1 Introduction↩︎

1.1 Toward a q-Deformed Quantization Formalism↩︎

Topological recursion [1] can be viewed as a powerful quantization formalism [2] that associates a wave-function \(\psi(\hslash,z)\) to a spectral curve \(P(x,y)=0\). In the classical differential setting, this wave-function is constructed from the multidifferentials \(\omega_{g,n}\) and is annihilated by a quantum curve \(\hat{P}(\hat{x},\hat{y})\psi=0\), where \([\hat{y}, \hat{x}]=\hslash\). However, the non-commutativity of the operators \(\hat{x}\) and \(\hat{y}=\hslash\frac{d}{dx}\) implies that the choice of \(\hat{P}\) is not unique. While topological recursion selects specific quantizations [3] and [4], these often appear algebraically strange or restrictive, and the traditional freedom in choosing the integration divisor \(D\) is insufficient to capture all possible quantum orderings.

The shift to the \(q\)-deformed setting introduces a more fundamental non-commutativity. Here, the differential relation is replaced by a \(q\)-difference relation, leading to the \(q\)-commutation rule: \[\hat{y}\hat{x} = q \hat{x}\hat{y}, \quad \text{where } q = e^{\hslash}.\] In this context, a \(q\)-deformed spectral curve \(P(x,y)=0\) is quantized into a linear \(q\)-difference system. For example, the \((r,1)\) model \(x^{r-1}y^r-1=0\) classically leads to a specific differential operator \((\hat{y}\hat{x})^{r-1}\hat{y}-1=0\). In the \(q\)-deformed case, the challenge is amplified: how can we modify the recursive machinery to reconstruct wave-functions for the full family of \(q\)-orderings and \(q\)-difference connections?

The motivation of this paper is to address this limitation. We propose that the correct way to obtain more general \(q\)-quantizations is to introduce a shifted version of \(q\)-topological recursion. By allowing non-zero highest weights in the underlying Airy structures which we identify as \(q\)-Casimirs, we gain the necessary algebraic freedom to reconstruct WKB solutions for a broad class of \(q\)-quantum curves. This approach not only provides a systematic \(q\)-quantization procedure but also explains why the classical recursion was restricted to a specific subset of operators.

1.2 A Triple Perspective on the q-Deformed Framework↩︎

The \(q\)-deformation of the \((r,s)\) model allows us to unify three fundamental perspectives on topological recursion, each revealing a different facet of the shifted \(q\)-deformed loop equations.

The \(q\)-WKB Perspective addresses the quantization of the \(q\)-spectral curves from the point of view of integrability: This is via the WKB analysis of \(q\)-difference systems. While the topological type property was established in the classical case to guarantee that WKB solutions are governed by topological recursion [5], [6] and [7], [8], we extend this to the \(q\)-deformed setting. By sharpening the assumptions of the topological type framework, we show that a broader class of \(q\)-quantum curves admits \(q\)-WKB solutions, provided they are reconstructed via the shifted \(q\)-topological recursion introduced in this work.

Complementary to this, the Algebraic Perspective via \(q\)-Airy Structures ensures the mathematical consistency of the formalism. The \((r,s)\) curves are parametrized by \(x(z)=z^r\) and \(y(z)=z^{s-r}\), for \(r \geq 2\), \(s \geq 1\), and these structures are representations of \(\mathcal{W}(\mathfrak{gl}_r)\)-algebras where the partition function is a vacuum vector of weight zero [9]. Our \(q\)-deformation naturally emerges by relaxing this constraint: by considering highest weight vectors with non-zero weights, we obtain shifted Airy structures. This algebraic shift is precisely what encodes the \(q\)-difference nature of the system.

Finally, this unified geometric framework provides the synthesis of these approaches. The strength of this triple correspondence lies in its internal consistency. The Airy structure formulation provides a rigorous and immediate proof of the \(q\)-correlators \(W^q_{g,n}\)-a property that is notoriously difficult to establish directly from the recursive formulas. By bridging \(q\)-WKB analysis, \(\mathcal{W}\)-algebra representations, and \(q\)- deformed loop equations, we provide a complete description of the \((r,s,q)\)-deformed system, where the classical \((r,s)\) constraints \((r\equiv\pm 1 \pmod s)\) are reinterpreted through the lens of \(q\)-algebraic rigidity.

1.3 Generalization to the q-deformed setting↩︎

To generalize our results, we follow the methodology introduced by Belliard et al in [10]. This allows us to consistently define the \(q\)-analogue of our model.

The main objective of this paper is to bridge the gap between \(q\)-difference systems and topological recursion through the lens of Airy structures. While the classical \((r,s)\) models are well-understood in the differential context, their \(q\)-analogues introduce non-local effects that require a fundamental shift in the recursive machinery. Our results can be summarized across three main axes: the algebraic construction of shifted Airy structures, the derivation of \(q\)-loop equations, and the reconstruction of \(q\)-quantum curves.

To establish this generalization, we focus on the \((r,s,q)\)-spectral curves, which are \(q\)-deformations of the local models for ramification points. Classically, topological recursion is well-behaved on these curves only if \(r = \pm 1 \pmod{s}\)[9]. We investigate how this condition evolves when the system is governed by \(q\)-difference operators.

Our main finding is that for \(r\equiv 1\pmod s\), we can construct large families of \(q\)-quantum curves. Their solutions are calculated using a new formula we call shifted \(q\)-topological recursion. This formula produces multidifferentials by incorporating the \(q\)-shifts directly into the recursive structure.

More precisely, the scope of the \(q\)-deformation depends on the algebraic constraints of the \((r,s)\) system. For the cases \((r,s)=(r,1)\), the standard partition function is annihilated by all non-negative modes \(W^i_k\), \(k \geq 0\), \(1 \leq i \leq r\) of the \(\mathcal{W}(\mathfrak{gl}_r)\)-algebra generators, defining a highest weight vector of weight zero. We show that we can generalize this to the \(q\)-deformed setting by constructing Airy structures from highest weight vectors with non-zero weights. These weights, corresponding to the zero modes \(W^i_0\), manifest in the \(q\)-difference operators as \(r\) independent scalars \(S_i \in \hslash \mathbb{C}[ \hslash ]\), for \(1 \leq i \leq r\), providing the maximal degrees of freedom for quantization.

For the cases \(r = 1 \pmod{s}\) with \(s \geq 2\), the \(q\)-deformation is more constrained as the partition function must be annihilated by certain negative modes in addition to the non-negative ones. In this regime, although we can still construct \(q\)-Airy structures, the only available freedom is to assign a non-zero weight \(S_1 \in \hslash \mathbb{C}[ \hslash ]\) to the zero mode \(W^q_{0,1}\) of the conformal weight 1 generator. Finally, for \(r =-1 \pmod{S}\) with \(s \geq 3\), we prove a rigidity result: no non-zero weights can be introduced within the Airy structure framework, implying that these systems do not admit such \(q\)-shifted deformations.

Our investigation begins by constructing a \(q\)-deformed framework for Airy structures.

In Section 2, we show that it is possible to define more general families, which we call shifted \(q\)-Airy structures, by allowing the partition functions to correspond to highest weight vectors with non-zero weights (see Theorem 17). These non-zero weights act as the algebraic engine of the \(q\)-deformation, capturing the non-local shifts of the \(q\)-difference system. In this setting, the \((r,s)\)-Airy structures associated with classical topological recursion, constructed in [9] as representations of \(\mathcal{W}(\mathfrak{gl}_r)\)-algebras with vacuum partition functions (weight zero), are naturally recovered in the limit where \(q \to 1\) and where these weights vanish.

Finally in Section 3, we extend the correspondence between Airy structures and topological recursion to the \(q\)-deformed setting. We show that shifted \((r,s,q)\)-Airy structures are equivalent to a specific modification of the loop equations for the \(q\)-correlators \(W^q_{g,n}\), which we define as shifted \(q\)-deformed loop equations (see Proposition 31). These equations can be solved recursively, leading to a variation of the recursion formula that we term shifted \(q\)-topological recursion (see Theorem 32). In this framework, the non-zero highest weights (the \(q\)-Casimirs) introduce explicit corrections to the \(W^q_{g,1}\) disk amplitudes. Crucially, as this recursion originates from the Airy structure formalism, the resulting multidifferentials are guaranteed to satisfy the required symmetry properties.

With these shifted \(q\)-deformed loop equations in hand, we investigate whether the associated quantization formalism produces wave-functions for a broader class of \(q\)-deformed spectral curves.

Beyond these results, this paper also serves an expository purpose. We unify several fundamental perspectives on \(q\)-topological recursion: the geometric approach via residues, the algebraic formulation through \(q\)-Airy structures, and the integrable perspective via the WKB analysis of \(q\)-connections of topological type. The central pillar connecting these viewpoints is the concept of \(q\)-deformed loop equations, which appear as the common thread throughout this work.

1.4 Notation↩︎

We introduce \(\hslash\) along the conventions in [11].

We use the convention that \(\mathbb{N} = \{0,1,2,\ldots \}\) and \(\mathbb{N}^* = \{1,2,3,\ldots \}\). We write \([r] = \{1,\ldots, r\}\). For a set \(N\) and a variable \(z\), we write \(z_N = \{ z_n \, | \, n \in N \}\). Moreover, the \(q\)-deformed number and the \(q\)-deformed factorial are defined respectively by[12]: \[[n]_q=\frac{q^n-q^{-n}}{q-q^{-1}},\quadand\quad [n]_q!=\prod_{k=1}^{n}[k]_q.\]

We consider fields in vertex operator algebras as differential forms of degree equal to the conformal weight of the state. I.e. if in a VOA \(V\), the state \(v \in V\) has conformal weight \(\Delta\), then we index its field by \[Y(v;x) = \sum_{k \in \mathbb{Z}} v_k \frac{(dx)^\Delta}{x^{\Delta + k}}\,.\] We use \(x\) for the variable instead of the conventional \(z\), because this conforms with our interpretation via the spectral curve of topological recursion [13].

When considering a spectral curve with local coordinate \(z\), and functions \(x(z)\) and \(y(z)\), we may write \(x_j = x(z_j)\) and \(y_j = y(z_j)\) to lighten notation.

While the connection between Airy structures and topological recursion is well-established for classical \(\mathcal{W}\)-algebras [9], [13], the extension to \(q\)-deformed structures requires a careful treatment of the shifted loop equations and the modified basis of differentials.

Our approach throughout this work is largely based on the framework established by Belliard et al in [10], which we extend here to the \(q\)-deformed case.

2 Deformed shifted \((r,s,q)\)-Airy structures and highest weight vectors↩︎

In this section, we illustrate how higher quantum Airy structures in the sense of [9], [14] or the related Airy ideals [11], may be used to rebuild highest weight vectors for \(q\)-deformed \(\mathcal{W}(\mathfrak{gl}_r)\) at self-dual level. This concerns a \(q\)-deformation of the \((r,s)\)-Airy structures investigated in [9], which we call \(q\)-deformed shifted \((r,s)\)-Airy structures, denoted \((r,s,q)\). Here, we recall only main results, and follow the concepts of Airy structures introduced in [11], [15].

2.1 \(q\)-deformed Airy structures↩︎

Let us start by giving the definition of the \(q\)-deformed Airy structures (also called \(q\)-deformed Airy ideals). We follow the results presented in [15] to investigate their \(q\)-deformation.

2.1.1 The Rees \(q\)-deformed Weyl algebra↩︎

Let \(A\) be a finite or countably infinite index set. We exploit the notation \(x_A\) for the set of variables \(\{x_a \}_{a \in A}\), and \(\mathcal{D}_{q,A}\) for the set of \(q\)-deformed differential operators \(\left\{ \mathcal{D}_{q,A} \right \}_{a \in A}\), where the action of \(\mathcal{D}_{q,A}\) on a function \(f(x_a)\) is defined by: \[\mathcal{D}_{q,A}f(x_a)=\frac{f(q\,x_a)-f(q^{-1}\,x_a)}{x_a(q-q^{-1})}.\] The \(q\)-deformed Weyl algebra \(\mathbb{C}[x_A]\langle \mathcal{D}_{q,A} \rangle\) is the algebra of \(q\)-deformed differential operators with polynomial coefficients defined by the commutation relations: \[\mathcal{D}_{q,a} \, x_b = \delta_{ab} \mathbf{M}_{q,a} + q^{\delta_{ab}} x_b S_{q,a}^{-1} \mathcal{D}_{q,a} \,,\] where \(\delta_{ab}\) is the Kronecker symbol , \(S_{q,a}\) the \(q\)-shift operator \(S_{q,a} f(x_a) = f(q x_a)\), and \(\mathbf{M}_{q,a}\) the symmetric mean \(q\)-deformed operator (or \(q\)-average): \[\mathbf{M}_{q,a} f(x_a) = \frac{q f(q x_a) + q^{-1} f(q^{-1} x_a)}{q + q^{-1}} \,.\] We also define the completed \(q\)-deformed Weyl algebra as the completion of the \(q\)-Weyl algebra, allowing infinite sums in the \(\mathcal{D}_{q,a}\)-derivatives (for a countably infinite index set \(A\)) but restricted to finite polynomials in the variables \(x_A\). The \(q\)-deformed operator \(\mathcal{D}_{q,A}\) has many filtrations, one of which is the Bernstein filtration (for the classical case, the reader is refered to Definition 2.3 in [15]). From this filtration, we build a graded algebra via the Rees construction.

When \(A\) is countably infinite, we want to be able to take infinite linear combinations of operators \(P_a\) without divergent sums appearing. To this end, we define the notion of a bounded collection of differential operators:

For any bounded collection \(\{P_i\}_{i \in I}\), the linear combinations \(\sum_{i \in I} c_i P_i\) for any \(c_i \in \mathcal{D}_{q,A}^\hslash\) are well defined operators in \(\mathcal{D}_{q,A}^\hslash\), regardless of whether \(I\) is finite or countably infinite.

2.1.2 \(q\)-deformed Airy ideals↩︎

Here, we define the concepts of \(q\)-deformed Airy ideals or \(q\)-deformed Airy structures, which is a particular class of left ideals in \(\mathcal{D}_{q,A}^\hslash\).

2.1.3 \(q\)-deformed partition function↩︎

The main reason that Airy ideals are interesting is because they are annihilator ideals for some partition functions.

Recall the definition of annihilator ideal:

Before stating the main result, we establish the integrability condition for \(q\)-difference systems. Let \(\mathbb{K}\) be a field of characteristic zero (typically \(\mathbb{C}\)). We denote by \(\mathbb{K}[[x_A]]\) the ring of formal power series in the variables \(\{x_a\}_{a \in A}\) with coefficients in \(\mathbb{K}\) and \(\mathcal{D}_{q,i}\) denote the \(q\)-derivative operator acting on the \(i\)-th variable.

With this tool, we can now characterize the partition function associated with a \(q\)-deformed Airy ideal. The result regarding the theory of \(q\)-deformed Airy structures is given in the following proposition. The classical case is furnished in the theorem of the work[14].

2.1.4 \(q\)-deformed Airy ideals in universal enveloping algebras↩︎

Following the framework developed in [9], we extend the construction of Airy structures to the \(q\)-deformed setting. These structures are often realized via representations of quantum Lie algebras or \(q\)-deformed non-linear algebras (e.g., \(W_q\)-algebras).

Let \(\mathfrak{g}_q\) be either a quantum Lie algebra or a non-linear quantum Lie algebra and \(U_q(\mathfrak{g}_q)\) the \(q\)-deformed universal enveloping algebra. Given an exhaustive ascending filtration on \(U_q(\mathfrak{g}_q)\) (such as the filtration by conformal weight); then according to Definition 1, we construct the \(q\)-deformed Rees universal enveloping algebra as follows: \[U_q^\hslash(\mathfrak{g}_q) = \bigoplus_{n \in \mathbb{N}} \hslash^n F_n U_q(\mathfrak{g}_q).\]

The construction of \(q\)-deformed Airy ideals proceeds as follows:

Then we have the following simple lemma:

Thus any left ideal \(\mathcal{I}_{U^\hslash_q} \subseteq U^\hslash_q(\mathfrak{g})\) that is obtained as the homogenization of a left ideal in \(U_q^\hslash(\mathfrak{g})\) automatically satisfies \([\mathcal{I}_{U^\hslash_q},\mathcal{I}_{U^\hslash_q}] \subseteq \hslash^2\mathcal{I}_{U^\hslash_q}\). We summarize the construction of \(q\)-Airy ideals from universal enveloping algebras in three steps:

1. We start with a left ideal \(\mathcal{I}_{U_q^\hslash} \subseteq U_q^\hslash(\mathfrak{g})\) or, equivalently, a cyclic left module \(M \simeq U_q^\hslash(\mathfrak{g})/ \mathcal{I}_{U_q^\hslash}\) generated by a vector \(v\) whose annihilator is \(\mathcal{I}_{U_q^\hslash} = \text{Ann}_{U_q^\hslash(\mathfrak{g})}(v)\).

2. We construct the homogenization \(\mathcal{I}_{U^\hslash_q}= h_q(\mathcal{I}_{U_q^\hslash})\), which is a left ideal in \(U^\hslash_q(\mathfrak{g})\). By construction, we know that \([\mathcal{I}_{U^\hslash_q}, \mathcal{I}_{U^\hslash_q}] \subseteq \hslash^2 \mathcal{I}_{U^\hslash_q}\). From the point of view of modules, we obtain a cyclic left module \(M[\hslash] \simeq U^\hslash_q(\mathfrak{g})/\mathcal{I}_{U^\hslash_q}\) generated by the vector \(v\) and where \(\hslash\) acts by multiplication; the annihilator of \(v\) in \(U^\hslash_q(\mathfrak{g})\) is \(\mathcal{I}_{U^\hslash_q} = \text{Ann}_{U^\hslash_q(\mathfrak{g})}(v)\).

3. Establish a representation \(\rho_q: U_q^\hslash(\mathfrak{g}_q) \to \mathcal{D}_{q,A}^\hslash\) into the \(q\)-Weyl algebra such that there exists a bounded generating set \(\{H_a\}_{a \in A}\) for \(\mathcal{I}_{U_q^\hslash}\) satisfying: \[\rho_q(H_a) = \hslash \mathcal{D}_{q,a} + \hslash p_a(x_A) + O(\hslash^2) \,.\]

By Lemma 2, the ideal \(\mathcal{I} = \mathcal{D}_{q,A}^\hslash \rho_q(\mathcal{I}_{U_q^\hslash})\) is a \(q\)-deformed Airy ideal, uniquely determining a symmetric \(q\)-partition function \(Z_q\) via the recursion.

2.2 The \((r,s,q)\)-Airy structures↩︎

Here, we apply these principles to construct \(q\)-deformed Airy ideals from the universal enveloping algebra of the modes of the strong generators of the \(q\)-deformed \(\mathcal{W}(\mathfrak{gl}_r)\)-algebra at the self-dual level.

Following the three-step approach generalized for the \(q\)-deformed setting, the construction is then lifted to the quantum setting as follows:

1. Defining the \(q\)-deformed-Rees homogenization of the \(q\)-deformed-\(\mathcal{W}\) generators, where the filtration is governed by the \(q\)-deformed-conformal weight.

2. Ensuring the \(q\)-deformed-integrability of the ideal through the commutation relations in the Rees algebra: \(\big[ h_q(\mathcal{I}), h_q(\mathcal{I}) \big] \subseteq \hslash^2 h_q(\mathcal{I})\). This closure property is guaranteed by the \(q\)-W-algebra structure constants being appropriately scaled by \(\hslash\).

3. Implementing a representation \(\rho_q\) in the \(q\)-deformed Rees Weyl algebra \(\mathcal{D}_{q,A}^\hslash\), where the leading terms of the generators are recovered as \(q\)-derivatives \(\hslash \mathcal{D}_{q,a}\) (or their appropriate linear combinations).

2.2.1 The \(q\)-deformed \(\mathcal{W}(\mathfrak{gl}_r)\)-algebra at self-dual level↩︎

Let us introduce the \(q\)-deformed \(\mathcal{W}(\mathfrak{gl}_r)\)-algebra at self-dual level via its realization as a subalgebra of the \(q\)-deformed Heisenberg VOA \(\mathcal{H}(\mathfrak{gl}_r)\).

Let \(\mathfrak{h} \subset \mathfrak{gl}_r\) be the Cartan subalgebra with orthogonal canonical basis \(\{\chi^j \}_{j=1}^r\). We denote by \(\mathcal{H}_q\) the \(q\)-deformed vertex algebra which is the \(q\)-analogue of the Heisenberg VOA. It is generated by modes of the form \(\{J^{j}_n\}_{j\in [r],\,n\in\mathbb{Z}}\) obeying the following \(q\)-deformed commutation relation. \[[J^{i}_n,\,J^{k}_m]_q=\frac{[n]_q}{n}\delta_{i,k}{\delta_{n+m,0}}.\] We define the \(q\)-deformed fields as follows by preserving the classical differential form notation to ensure a smooth contact with the topological recursion limit: \[J^j(z) = Y(\chi_{-1}^j |0 \rangle, z) = \sum_{n \in \mathbb{Z}} J_n^j \frac{dz}{z^{n+1}}.\] However, unlike the classical case, the modes \(J_n^j\) verify the \(q\)-deformed Heisenberg commutation relations, and the differential \(dz\) is understood to be compatible with the \(q\)-deformed-shifted geometry of the \(\mathcal{W}\)-algebra.

The \(q\)-deformed-\(\mathcal{W}(\mathfrak{gl}_r)\)-algebra at self-dual level is the quantum VOA strongly freely generated by the \(q\)-analogs vectors \[\label{WrGenerators} w^j_q = \sum_{1 \leq i_1 < i_2 < \dots < i_j \leq r} \chi^{i_1}_{-1} \chi^{i_2}_{-1} \dots \chi^{i_j}_{-1} |0\rangle_q , \qquad j \in [r].\tag{5}\]

The corresponding fields take the following form: \[\begin{align} \label{eq:expl1} W^j(z) :&= Y_q\left( \sum_{1 \leq i_1 < \dots < i_j \leq r} \chi^{i_1}_{-1} \chi^{i_2}_{-1} \dots \chi^{i_j}_{-1} |0 \rangle_q, z \right)\nonumber \\ &= \sum_{1 \leq i_1 < \dots < i_j \leq r} : J^{i_1}(z) J^{i_2}(zq^{-2}) \dots J^{i_j}(zq^{-2(j-1)}) : \frac{(dz)^j}{z^j}\nonumber \\ &= \sum_{n \in \mathbb{Z}} W^j_n \frac{(dz)^j}{z^{n+j}}. \end{align}\tag{6}\] This gives the explicit relation \[\label{eq:explicit} W^j_n = \sum_{1 \leq i_1 < \ldots < i_j \leq r} \sum_{m_1 + \ldots + m_j = n} \left( \prod_{k=1}^j q^{2(k-1)(m_k+1)} \right) J^{i_1}_{m_1} J^{i_2}_{m_2} \dots J^{i_j}_{m_j}.\tag{7}\] The modes \(\{W^j_n\}_{j \in [r], n \in \mathbb{Z}}\) defined in the relation 7 span the \(q\)-deformed non-linear Lie algebra or \(q\)-deformed \(\mathcal{W}\)-algebra. Let \(U_{r,q}\) denote the \(q\)-deformed universal enveloping algebra of the modes. Despite the non-local shifts in the parameter of deformation \(q\), the algebra inherits a natural filtration by conformal weight (or spin), which remains the fundamental grading for the Airy structure.

We define the filtration by assigning the degree \(j\) to the mode \(W^{j}_{n}\), regardless of the mode index \(n\) or the \(q\)-deformed factors involved. Specifically, the subspace \(F_nU_{r,q}\) is spanned by monomials of the form: \[W^{j_1}_{n_1} W^{j_2}_{n_2} \cdots W^{j_k}_{n_k} \quad \text{such that} \quad \sum_{p=1}^{k} j_p \leq n.\] The \(q\)-deformed Rees algebra is built by introducing the formal quantum parameter \(\hslash\) to keep track of this filtration. This is equivalent to the homogenization: \[\mathcal{W}^j_n := \hslash^j W^j_n.\] By the construction, these operators satisfy the \(q\)-deformed commutation relations: \[\big[\mathcal{W}^j_n, \mathcal{W}^l_m\big] = \hslash^{j+l-1} \sum_{p, s} f_{n,m,s}^{j,l,p}(q) \mathcal{W}^p_s,\] where the structure constants \(f(q)\) are rational functions of \(q\). Since the right-hand side is \(O(\hslash^2)\) for \(j,l \geq 1\) (except for the \(q\)-center), the ideal generated by these modes satisfies the integrability condition of Definition 3.

2.2.2 Preliminary \(q\)-lemmas↩︎

We establish several preliminary \(q\)-lemmas, which will be useful throughout this part. We first prove a simple result about partitions and elementary polynomials. To construct the \(q\)-analogue Airy ideal, we must organize the generators of the \(\mathcal{W}\)-algebra according to a block structure compatible with the decomposition of the quantum spectral curve. In the classical case, this organization is dictated by the branching of the curve into several sheets. In our \(q\)-framework, this structure is preserved by the combinatorics of integer partitions, which allows us to define a suitable polarization for the Heisenberg modes. This structure is formally captured by the following definition:

The realization of the \(q\)-deformed \(\mathcal{W}(\mathfrak{gl}_r)\)-algebra at the self-dual level allows for a natural embedding into the product of its \(q\)-deformed components.

Next we introduce a few simple definitions:

To put it simply, a mode \(W^j_m\) with \(m<0\) is non-negative of level \(d\) with respect to \(\lambda\) if all monomials in the sum 8 contain either one positive mode or at least \(d\) non-trivial zero modes.

The notions are related as follows:

We can rewrite the condition above in terms of a new partition of \(r\).

2.2.3 Constructing quantum left ideals \(\mathcal{I}_{U_{r,q}}(\lambda) \subset U_{r,q}\)↩︎

We start with the construction of the Airy ideals in the framework of \(q\)-deformed algebra. The first step is to construct a family of quantum proper left ideals \(\mathcal{I}_{U_{r,q}} \subset U_{r,q}\) in the universal enveloping algebra of modes associated to partitions of \(r\). The construction presented in this part is the generalization in the sense of \(q\)-deformation of the results investigated in[9]. We provide a proof of the main result so that we can generalize it in the next section.

2.2.4 Determining the \(q\)-deformed homogenization of \(\mathcal{I}_{U_{r,q}}(\lambda)\)↩︎

Associated to a partition \(\lambda\) of \(r\) we constructed a left ideal \(\mathcal{I}_{U_{r,q}}(\lambda)\) in the universal enveloping algebra of modes. The homogenization of \(\mathcal{I}_{U^\hslash_{r,q}}(\lambda)\) is obtained by homogenizing all elements of \(\mathcal{I}_{U_{r,q}}(\lambda)\). By Lemma 3, we know that \[[ \mathcal{I}_{U^\hslash_{r,q}}(\lambda), \mathcal{I}^\hslash_{U_{r,q}}(\lambda)] \subseteq \hslash^2 \mathcal{I}_{U^\hslash_{r,q}}(\lambda).\] For the \(q\)-ideals that we constructed above, the homogenization is easy to describe. Since the modes \(W^j_m\) form a PBW basis for \(U_{r,q}\), and \(W^j_m \in \mathcal{I}_{U_{r,q}}(\lambda)\) for \(m \geq 1- \lambda(j)\) but \(W^j_m \notin \mathcal{I}_{U_{r,q}}(\lambda)\) for \(m < 1- \lambda(j)\), we conclude that the \(q\)-deformed homogenization \(\mathcal{I}_{U^\hslash_{r,q}}(\lambda) \subset U^\hslash_{r,q}\) is generated by the homogenization of the modes, that is, by \(W^{\hslash,j}_m := \hslash^j W^j_m\) for \(j \in [r]\) and \(m \geq 1-\lambda(j)\).

2.2.5 Representing \(U^\hslash_{r,q}\) in \(\mathcal{D}_{A,q}^\hslash\)↩︎

To a partition \(\lambda\) of \(r\) we constructed a left ideal \(\mathcal{I}_{U_{r,q}^\hslash}(\lambda) \in U^\hslash_{r,q}\) that satisfies the condition \([ \mathcal{I}_{U_{r,q}^\hslash}(\lambda), \mathcal{I}_{U_{r,q}^\hslash}(\lambda)] \subseteq \hslash^2 \mathcal{I}_{U_{r,q}^\hslash}(\lambda)\). For each of those, can we find a representation \(\rho_q: U^\hslash_{r,q} \to \mathcal{D}_{A,q}^\hslash\), for some index set \(A\), such that there exists a generating set \(\{H_a\}_{a \in A}\) for \(\mathcal{I}_{U^\hslash_{r,q}}\) with \(\rho_q(H_a) = \hslash \mathcal{D}_{a,q} + O(\hslash^2)\) and the collection \(\{\rho_q(H_a)\}_{a \in A}\) bounded?

One way to do that for a subset of those ideals is to consider representations of \(U_{r,q}^\hslash\) that come from \(\mathbb{Z}_r\)-twisted representations for the \(q\)-deformed Heisenberg VOA \(\mathcal{H}_q(\mathfrak{gl}_r)\). For more simplicity, we present the final result in the following Proposition.

This is not yet in the form required for an \(q\)-deformed Airy structure, since for \(i \geq 2\) the generators \(\mu_{q}( W^{\hslash,i}_k)\) are of order \(O(\hslash^i)\). To recover a linear term in \(\hslash\), we perform a polynomial shift of the Heisenberg partition function. For \(s \in [r+1]\) and \(s\) coprime with \(r\), we may define a new representation also called shifted representation \(\rho^q\) via conjugation: \[\label{eq:rep} \rho_q( W^{\hslash,i}_k) = \hat{T}^q_s \mu_q(W^{\hslash,i}_k) (\hat{T}^q_s)^{-1},\tag{11}\] where \(\hat{T}^q_s = \exp \Big( -\frac{J^{(q)}_s}{[s]_q\hslash} \Big).\) Under this transformation, the leading order term of each generator becomes linear in \(\hslash\), verifying the following relation: \[\rho_q( W^{\hslash,i}_k) = \hslash J^{(q)}_{r k + s (i-1)} + O(\hslash^2),\] provided \(k \geq - \lfloor \frac{s(i-1)}{r} \rfloor\).

Combining the two Subsubsections 2.2.4 and 2.2.5, we need to find partitions \(\lambda\) of \(r\) such that \[1-\lambda(i) = - \lfloor \frac{s(i-1)}{r} \rfloor.\] We have a valid partition if and only if \(r \equiv \pm 1 \pmod{s}\) (for \(s \geq 2\)). More precisely, the partition \(\lambda\) is determined as follows:

• For \(s=1\): The condition is trivially satisfied, and the partition is given by a single row, \(\lambda = (r)\).

• For \(2 \leq s \leq r-1\): The condition \(r \equiv \pm 1 \pmod{s}\) allows us to uniquely write the division \(r = r' s + r''\) with \(r' \geq 1\) and a remainder \(r'' \in \{1, s-1\}\). The partition is then given by \(\lambda = (\lambda_1, \dots, \lambda_s)\) with: \[\lambda_1 = \dots = \lambda_{r''} = r'+1, \qquad \lambda_{r''+1} = \dots = \lambda_s = r'.\]

• For \(s=r+1\): We have \(r \equiv -1 \pmod{s}\), and the partition is a single column of ones, \(\lambda = (1, 1, \dots, 1)\).

We can summarize this in the following theorem:

2.2.6 Generalized quantum deformed representations↩︎

We can generalize the construction of the \(q\)-deformed Airy ideals in Theorem 11 by constructing more general representations \(\rho_q: U^\hslash_{r,q} \to \mathcal{D}_{q}^\hslash\). The idea is to conjugate by more complicated deformed operators, instead of conjugating by \(T^q_s\) given in 11 .

We consider a collection of \(q\)-deformed complex numbers given by: \[\label{eq:complex} F^{q}_{0,1}[-k]\,\text{ for } k \geq \min \{ s, r\} \,, \quad F^{q}_{\frac{1}{2},1}[-k]\,\text{ for } k > 0\,, \quad F^{q}_{0,2}[-k,-l] \,\text{ for } k,l > 0,\tag{13}\] with the following conditions \(F^{q}_{0,1}[-s] \neq 0\) and \(F^{q}_{0,2}[-k,-l] = F^{q}_{0,2}[-l,-k]\).

Note that for \(k,l > 0\), the modes \(J^{(q)}_k\) and \(J^{(q)}_l\) commute, making the symmetrized product implicit. We define a generalized representation \(\rho_q': U^{q}_{\hslash, r} \to \mathcal{D}^{q}_{\mathbb{N}^*}\) via conjugation: \[\label{eq:repp} \rho_q'( W^{\hslash,i}_k) = \hat{\Phi}^{q} \hat{T}^{q} \mu_{q}(W^{\hslash,i}_k) \hat{T}^{q})^{-1} (\hat{\Phi}^{q})^{-1}.\tag{15}\]

Then it is not too difficult to show that Theorem 11 generalizes to this new class of representations:

2.3 Shifted \((r,s,q)\)-Airy structures↩︎

The construction of the previous section can be naturally generalized by starting with highest weight vectors with non-zero weights. This gives rise to new quantum left ideals that can be used to construct quantum Airy structures. We continue by using the three-step approach.

2.3.1 Construction of quantum left ideals \(\mathcal{I}_{U_{r,q}}(\lambda; S)\)↩︎

We start by generalizing the previous construction, but now starting with highest weight vectors.

Looking at the statement of the lemma, we notice that in the end, the ideal \(\mathcal{I}_{U_{r,q}}(\tilde{\lambda}; S)\) is almost entirely defined in terms of the new partition \(\tilde{\lambda}\); the original partition only appears in the choice of non-zero weights. So we can think of the partition \(\tilde{\lambda}\) as our starting point. After renaming \(\tilde{\lambda}\) as \(\lambda\) for simplicity, this leads to the following simpler reformulation of the result

2.3.2 Determination of the quantum homogenization of \(\mathcal{I}_{U_{r,q}}(\lambda;S)\)↩︎

Just as for the \((r,s,q)\)-Airy ideals, the homogenization is easy to obtain. By the same argument as before, we conclude that the quantum homogenization \(\mathcal{I}_{U^\hslash_{r,q}}(\lambda;S) \subset U^\hslash_{r,q}\) is generated by the homogenization of the shifted modes, that is, by \(W^{\hslash,j}_m(S) := \hslash^j (W^j_m- \delta_{m,0} S_j)\) for \(j \in [r]\) and \(m \geq 1-\lambda(j)\). Therefore, by Lemma 3, we conclude that: \[\big[ \mathcal{I}_{U_{r,q}^\hslash}(\lambda;S), \mathcal{I}_{U_{r,q}^\hslash}(\lambda;S)\big] \subseteq \hslash^2 \mathcal{I}_{U_{r,q}^\hslash}(\lambda;S).\]

2.3.3 Representation of \(U^\hslash_{r,q}\) in \(\mathcal{D}_{A,q}^\hslash\)↩︎

For the modes \(W^{\hslash,j}_m(S)\) with \(m >0\), we use the same representation \(\mu_q: U^\hslash_{r,q}\to \mathcal{D}_{q}^\hslash\) as before from ?? . We extend this representation to the shifted zero-modes by setting: \[\mu_q(W^{\hslash,i}_0(S) ) = \mu_q(W^{\hslash,i}_0) - \sum_{n=1}^\infty \hslash^n S_{i,n},\] where the \(S_{i,n} \in \mathbb{C}\) are constants for \(i \in [\lambda_1-\lambda_2]\) and \(S_{i,n} = 0\) for \(i > \lambda_1-\lambda_2\). One can easily verify that mapping the formal shifts to the power series \(\sum_{n=1}^\infty \hslash^n S_{i,n}\) preserves the algebraic structure, thereby yielding a well-defined representation of the universal enveloping algebra \(U^\hslash_{r,q}\).

As before, for \(s \in [r+1]\) and \(s\) coprime with \(r\), we define a generalized representation \(\rho_q: U^\hslash_{r,q} \to \mathcal{D}_{q}^\hslash\) via conjugation by the \(q\)-deformed shift operator: \[\label{eq:rep2} \rho_q( W^{\hslash,i}_k(S)) = T^q_s \mu_q(W^{\hslash,i}_k(S)) (T^q_s)^{-1}, \quad where \quad T^q_s = \exp \Big( -\frac{J^{(q)}_s}{[s]_q\hslash} \Big).\tag{17}\] By restricting to modes where \(k \geq - \lfloor \frac{s(i-1)}{r} \rfloor\), the expansion of the generators yields: \[\rho_q( W^{\hslash,i}_k) = \hslash J^{(q)}_{r k + s (i-1)} + \hslash S_{i,1} + O(\hslash^2),\] which matches the required canonical form for an Airy structure.

A partition \(\lambda\) of \(r\) satisfying \(1-\lambda(i) = - \lfloor \frac{s(i-1)}{r} \rfloor\) exists if and only if \(r = \pm 1 \pmod{s}\).

• For \(s=1\), the partition is simply \(\lambda=(r)\).

• For \(2 \leq s \leq r-1\), let \(r = r' s + r''\) with \(r'' \in \{1, s-1\}\).The partition is \(\lambda = (\lambda_1,\ldots, \lambda_s)\) with \[\lambda_1 = \ldots = \lambda_{r''} = r'+1, \qquad \lambda_{r''+1} = \ldots = \lambda_s = r'.\]

• For \(s=r+1\), we obtain the partition \(\lambda = (1,1,\ldots,1)\).

We observe that, for \(s \geq 2\), the shifting freedom depends on the residue of \(r\pmod{s}\).

• If \(r= 1 \pmod{s}\), then \(\lambda_1 = \lambda_2+1\), implying that only the shift \(S_{1,n}\) can be non-zero. Consequently, we can only shift the first zero-mode \(W^1_0\).

• if \(r=-1 \pmod{s}\), with \(s \geq 3\). Then, \(\lambda_1 = \lambda_2\). In this case, all shift parameters \(S_{i,n}\) must vanish, and we recover the standard \((r,s,q)\)-Airy structures.

The case \(s=1\) is of particular interest. The partition is \(\lambda = (r)\), which allows for the shifting of all zero-modes; thus, \(S_{i,n}\) may be non-zero for all \(i \in [r]\) and \(n \geq 1\).

To summarize these conditions, we define the notion of a set of \(s\)-consistent shifts:

We then obtain the following theorem:

The \(s=1\) case is particularly interesting. Since \(\mathcal{I}^q(S)\) is an \(q\)-deformed Airy ideal, there exists a unique partition function \(Z_q\) such that \(\mathcal{I}^q(S) Z_q = 0\). Explicitly, the annihilation conditions are: \[\rho_q(W^{\hslash,j}_m(S)) Z_q = 0 \qquad for j\in[r], m \geq 0.\] Expanding this using the definition of the shifted modes, we obtain: \[\rho_q(W^{\hslash,j}_m) Z_q = \left(\sum_{n=1}^\infty \hslash^n S_{j,n} \right) Z_q \qquad for j\in[r],\, m \geq 0.\] Consequently, we can interpret the \(q\)-deformed partition function \(Z_q\) for the shifted \((r,s,q)\)-Airy structures as being a highest weight vector for for the \(q\)-deformed \(\mathcal{W}\)-algebra \(\mathcal{W}_q(\mathfrak{gl}_r)\) at self-dual level. In this representation, the zero-modes \(W^j_0\) do not simply annihilate the vacuum but rather act diagonally, with eigenvalues prescribed by the \(s\)-consistent shift parameters \(S_{j,n}\).

2.3.4 Extended \(q\)-deformed representations↩︎

Just as with the standard \((r,s,q)\)-Airy structures, we can construct more general shifted \((r,s,q)\)-Airy structures via conjugation. We define the generalized representations \(\rho_q': U^\hslash_{r,q} \to \mathcal{D}_{q}^\hslash\) through the composite conjugation: \[\label{eq:rep2p} \rho_q'( W^{\hslash,i}_k(S^q)) = \hat{\Phi}^q\hat{T}^q \mu^q(W^{\hslash,i}_k(S^q)) (\hat{T}^q)^{-1} (\hat{\Phi}^q)^{-1}.\tag{18}\] where \(\hat{T}^q\) and \(\hat{\Phi}^q\) are the \(q\)-deformed operators defined in 14 .

Following the logic established in [9] and [13], the results of Theorem 17 extend to this broader class of representations:

2.4 Additional shift parameters↩︎

In the previous section we showed that we can shift some zero modes to get new shifted \((r,s,q)\)-Airy structures. But are we allowed to shift other modes that are not zero modes? The answer is no, because of the following simple lemma.

It follows from this simple lemma that the homogenization of \(\mathcal{I}^q\) is the whole \(q\)-Rees universal enveloping algebra \(U_{r,q}^\hslash\). Consequently, it is impossible to find a representation that maps its generators to operators of the required form in a \(q\)-Rees Weyl algebra, leading us to conclude that we cannot obtain Airy ideals in this way.

3 Shifted \(q\)-loop equations and \(q\)-topological recursion↩︎

In this section, we demonstrate that the defining constraints of these structures are equivalent to shifted \(q\)-loop equations for a system of \(q\)-correlators defined on a new family of shifted \((r,s,q)\)-spectral curves.

Furthermore, we derive a recursive formula that solves these shifted \(q\)-loop equations. While this formula mirrors the structure of the \(q\)-deformed Topological Recursion, it incorporates appropriate shifts in the initial data and the resulting correlators. Consequently, we refer to this framework as Shifted \(q\)-Topological Recursion.

3.1 \(q\)-Spectral curves, \(q\)-loop equations and \(q\)-topological recursion↩︎

The construction of \(q\)-spectral curves, \(q\)-loop equations, and \(q\)-topological recursion constitutes the core of this section. These structures are built to be consistent with their classical counterparts, recovering the results of [15], [17] in the limit \(q \to 1\).

We begin with the general definition of the local data:

Given a \(q\)-deformed admissible local spectral curve, we construct a specific basis of one-forms that facilitates the decomposition of higher-order correlators.

We also introduce the notation:

The main object of study is a system of correlators.

We will single out particular systems of correlators that satisfy the projection property.

It is straightforward to see that the basis of one-forms introduced in Definition 14 is naturally suited for studying systems of correlators that satisfy the projection property.

To formulate the loop equations, we define symmetric combinations of correlators that account for the different ways to distribute topological data \((g,n)\) over multiple sheets.

For each component \(C_j\), with \(j \in [N]\), and for each \(i \in [ r_j]\), we also define the symmetric \(i\)-sheeted correlator combinations: \[{\mathcal{E}^{i, (j)}_{g,n}(x; z_{[n]}) = \sum_{\substack{Z \subseteq \mathfrak{f}(z) \\ |Z| = i}} \mathcal{W}^{q}_{g,i,n}(Z; z_{[n]}).}\] We can now define so-called \(q\)-loop equations, which are particular equations satisfied by systems of \(q\)-correlators.

The primary result of relevance is that the data of a \(q\)-deformed admissible local spectral curve \(\mathcal{S}_q\) uniquely determines its correlators through a recursive process.

3.2 The \((r,s,q)\)-spectral curves↩︎

From now on we will focus on admissible local spectral curves with only one component (\(N=1\)); we will therefore drop the superscript \(^{(j)}\) from the various expressions.

We start by define the \(q\)-deformed Bergman kernel as: \[\label{qB} B_q(z_1,z_2):= \frac{1}{(z_1 - qz_2)(z_1 - q^{-1}z_2)}.\tag{22}\] It is easy to see that the \(q\)-deformed Bergman kernel obey the relations \(B_q(z_1,z_2)=B_q(z_2,z_1)\) and \(\oint_C B_q(z_1,z_2) f(z_1) \frac{dz_1}{2\pi i} = \mathcal{D}_q f(z_2)\).

A special example of the construction may be derived from the \((r,s,q)\)-Airy structures of Section 2.2. We can show that finding the partition function of the \((r,s,q)\)-Airy structures of Theorem 11 is equivalent to topological recursion on the \((r,s,q)\)-spectral curve which is defined as follows:

If we define the meromorphic function \(y\) on \(C\) by the relation \(\omega^{q}_{0,1} = y\;dx\), where \(x=z^r\) is the standard coordinate on the base, we find \(y(z) = \frac{s}{[s]_q} z^{s-r}\). This allows us to interpret \(x(z)\) and \(y(z)\) as a parametrization of the following \(q\)-deformed algebraic curve:

• For \(s\in[r-1]\), we obtain: \[\label{eq:rsac1} C^q\,\cdot x^{r-s} y^r - 1 = 0,\tag{23}\] where \(C^q=\big([s]_q\big)^{r}\,s^{-r}\) is a \(q\)-dependent constant.

• For \(s=r+1\), we get the \(q\)-deformed \(r\)-Airy algebraic curve \[\label{eq:rsac2} y^r - \bigg(\frac{r+1}{[r+1]_q}\bigg)^{r}\cdot x =0.\tag{24}\]

We call these algebraic curves the \((r,s,q)\)-algebraic curves.

In fact, the correspondence applies more generally to the deformed \((r,s,q)\)-Airy structures of Proposition 14, so let us explain it in this more general setting. We define the \(q\)-deformed \((r,s,q)\)-spectral curves in terms of the data introduced in 13 .

To extract the \(q\)-loop equations from the deformed \((r,s,q)\)-Airy structure, we follow the correspondence established in Proposition 14. The claim is that the differential constraints for the partition function \(Z_q\) of the deformed \((r,s,q)\)-Airy structure can be recast as a system of \(q\)-correlators \(\omega^q_{g,n}\) on the deformed \((r,s,q)\)-spectral curve. These deformed correlators must satisfy both the \(q\)-loop equations and the \(q\)-projection property inherited from the \(q\)-deformed \(\mathcal{W}\)-algebra.

For clarity of notation, let us denote the generators of the deformed \((r,s,q)\)-Airy structure as: \[\label{eq:unshiftedH} H^{i,q}_k := \rho_q'( W^{\hslash,i}_k),\tag{31}\] following the representation introduced in 15 . The \(q\)-deformed partition function \(Z_q\) is then uniquely determined by the set of \(q\)-differential constraints: \[H^{i,q}_k Z_q = 0 \,, \quad i \in [r], \quad k \geq - \lfloor \frac{s(i-1)}{r} \rfloor\,.\] To translate these algebraic relations into geometry, we introduce the higher-order quantum fields \(H^{i,q}(x)\) constructed from the operators \(H^{i,q}_k\). These fields are defined as formal series of \(i\)-differentials on the base of the spectral curve: \[H^{i,q}(x) = \sum_{k \in \mathbb{Z}_q} H^{i,q}_k \frac{dx^i}{x^{k+i}},\] where the summation index \(k\) respects the bounds imposed by the Airy structure. These fields represent the quantum version of the elementary polynomials of the spectral curve.

We decompose the \(q\)-deformed currents into their positive and negative frequency parts. Recalling the definition of the modes \(J^{(q)}_k\) in ?? and the basis of one-forms \(\xi^{(j,q)}_{-k}(z)=z^{k-1}dz\) from Definition 14, we define: \[\begin{align} \mathcal{J}^{(q)}_-(z) &= \sum_{k>0} J^{(q)}_k \xi^q_{-k}(z)\tag{32}\,, \\ \mathcal{J}^{(q)}_+(z) &= \sum_{k>0} J^{(q)}_{-k} \xi^q_k (z)\tag{33}\,. \end{align}\]

 

Using this notation, we may rewrite the \(q\)-deformed differential operators \(H^{i,q}_k\) more explicitly in terms of the data of the deformed \((r,s,q)\)-spectral curve from Definition 22.

Note that the results given in [13], Section 4.1 can be obtained in the classical limit of the parameter \(q\).

While the expression in Proposition 27 may appear involved at first, it provides the precise framework needed to extract the \(q\)-loop equations.

From the \(q\)-deformed partition function \[Z_q = \exp \bigg( \sum_{ \substack{g \in \frac{1}{2} \mathbb{N}, n \in \mathbb{N}^* \\ 2g-2 + n > 0}} \frac{\hslash^{2g-2+n}}{[n]_q!} F^{q}_{g,n}[k_1, \dotsc, k_n] \prod_{j=1}^n x_{k_j} \bigg)\] associated with the deformed \((r,s,q)\)-Airy structure, we construct a system of \(q\)-correlators on the deformed \((r,s,q)\)-spectral curve.

Since the \(q\)-correlators have finite expansions in the basis \(\xi^q_{-k_j}(z_j)\) for \(k_j >0\), the system of \(q\)-correlators naturally satisfies the \(q\)-projection property (see Lemma 11):

The key step is to show that this system also satisfies the \(q\)-loop equations:

In summary, we have demonstrated that from the data of the partition function \(Z_q\) associated with the deformed \((r,s,q)\)-Airy structure, one can can construct a system of \(q\)-correlators on the deformed \((r,s,q)\)-spectral curve that satisfies both the \(q\)-loop equations and the \(q\)-projection property. Since these two conditions uniquely characterize the \(q\)-correlators, the system is entirely determined by the geometry of the \(q\)-spectral curve through the \(q\)-topological recursion formula (see Theorem 21).

3.3 The q-shifted loop equations and q-topological recursion↩︎

We now investigate the consequences of considering the shifted \((r,s,q)\)-Airy structures introduced in Theorem 17 or, more generally, the \(q\)-deformed and shifted versions from Proposition 19.

We will demonstrate that, from the partition function of these deformed and shifted structures, one can construct a new system of correlators on a shifted version of the deformed \((r,s,q)\)-spectral curve (see Definition 22). While this system still satisfies the \(q\)-projection property, it no longer obeys the standard loop equations. Instead, it satisfies a modified set of constraints which we call shifted \(q\)-loop equations. We further show that there exists a unique system of \(q\)-correlators satisfying both the shifted \(q\)-loop equations and the \(q\)-projection property, which can be constructed recursively via a shifted version of the \(q\)-topological recursion formula.

Using the notation from the previous section, we first define the shifted deformed \((r,s,q)\)-spectral curve:

Note that the primary departure from Definition 22 lies in \(\omega^{q}_{\frac{1}{2},1}(z)\), which is now shifted by terms linear in the constants \(S_{i,1}\).These constants correspond to the \(\mathcal{O}(\hslash)\) terms in the shifts of the differential operators of the underlying \(q\)-Airy structure. We define the shifted \((r,s,q)\)-spectral curve as the specific case where: \[F^{q}_{0,1} [-k] = [r]_q \delta_{k,s}, \qquad F^{q}_{\frac{1}{2},1}[-k] = 0, \qquad F^{q}_{0,2}[-k,-l] =0.\]

In this setting, the deformations vanish and we recover a shifted version of the original \((r,s,q)\)-spectral curve from Definition 20. This shifted curve can still be interpreted as a parametrization of the algebraic curves in 23 and 24 , but with a non-trivial \(\omega^{q}_{\frac{1}{2},1}(z)\) introduced by the shifts.

We begin by considering the differential constraints from Proposition 19. To simplify the notation, we write the shifted operators as: \[H^{i,q}_k := \rho_q'(W^{\hslash,i}_k(S)) =(H^{i,q}_k)^{\mathrm{unshifted}} -\delta_{k,0} \sum_{\ell=1}^\infty \hslash^\ell S_{i,\ell},\] where the set of shifts is assumed to be \(s\)-consistent. Here, \((H^{i,q}_k)^{\mathrm{unshifted}}\) denotes the unshifted differential operators of 31 . This notation highlights that the primary difference from the standard \((r,s,q)\) case lies in the modification of the zero-modes \(H^{i,q}_0\) by the \(s\)-consistent shifts \(\{S_{i,\ell} \}_{i \in [r], \ell \in \mathbb{N}^*}\).

The \(q\)-partition function associated with this deformed and shifted \((r,s,q)\)-Airy structure is given by: \[Z_q = \exp \bigg( \sum_{ \substack{g \in \frac{1}{2} \mathbb{N}, n \in \mathbb{N}^* \\ 2g-2+n > 0}} \frac{\hslash^{2g-2+n}}{[n]_q!} F^{q}_{g,n}[k_1, \dotsc, k_n] \prod_{j=1}^n x_{k_j} \bigg)\] and satisfies the differential constraints: \[H^{i,q}_k Z_q = 0 \,, \quad i \in [r]\,, \quad k \geq - \lfloor \frac{s(i-1)}{r} \rfloor\, .\] Following the construction in in Definition 24, we use this \(q\)-partition function to define a system of \(q\)-correlators on the shifted deformed \((r,s,q)\)-spectral curve. As in the unshifted case, the system of \(q\)-correlators naturally satisfies the \(q\)-projection property:

The remaining question is whether this system satisfies the \(q\)-loop equations, which is addressed in the following proposition.

We have established that the differential constraints of the deformed and shifted \((r,s,q)\)-Airy structures are equivalent to the existence of a system of \(q\)-correlators on the shifted deformed \((r,s,q)\)-spectral curve, satisfying both the \(q\)-projection property and the shifted \(q\)-loop equations. Since these shifted \(q\)-loop equations differ from the standard ones, it is natural to investigate whether these \(q\)-correlators can still be reconstructed recursively via a modification of the \(q\)-topological recursion formula.

We begin with the following combinatorial lemma, which is essential for proving the recursion.

In the proof of Theorem 21, a key step relies on the right-hand side of 34 having a sufficient vanishing order, causing it to vanish under the residue formula. However, for the shifted \(q\)-loop equations ?? , this property no longer holds due to the additional shift terms. Consequently, these shifts must be explicitly incorporated into the \(q\)-topological recursion formula.

4 Concluding remarks↩︎

The synthesis of our results can be summarized as follows. From the partition function of the deformed \((r,s,q)\)-Airy structure, one constructs a system of \(q\)-correlators on the deformed \((r,s,q)\)-spectral curve that satisfies the standard topological recursion formula. Conversely, the partition function of the shifted and deformed \((r,s,q)\)-Airy structure yields a system of \(q\)-correlators on a shifted \(q\)-spectral curve, which shares the same underlying geometry but features a modified initial condition \(\omega^{q}_{0,1}(z)\).

The fundamental distinction lies in the recursion itself: the shifted \(q\)-topological recursion formula incorporates additional source terms that directly shift the \(n=1\) \(q\)-correlators \(\omega^{q}_{g,1}(z)\). These shifts propagate through the recursive steps, resulting in a distinct global system of \(q\)-correlators. Notably, both shifted and unshifted systems consistently satisfy the \(q\)-projection property.

In the specific case \(s=1\), where all shifts are permissible, we have established that the partition function \(Z_q\) acts as a highest-weight vector for the \(q\)-deformed \(\mathcal{W}(\mathfrak{gl}_r)\)-algebra at the self-dual level. We have demonstrated that these highest-weight conditions manifest within the \(q\)-topological recursion framework as explicit shifts of the \(q\)-correlators \(\omega^{q}_{g,1}(z)\). The precise mapping between this \(q\)-deformed algebraic structure and the shifted geometry of the spectral curve confirms the robustness and consistency of this \(q\)-deformed topological recursion framework.

Our findings open several avenues for future research. First, the precise mapping between the \(q\)-deformed algebraic structure and the shifted geometry of the spectral curve established herein provides the necessary foundation to define and study the corresponding quantum behavior. The geometric construction of these \(q\)-deformed quantum curves will be addressed in detail in an upcoming companion paper.

Furthermore, it would be highly interesting to extend these shifted structures to higher \(s\), or to explore their potential connections to broader classes of \(q\)-integrable systems.

5 Acknowledgements:↩︎

FM was supported¹ through the Georg Forster Fellowship. RW was supported² by the Cluster of Excellence Mathematics Münster and the CRC 1442 Geometry: Deformations and Rigidity. The authors thank Jakob Lindner for useful discussions.

6 Appendix↩︎

Before stating the main theorem, we recall the necessary ingredients for the construction of the basis in the \(q\)-deformed case.

• For each simple root \(\alpha_i\), there exists an automorphism \(T_i\) of \(U_q(\mathfrak{g})\) satisfying the braid relations. These are used to define the quantum root vectors \(E_{\beta}\) for any \(\beta \in \Delta^+\).

• This formula provides the \(q\)-commutation relation between two root vectors \(E_{\beta_i}\) and \(E_{\beta_j}\) with \(i < j\): \[E_{\beta_j} E_{\beta_i} - q^{(\beta_i, \beta_j)} E_{\beta_i} E_{\beta_j} = \text{lower terms in the convex ordering}.\] It ensures that the algebra can be spanned by ordered monomials.

The Poincaré-Birkhoff-Witt (PBW) theorem is a foundational result in the theory of Lie algebras and their deformations. It provides a structured basis for the universal enveloping algebra, ensuring that the non-commutative structure does not collapse the underlying vector space.

References↩︎

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1. “Funded by the Alexander von Humboldt Foundation↩︎

2. “Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 427320536 – SFB 1442, as well as under Germany’s Excellence Strategy EXC 2044/2 – 390685587, Mathematics Münster: Dynamics – Geometry – Structure.”↩︎