots & \mathbf{I(m_{k + 1, k + 1}, m_{k, k})} \end{array}\right)\,,\\ \\ &\overline{B}_{k} = \left(\begin{array}{c|c|c|c|c} O(m_{a[k], k}, m_{a[k + 1], k + 1}) & O(m_{a[k], k}, m_{a[k + 1] + 1, k + 1}) & \cdots & O(m_{a[k], k}, m_{k, k + 1}) & O(m_{a[k], k}, m_{k + 1, k + 1}) \\ \hline \mathbf{I(m_{a[k] + 1, k}, m_{a[k + 1], k + 1})} & O(m_{a[k] + 1, k}, m_{a[k + 1] +1, k + 1}) & \cdots & O(m_{a[k] + 1, k}, m_{k, k + 1}) & O(m_{a[k] + 1, k}, m_{k + 1, k + 1}) \\ \hline \vdots & \ddots & \ddots & \vdots & \vdots \\ \hline O(m_{k - 1, k}, m_{a[k + 1], k + 1}) & O(m_{k - 1, k}, m_{a[k + 1] + 1, k + 1}) & \cdots & O(m_{k - 1, k}, m_{k, k + 1}) & O(m_{k - 1, k}, m_{k + 1, k + 1}) \\ \hline O(m_{k, k}, m_{a[k + 1], k + 1}) & O(m_{k, k}, m_{a[k + 1] + 1, k + 1}) & \cdots & \mathbf{I(m_{k, k}, m_{k - 1, k})} & O(m_{k, k}, m_{k + 1, k + 1})\\ \end{array}\right)\,. \end{align} }\]

3. The Third Case \(k \geqslant n - p\): This is the final area in 152 . The ranges read: \[\begin{align} &a[k] = k - p + 1, \quad b[k] = n - p = \tilde{p},\\ &a[k + 1] = k - p + 2 \quad b[k + 1] = n - p = \tilde{p}. \end{align}\] This sector is connected to the first sector via quiver symmetry 42 . This means that we can get the vacuum expectation values by interchanging the fields \(A_{k}\) and \(B_{k}\) from the first area and applying the corresponding change of the dimension parameters \(m_{i, k}\).

   Finally, we end up with the result: \[{!}{ \begin{align} &\overline{A}_{k} = \left(\begin{array}{c|c|c|c|c} \mathbf{I(m_{a[k + 1], k + 1}, m_{a[k], k})} & O(m_{a[k + 1], k + 1}, m_{a[k] + 1, k}) & \cdots & O(m_{a[k + 1], k + 1}, m_{\tilde{p} - 1, k}) & O(m_{a[k + 1], k + 1}, m_{\tilde{p}, k})\\ \hline O(m_{a[k + 1] + 1, k + 1}, m_{a[k], k}) & \mathbf{I(m_{a[k + 1] + 1, k + 1}, m_{a[k] + 1, k})} & \cdots & O(m_{a[k + 1] + 1, k + 1}, m_{\tilde{p} - 1, k}) & O(m_{a[k + 1] + 1, k + 1}, m_{\tilde{p}, k})\\ \hline \vdots & \ddots & \ddots & \vdots & \vdots \\ \hline O(m_{\tilde{p}, k + 1}, m_{a[k], k}) & O(m_{\tilde{p}, k + 1}, m_{a[k] + 1, k}) & \cdots & \mathbf{I(m_{\tilde{p}, k + 1}, m_{\tilde{p} - 1, k})} & O(m_{\tilde{p}, k + 1}, m_{\tilde{p}, k}) \end{array}\right)\,,\\ \\ &\overline{B}_{k} = \left(\begin{array}{c|c|c|c|c} O(m_{a[k], k}, m_{a[k + 1], k + 1}) & O(m_{a[k], k}, m_{a[k + 1] + 1, k + 1}) & \cdots & O(m_{a[k], k}, m_{\tilde{p} - 1, k + 1}) & O(m_{a[k], k}, m_{\tilde{p}, k + 1}) \\ \hline \mathbf{I(m_{a[k] + 1, k}, m_{a[k + 1], k + 1})} & O(m_{a[k] + 1, k}, m_{a[k + 1] + 1, k + 1}) & \cdots & O(m_{a[k] + 1, k}, m_{\tilde{p} - 1, k + 1}) & O(m_{a[k] + 1, k}, m_{\tilde{p}, k + 1}) \\ \hline \vdots & \ddots & \ddots & \vdots & \vdots \\ \hline O(m_{\tilde{p} - 2, k}, m_{a[k + 1], k + 1}) & O(m_{\tilde{p} - 2, k}, m_{a[k + 1] + 1, k + 1}) & \cdots & O(m_{\tilde{p} - 2, k}, m_{\tilde{p} - 1, k + 1}) & O(m_{\tilde{p} - 2, k}, m_{\tilde{p}, k + 1}) \\ \hline O(m_{\tilde{p} - 1, k}, m_{a[k + 1], k + 1}) & O(m_{\tilde{p} - 1, k}, m_{a[k + 1] + 1, k + 1}) & \cdots & \mathbf{I(m_{\tilde{p} - 1, k}, m_{\tilde{p} - 1, k + 1})} & O(m_{\tilde{p} - 1, k}, m_{\tilde{p}, k + 1})\\ \hline O(m_{\tilde{p}, k}, m_{a[k + 1], k + 1}) & O(m_{\tilde{p}, k}, m_{a[k + 1] + 1, k + 1}) & \cdots & O(m_{\tilde{p}, k}, m_{\tilde{p} - 1, k + 1}) & \mathbf{I(m_{\tilde{p}, k}, m_{\tilde{p}, k + 1})} \end{array}\right)\,. \end{align} }\]

Next, we apply the algorithm described in section 2.6 using the vacuum expectation values of the fields at the fixed points defined above.

First, we point out that: \[{\boldsymbol{E}}_{\Upsilon_{p,\lambda}}[\mu \to \mu_{p, p + 1}^{1}] = -\frac{1}{\epsilon}\,,\] which is a result of the choice of the normalization that is provided by equivariant integration. For later convenience we introduce shifted dimensions following [44]: \[l_{i, k} = m_{i, k} - i\,.\] Then, after calculating the Euler classes, we end up with the following matrix coefficients for \(k \ne p\): \[\label{sl40n4132equiv32coefficients32k32336132p} \begin{align} &{\boldsymbol{E}}_{\Upsilon_{p,\lambda}}[\mu \to \mu_{k, k + 1}^{j}] = \frac{\prod\limits_{i = 1}^{j}(l_{i, k + 1} - l_{j, k})\prod\limits_{i = 1}^{j - 1}(l_{i, k - 1} - l_{j, k} - 1)}{\prod\limits_{i = 1}^{j - 1}(l_{i, k} - l_{j, k})(l_{i,k} - l_{j, k} - 1)}\frac{1}{l_{j, k}\epsilon + a[k]\epsilon - |k - p|\frac{\epsilon}{2}}\,,\\ &{\boldsymbol{F}}_{\Upsilon_{p,\lambda}}[\mu \to \mu_{k, k - 1}^{j}] = \frac{(-1)\prod\limits_{i = j + 1}^{k + 1}(l_{i, k + 1} - l_{j, k} + 1)\prod\limits_{i = j}^{k - 1}(l_{i, k - 1} - l_{j, k})}{\prod\limits_{i = j + 1}^{k}(l_{i, k} - l_{j, k} + 1)(l_{i, k} - l_{j, k})}\Bigl((l_{j, k} - 1)\epsilon + a[k]\epsilon - |k - p|\frac{\epsilon}{2}\Bigr)\,. \end{align}\tag{176}\] For the case \(k = p\) we have: \[\label{sl40n4132equiv32coefficients32k326132p} \begin{align} &{\boldsymbol{E}}_{\Upsilon_{p, \lambda}}[\mu \to \mu_{p, p + 1}^{j}] = -\frac{\prod\limits_{i = 2}^{j}(l_{i, p + 1} - l_{j, p})\prod\limits_{i = 1}^{j - 1}(l_{i, p - 1} - l_{j, p} - 1)}{\prod\limits_{i = 1}^{j - 1}(l_{i, p} - l_{j, p})(l_{i, p} - l_{j, p} - 1)}\frac{1}{\epsilon}\,,\\ &{\boldsymbol{F}}_{\Upsilon_{p, \lambda}}[\mu \to \mu_{p, p - 1}^{j}] = \frac{(l_{1, p + 1} - l_{j, p})\prod\limits_{i = j + 1}^{p + 1}(l_{i, p + 1} - l_{j, p} + 1)\prod\limits_{i = j}^{p - 1}(l_{i, p - 1} - l_{j, p})}{\prod\limits_{i = j + 1}^{p}(l_{i, p} - l_{j, p} + 1)(l_{i, p} - l_{j, p})}\epsilon\,. \end{align}\tag{177}\]

Again, we can normalize the coefficients using the formula 30 . At the zero level of the algebra we get: \[\begin{align} &e^{(k)}_{0}|\mu\rangle = \sum_{j = a[k]}^{b[k]}{\boldsymbol{E}}^{(root)}_{\Upsilon_{p,\lambda}}[\mu \to \mu_{k, k + 1}^{j}]\,|\mu_{k, k + 1}^{j}\rangle = \sum_{j = a[k]}^{b[k]}a_{k, k + 1}^{j}\,|\mu_{k, k + 1}^{j}\rangle\,,\\ &f^{(k)}_{0}|\mu\rangle = \sum_{j = a[k]}^{b[k]}{\boldsymbol{F}}^{(root)}_{\Upsilon_{p, \lambda}}[\mu \to \mu_{k, k - 1}^{j}]\,|\mu_{k, k - 1}^{j}\rangle = \sum_{j = a[k]}^{b[k]}b_{k, k - 1}^{j}\,|\mu_{k, k - 1}^{j}\rangle\,, \end{align}\] where: \[\label{gelfand-coefficients} \begin{align} &a_{k, k + 1}^{j} = \left[(-1)\frac{\prod\limits_{i = 1}^{k+1}(l_{i, k+1} - l_{j, k})\prod\limits_{i = 1}^{k-1}(l_{i, k-1} - l_{j, k} - 1)}{\prod\limits_{i\ne j}^{k}(l_{i, k} - l_{j, k})(l_{i, k} - l_{j, k} - 1)}\right]^{\frac{1}{2}}\,, \\ &b_{k, k - 1}^{j} = \left[(-1)\frac{\prod\limits_{i = 1}^{k + 1}(l_{i, k + 1} - l_{j, k} + 1)\prod\limits_{i = 1}^{k-1}(l_{i, k-1} - l_{j, k})}{\prod\limits_{i\ne j}^{k}(l_{i, k} - l_{j, k} + 1)(l_{i, k} - l_{j, k})}\right]^{\frac{1}{2}}\,. \end{align}\tag{178}\] These formulas are precisely the formulas introduced by Gelfand in [44] for the representations of \(\mathfrak{s}\mathfrak{l}_{n}\) algebras.

We still need to check the hysteresis relations 24 to prove that the coefficients 176 and 177 satisfy the Yangian algebraic relations 134 . Empirically, these relations are satisfied in concrete examples for low \(n\) and take the form: \[\label{sl40n4132hysteresis} \begin{align} {\boldsymbol{E}}_{\Upsilon_{p,\lambda}}[\mu \to \mu_{k, k + 1}^{j}]{\boldsymbol{F}}_{\Upsilon_{p,\lambda}}[\mu_{k, k + 1}^{j} \to (\mu_{k, k + 1}^{j})^{j}_{k, k - 1}] &= (-1)\frac{\prod\limits_{i = 1}^{k+1}(l_{i, k+1} - l_{j, k})\prod\limits_{i = 1}^{k-1}(l_{i, k-1} - l_{j, k} - 1)}{\prod\limits_{i\ne j}^{k}(l_{i, k} - l_{j, k})(l_{i, k} - l_{j, k} - 1)} = \\ &= \mathop{\rm res}_{z = l_{j, k}\epsilon + a[k]\epsilon - |k - p|\frac{\epsilon}{2}}\Psi_{\mu, \Upsilon_{p,\lambda}}^{(k)}(z)\,,\\ \frac{{\boldsymbol{E}}_{\Upsilon_{p, \lambda}}\bigl[\mu \to \mu_{k, k + 1}^{i}\bigr]{\boldsymbol{E}}_{\Upsilon_{p,\lambda}}\bigl[\mu_{k, k + 1}^{i} \to (\mu_{k, k + 1}^{i})_{k, k + 1}^{j}\bigr]}{{\boldsymbol{E}}_{\Upsilon_{p,\lambda}}\bigl[\mu \to \mu_{k, k + 1}^{j}\bigr]{\boldsymbol{E}}_{\Upsilon_{p, \lambda}}\bigl[\mu_{k, k + 1}^{j} \to (\mu_{k,k + 1}^{j})_{k, k + 1}^{i}\bigr]} &= \frac{l_{i, k} - l_{j, k} + 1}{l_{i, k} - l_{j, k} - 1} = \\ &= \varphi_{k, k}\Bigl(l_{j, k}\epsilon + a[k]\epsilon - |k - p|\frac{\epsilon}{2} - l_{i, k}\epsilon - a[k]\epsilon + |k - p|\frac{\epsilon}{2}\Bigr)\,,\\ \frac{{\boldsymbol{E}}_{\Upsilon_{p, \lambda}}\bigl[\mu \to \mu_{k, k + 1}^{i}\bigr]{\boldsymbol{E}}_{\Upsilon_{p,\lambda}}\bigl[\mu_{k, k + 1}^{i} \to (\mu_{k, k + 1}^{i})_{t, t + 1}^{j}\bigr]}{{\boldsymbol{E}}_{\Upsilon_{p,\lambda}}\bigl[\mu \to \mu_{t, t + 1}^{j}\bigr]{\boldsymbol{E}}_{\Upsilon_{p, \lambda}}\bigl[\mu_{t, t + 1}^{j} \to (\mu_{t, t + 1}^{j})_{k, k + 1}^{i}\bigr]} &= \frac{l_{j, t} + a[t] - |t - p|\frac{1}{2} - l_{i, k} - a[k] + |k - p|\frac{1}{2} + \frac{1}{2}}{l_{j, t} + a[t] - |t - p|\frac{1}{2} - l_{i, k} - a[k] + |k - p|\frac{1}{2} - \frac{1}{2}} = \\ &= \varphi_{t, k}\Bigl(l_{j, t}\epsilon + a[t]\epsilon - |t - p|\frac{\epsilon}{2} - l_{i, k}\epsilon - a[k]\epsilon + |k - p|\frac{\epsilon}{2}\Bigr)\,,\\ \end{align}\tag{179}\] where \(k,\,t\in\, Q_{0}\), \(k \ne t\). The relations involving \({\boldsymbol{F}}_{\Upsilon_{p,\lambda}}[\mu \to \mu_{k, k - 1}^{j}]\) take a similar form; therefore, we do not present them here.

3.4 General Comments↩︎

In this section, we give a few closing comments regarding the algebras \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) and their representations.

3.4.1 Connection to Drinfeld Yangians↩︎

Having constructed the algebras \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) combining the quiver approach and equivariant integration, we are now interested in how these Yangians are connected to the ones introduced by Drinfeld in [1], [2].

One might notice that our algebras are one-parametric: \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n}) = \mathsf{Y}_{\epsilon}(\mathfrak{s}\mathfrak{l}_{n})\). We claim, however, that one could scale away this parameter completely. Indeed, let us consider the algebra isomorphism 7 and choose the parameter of the symmetry \(\sigma\) to be: \[\begin{align} &\sigma = \frac{\epsilon}{\epsilon'}\,,\\ &e^{(a)}_{n} \to \left(\dfrac{\epsilon}{\epsilon'}\right)^{n}e^{(a)}_{n}\,,\\ &f^{(a)}_{n} \to \left(\dfrac{\epsilon}{\epsilon'}\right)^{n}f^{(a)}_{n}\,,\\ &\psi^{(a)}_{n} \to \left(\dfrac{\epsilon}{\epsilon'}\right)^{n}\psi^{(a)}_{n}\,. \end{align}\] This transformation is an isomorphism of the algebras \(\mathsf{Y}_{\epsilon}(\mathfrak{s}\mathfrak{l}_{n})\) and \(\mathsf{Y}_{\epsilon'}(\mathfrak{s}\mathfrak{l}_{n})\). Therefore, one can always fix the parameter to be \(\epsilon' = 1\), provided that \(\epsilon \ne 0\). This also proves that the Yangians we have constructed are isomorphic to Drinfeld Yangians for the algebras \(\mathfrak{s}\mathfrak{l}_{n}\).

In fact, one could include the deformation parameter into the Drinfeld definitions directly. In the same way, the resulting algebra is essentially independent of this parameter [3]. In the physical literature, however, the equivariant parameters are considered as complex mass (flavor fugacity) parameters, which means \(\epsilon\) is a dimensional parameter; see, for example, [45], [61]. Therefore, one should in general keep \(\epsilon \ne 1\).

Now, let us return to the parameters of the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\). To preserve the crystal structure of the states, we have included the second parameter \(h\). We, however, treated it as an effective additional parameter and always set it to be \(0\) in the calculations. One could ask if this parameter adds a new structure to the algebra.

Indeed, the algebra relations 134 modify. We present a few examples of these relations: \[\begin{align} &[e_{n + 1}^{(a)}, e^{(a)}_{m}] - [e^{(a)}_{n}, e^{(a)}_{m + 1}] = \epsilon\{e_{n}^{(a)}, e_{m}^{(a)}\}\,,\\ &[e_{n + 1}^{(a)}, e_{m}^{(a + 1)}] - [e^{(a)}_{n}, e^{(a + 1)}_{m + 1}] = -\frac{\epsilon}{2}\{e^{(a)}_{n}, e^{(a + 1)}_{m}\} - h[e_{n}^{(a)}, e^{(a + 1)}_{m}]\,,\\ &[e_{n + 1}^{(a + 1)}, e^{(a)}_{m}] - [e^{(a + 1)}_{n}, e_{m + 1}^{(a)}] = -\frac{\epsilon}{2}\{e^{(a + 1)}_{n}, e^{(a)}_{m}\} + h[e^{(a + 1)}_{n}, e_{m}^{(a)}]\,, \end{align}\] where the similar behavior manifests in other relations as well. One could notice that these relations are not exactly of Drinfeld type and involve the commutator term. The commutator terms arise from the bond functions: \[\varphi_{a, a + 1}(z - w) = \varphi_{a, a - 1}(z - w) = \frac{z - w - \frac{\epsilon}{2} + h}{z - w + \frac{\epsilon}{2} + h}\,.\] One, however, could apply the spectral shift 9 in the position \(z \to z - h\) at a vertex \(a\) and completely eliminate the parameter \(h\) from these factors: \[\varphi_{a, a + 1}(z - w) = \varphi_{a, a - 1}(z - w) = \frac{z - w - \frac{\epsilon}{2}}{z - w + \frac{\epsilon}{2}}\,.\] In fact, these functions are exactly the ones we have used in our calculations 132 to construct the algebras \(\mathsf{Y}_{\epsilon}(\mathfrak{s}\mathfrak{l}_{n})\). This proves that the algebras \(\mathsf{Y}_{\epsilon, h}(\mathfrak{s}\mathfrak{l}_{n})\) and \(\mathsf{Y}_{\epsilon}(\mathfrak{s}\mathfrak{l}_{n})\) are isomorphic. This essentially means that the parameter \(h\) is indeed effective and does not contribute to the algebraic calculations, as was suggested in section 2.7.

3.4.2 Embedding structure↩︎

Mathematically, the triangular form of GT bases represents the hidden structure of an algebra. Namely, one should consider a given Lie algebra \(\mathfrak{A}_{n}\) not as a single object but as a part of a chain of subalgebras: \[\mathfrak{A}_{1} \subset \mathfrak{A}_{2} \subset \dots \subset \mathfrak{A}_{n}\,.\] This idea can also be applied to Yangian algebras [43], [82], [83]. Since we have introduced the Gelfand-Tsetlin bases in our construction as well, we expect that they capture the following embedding structure: \[\label{sl40n4132embeddings} \mathsf{Y}(\mathfrak{s}\mathfrak{l}_{2}) \subset \mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3}) \subset \mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4}) \subset \dots \subset \mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n}) \subset \dots\,.\tag{180}\] The explicit construction of the embeddings in the general case is quite bulky. Therefore, we limit ourselves in this text to the simplest example: \[\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{2}) \subset \mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\,,\] where we demonstrate how the embedding structure manifests in terms of the quiver diagrams and the corresponding GT bases.

Let us consider the quiver in fig. 2 that we used to study the representations \(\Upsilon_{1, \lambda}\) of the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\). Next, we remove the second vertex from this quiver, or more formally, delete the vertex and all the arrows incident to it. We depict the procedure in fig. 10, where we have erased the indices for the resulting quiver.

In fact, we end up with the Jacobian quiver [84], which indeed is used to describe the representations of the weight \(\lambda\) of the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{2})\); see, for example, [28], [41], [42].

Moreover, this representation can be derived from the representation \(\Upsilon_{1,\lambda}\) of \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\). The crystals 58 transform into: \[\begin{array}{c} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/snahtxwk.png}\label{bfacjmzg}\end{figure} \end{array}\tag{181}\] We can acquire the corresponding GT bases by removing the last row in \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\) GT bases 62 and setting \(n_{2} = 0\): \[\tag{182} |n_{1}, n_{2}\rangle = \quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/qbrjwefg.png}\tag{183}\end{figure} \qquad \to \qquad |n\rangle = \quad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ehvmrlfg.png}\tag{184}\end{figure}\,.\] The ansatz 64 reduces to: \[\label{Y40sl402414132Ansatz} \begin{align} &e^{[\lambda]}(z)|n\rangle = \dfrac{{\boldsymbol{E}}_{n, n+1}^{[\lambda]}}{z - n\epsilon}|n+1\rangle\,,\\ &f^{[\lambda]}(z)|n\rangle = \dfrac{{\boldsymbol{F}}_{n,n-1}^{[\lambda]}}{z - (n-1)\epsilon}|n-1\rangle\,,\\ &\psi^{[\lambda]}(z)|n\rangle = {\boldsymbol{\Psi}}_{n}^{[\lambda]}(z)|n\rangle\,, \end{align}\tag{185}\] where \({\boldsymbol{\Psi}}_{n}^{[\lambda]}(z)\) takes the form: \[\label{sl4024132eigenvalues} {\boldsymbol{\Psi}}_{n}^{[\lambda]}(z) = \frac{(z-\lambda\epsilon)(z+\epsilon)}{(z-n\epsilon)(z-(n-1)\epsilon)}\,.\tag{186}\] Notably, the Euler classes 72 also allow the reduction: \[\begin{align} &\text{Eul}_{(n_{1}, n_{2})}^{[\lambda, 0]} \quad \to \quad \text{Eul}_n^{[\lambda]} = \prod_{k = 1}^{n}(\lambda + 1 - k)\epsilon (n + 1 - k)\epsilon = \dfrac{n!\lambda!}{(\lambda - n)!}\epsilon^{2n}\,,\\ &\text{Eul}_{(n_{1}, n_{2}) \to (n_{1} + 1, n_{2})}^{[\lambda, 0]} \quad \to \quad \text{Eul}_{n,n+1}^{[\lambda]} =-\epsilon \prod_{k = 1}^{n}(k - \lambda - 1)\epsilon (k - n - 1)\epsilon = \dfrac{n!\lambda!}{(\lambda - n)!}(-\epsilon)^{2n + 1}\,,\\ &\text{Eul}_{(n_{1}, n_{2}) \to (n_{1}, n_{2} + 1)} \quad \to \quad 0\,, \end{align}\] which also affects the matrix coefficients 73 : \[\label{sl4024132coefficients} {\boldsymbol{E}}_{n, n+1}^{[\lambda]} = -\dfrac{1}{\epsilon}\,, \quad {\boldsymbol{F}}_{n, n - 1}^{[\lambda]} = -n(\lambda - n +1)\epsilon\,.\tag{187}\]

In general, a similar procedure can be implemented to acquire the following reduction: \[\begin{align} {\cal Q}_{n,p,\lambda} \quad &\to \quad {\cal Q}_{n-1,p,\lambda}\,,\\ \Upsilon_{p,\lambda}|_{\mathfrak{s}\mathfrak{l}_{n}} \quad &\to \quad \Upsilon_{p,\lambda}|_{\mathfrak{s}\mathfrak{l}_{n - 1}}\,. \end{align}\]

4 Conclusion↩︎

Simple Lie algebras remain one of the central objects in discussions of symmetries from various physical systems to mathematical objects. One of the natural generalizations of these algebras is Yangian algebras.

In this paper, we implement the quiver approach to investigate Yangians associated to Dynkin diagrams of A-type and their representations. While the approach reconstructs the known algebras [2], [3], it has its own specifics. For instance, we consider only the rectangular representations. This is tied to the fact that in order to construct more general representations of \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\), it seems that one should work with more complicated framings of the quivers. This naturally brings tensor product structure to the appearing representations [45], [60]. It results in more complicated representation theory, where one could get reducible representations instead of irreducible ones. Therefore, one needs to split them into a sum of the irreducible representations, and only after that could extract a required representation, which drastically complicates the analysis. The existence of the choice of framing suitable to describe an arbitrary irreducible representation remains unknown.

The quiver varieties for \({\cal Q}_{n,p,\lambda}\) are still toric, which allowed us to use powerful equivariant localization. While the construction is quite similar to the cases \(\mathsf{Y}(\widehat{\mathfrak{g}\mathfrak{l}}_{n})\) related to toric \({\boldsymbol{C}Y}_{3}\), we end up with only one equivariant parameter \(\epsilon\). However, to preserve the crystal structure of the states, we had to violate the gauge-fixing condition, namely, vertex constraints, prioritizing the no-overlap condition instead [52]. The generalization of the construction for other types of Dynkin diagrams is not straightforward. The quiver varieties, even for the graphs of D and E types, are no longer toric [39], [40], [52]. Therefore, the applicability of the approach and its generalization to the non-simply laced cases requires more careful consideration.

Notably, we showed that the quiver approach seems to inherit important information about the corresponding algebras \(\mathfrak{s}\mathfrak{l}_{n}\). For example, the crystal states resemble the famous Gelfand-Tsetlin bases. Namely, the GT patterns match the structure of different elements of Jacobian algebras of the quivers \({\cal Q}_{n,p,\lambda}\), which we have shown on various examples. Moreover, these bases highlight the embedding structure of the algebras and its natural correspondence to the quiver diagrams. As expected, the zero level of our Yangians restores the representations of the corresponding Lie algebras \(\mathfrak{s}\mathfrak{l}_{n}\). We also constructed the representations of the Yangian algebras themselves, providing the calculations with explicit examples.

This work is another step towards understanding the connection between quiver diagrams and Yangian algebras. While the question about what types of algebras can be constructed from quiver diagrams remains open, there are quite a few facts that fuel the investigation of this area:

• The quivers seem to be closely related to various Dynkin diagrams and encode the information about the Yangian algebras associated to the corresponding Lie algebras, as we have seen in this text;

• Drinfeld’s second realization of the Yangian algebras offers the description that resembles the Chevalley basis in simple Lie algebras;

• The quivers can be constructed for toric diagrams of \({\boldsymbol{C}Y}_{3}\) [11], which are already richer than Dynkin classification;

• There are generalizations of the quiver diagrams for \({\boldsymbol{C}Y}_{4}\) [52], [56], [58]. Moreover, [52] develops a generalization of Yangian algebras to double Yangians for these cases; see also the review [85].

That all gives us hope that the classification similar to Cartan exists for Yangian algebras as well. The roles of the effective gauge theories and their BPS states, Calabi-Yau manifolds, and the structure of simple Lie algebras, remain unclear in the general picture. However, it was proposed in [40] that the quiver Yangians for any quivers should still give rise to the BPS algebras.

There are still quite a few more open questions considering the algebras \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{n})\) aside from the ones mentioned above. For example, the lift to quantum toroidal and elliptic analogues of Yangians. Although the procedure is seemingly transparent [46], [86], the behavior of crystal representations with different values of the parameters requires an additional investigation.

Finally, we have not raised the questions about various stability chambers of the moduli space of quiver representations [14], [37], [87], [88] and limited ourselves only to the discussion of a particular chamber where we get crystal states. The questions about the structure of the states in various chambers, as well as the wall-crossing between different chambers, remain relevant.

Acknowledgments↩︎

We would like to express our gratitude to D. Galakhov, A. Morozov and N. Tselousov for enlightening discussions and suggestions regarding our work. We are indebted to D. Galakhov for providing valuable comments on the draft.

Our work is funded within the state assignment of NRC Kurchatov institute.

References↩︎

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1. e-mail: gavshin.an@phystech.edu↩︎

2. The expressions hold for so-called un-shifted Yangians. One could introduce the shift [45], [47], [48] in the mode expression of \(\psi^{(a)}(z)\). However, we do not cover this in our consideration.↩︎

3. We subtly changed the form of this symmetry compared to [11], [45] for our purposes. However, the concept behind it remains unchanged.↩︎

4. Partially ordered sets↩︎

5. We note that in our notation it is not a traditional molten crystal but rather its complement. One should keep in mind that the sets of paths in general can have complicated structure. We discuss later in the text why, in our case, we still consider the states as crystals.↩︎

6. We refer the interested reader to [37] for some details of the construction of state models in different stability chambers.↩︎

7. More generally, the McKay correspondence refers to the study of smooth resolutions of orbifold singularities \(\mathbb{C}^{n}/\Gamma\), for finite groups \(\Gamma \subset SL(n, \mathbb{C})\) [74]; \(\mathbb{C}^{n}\) can be replaced by a complex \(n\)-dimensional variety [73].↩︎

8. Including the trivial \(\rho_{0}\) for the affine cases↩︎

9. We avoid the superalgebras in this consideration. The question of constructing a quiver and assigning a superpotential for a given McKay quiver is more subtle in this case. We refer the interested reader to [39], [46] for more detailed reviews.↩︎

10. We encode the representations by the Dynkin labels of their highest weights for simplicity. For example, the notation \((\lambda, 0)\) is equivalent to \(\Upsilon_{1, \lambda}\) for the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{3})\). However, when we consider the representations of the algebra \(\mathsf{Y}(\mathfrak{s}\mathfrak{l}_{4})\), \(\Upsilon_{1, \lambda}\) corresponds to \((\lambda, 0, 0)\). Therefore, the notation \(\Upsilon_{p, \lambda}\) depends on the algebra indirectly.↩︎

11. Including also the representation \((0, 1, 0)\).↩︎

12. The presented pictures have a very loose scale. We highlight only the edges of the graph. The magenta-colored area represents the shape of the empty box.↩︎

13. One could depict the dimensions directly as the corresponding heights on the axis \(\epsilon\), therefore, had to draw the 3-dimensional space \(\Xi\). We believe that our notation is slightly more convenient for our purposes.↩︎