Abstract. We study spaces of conformal blocks associated with line bundles over elliptic curves, with coefficients in a vertex algebra. For vertex algebras satisfying suitable finiteness and semisimplicity conditions, which are met by all admissible affine vertex algebras as well as admissible \(W\)-algebras associated with nilpotent elements of standard Levi type, we prove the holonomicity of the sheaf of conformal blocks over the moduli space of bundles. Furthermore, we show that the space of flat sections of the associated Jacobi-invariant connection is spanned by trace functions on modules. This result provides a substantial generalization of the celebrated theorem of Yongchang Zhu to quasi-lisse vertex algebras. As a special case, we deduce that for affine vertex algebras at admissible level, the dimension of the space of conformal blocks coincides with the number of admissible weights at that level.

1 Introduction↩︎

Let \((V,\omega)\) be a conformal vertex algebra of central charge \(c\). In the seminal work [1] Zhu has proved that if \(V\) is lisse and rational, then the trace functions \(S_W\) on irreducible \(V\)-modules \(W\), defined by \[\begin{align} \label{eq:intro-zhu-trace} S_W(u,\tau)=\mathop{\mathrm{Tr}}_W u_0 q^{L_0-c/24}, \end{align}\tag{1}\] span the space of genus-one conformal blocks of \(V\), and constitute a finite dimensional vector valued modular form on \(SL_2(\mathbb{Z})\). Zhu’s theorem is a key part of the mathematical description of conformal field theories and has important applications to the structure theory of vertex algebras.

The purpose of this article is to establish an analogue of Zhu’s theorem for a broad class of quasi-lisse vertex algebras [2] by replacing ordinary conformal blocks on elliptic curves by conformal blocks associated with pairs \((E,\mathcal{L})\) consisting of an elliptic curve \(E\) together with a line bundle \(\mathcal{L}\) over \(E\). The corresponding trace functions depend on both the elliptic moduli parameter \(\tau\) and another parameter \(\alpha\) corresponding to the line bundle. The trace functions are equivariant under an action of the Jacobi group \(SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\) extending that of the modular group \(SL_2(\mathbb{Z})\).

Let \(h\) be an element of the weight-one space \(V_1\) of \(V\), and suppose that \(h_0\) acts on \(V\) semisimply with integer eigenvalues. We refer to such a vector as a current, and to the eigenspace decomposition with respect to \(h_0\) as the charge grading of \(V\). If \(X\) is a smooth complex algebraic curve and \(\mathcal{L}\) a holomorphic line bundle over \(X\), one may associate to \((V,\omega,h)\) a twisted chiral algebra over \(X\), and a sheaf over the moduli space of pairs \((X, \mathcal{L})\) which we refer to as charged conformal blocks [[3]][4]. In the genus-one case we consider the family \(T \rightarrow B\) of pairs \((E,\mathcal{L})_{(\alpha,\tau)}\) parametrised by \((\alpha,\tau) \in B= \mathbb{C}\times\mathcal{H}\), where \(\mathcal{H}\) is the complex upper half plane, \(E=E_\tau\) is the elliptic curve with modulus \(\tau\) and \(\mathcal{L}\) is the holomorphic line bundle of degree zero determined by \(\alpha\). See Section 2 below for details.

To formulate our results we introduce a charged analogue of the quasi-lisse condition. Let \(R_V\) denote Zhu’s Poisson algebra and let \(R_V^h\) be the Poisson algebra obtained as the quotient of \(R_V\) by the commutative algebra ideal generated by vectors of nonzero charge. We say that \((V,\omega,h)\) is stably quasi-lisse if the Poisson variety \(\operatorname{Spec}R_V^h\) has finitely many symplectic leaves. This condition is satisfied in many important classes of examples which we discuss below, including all quasi-lisse affine vertex algebras and all quasi-lisse \(W\)-algebras for an appropriate choice of current \(h\).

Our first result concerns the geometric structure of the sheaf of charged conformal blocks.

When \(h=0\) the charge grading is trivial and the condition of being stably quasi-lisse reduces to the quasi-lisse condition. In this case Theorem  1 recovers the finite dimensionality result of [2].

Our second result identifies charged conformal blocks with trace functions on an appropriate category of \(V\)-modules. Since quasi-lisse vertex algebras generally possess non ordinary modules, the traces 1 are not well defined in general. Instead we consider the trace functions \[\begin{align} \label{eq:trace46def46intro} S_W(u,\alpha,\tau)=\mathop{\mathrm{Tr}}_W u_0 y^{h_0}q^{L_0-c/24}, \end{align}\tag{2}\] where \(y = e^{2\pi i \alpha}\) and, as above, \(q = e^{2\pi i \tau}\). These trace functions converge for a much larger class of \(V\)-modules \(W\). Indeed in Section 7 below we introduce the notion of a stable \(V\)-module and a corresponding notion of stable rationality.

The restriction on the domain of convergence is quite natural and reflects the fact that the trace functions generally possess poles along the lattice \(N\alpha\in\mathbb{Z}+\mathbb{Z}\tau\). Nevertheless it is easy to see that every point in the complement of this lattice is equivalent under the action of the Jacobi group on \(B\) to a point in the domain of convergence, see Lemma 9.

Our third result establishes the Jacobi invariance of charged trace functions.

A few words about the proof of Theorem 2. Using the integrability of the connection \(\nabla\) we prove an isospectrality result in Section 6, which in turn guarantees that flat sections of \(\nabla\) possess series expansions in \(y\) and \(q\) of a particular form. Then in Section 7 we introduce a charged analogue of the Zhu algebra \(A(V)\), which we call the stable Zhu algebra \(A^{\operatorname{stab}}(V)\). This algebra plays for stable modules the same role that \(A(V)\) plays for positive energy modules. Assuming semisimplicity of \(A^{\operatorname{stab}}(V)\), we show that charged conformal blocks are exhausted by trace functions.

Our results apply in particular to admissible affine vertex algebras and admissible \(W\)-algebras associated with nilpotent elements of standard Levi type. Let \(\mathfrak{g}\) be a simple Lie algebra and \(L_k(\mathfrak{g})\) the simple affine vertex algebra at admissible level \(k\). Taking for \(h\) the Weyl vector \(\rho \in \mathfrak{g}= V_1\), one finds that the stable Zhu algebra \(A^{\mathrm{stab}}(L_k(\mathfrak{g}))\) is a finite dimensional semisimple quotient of the polynomial algebra \(\mathbb{C}[\mathfrak{h}]\) on the Cartan subalgebra \(\mathfrak{h}\) of \(\mathfrak{g}\). In particular \((L_k(\mathfrak{g}),\omega,\rho)\) is stably quasi-lisse and stably rational. Hence the corresponding trace functions span the space of charged conformal blocks. Restricting to \(u = \left|0\right\rangle\) we recover the fact, which follows easily from results of Kac and Wakimoto [5], that the graded dimensions of admissible \(\widehat\mathfrak{g}\)-modules of admissible level \(k\) constitute a vector valued Jacobi form of index \(\kappa = kh^\vee\dim(\mathfrak{g}) / 24\). In the case of the universal \(W\)-algebra \(\mathscr{W}^k(\mathfrak{g},f)\), whose Zhu algebra is the finite \(W\)-algebra \(U(\mathfrak{g},f)\), the stable Zhu algebra \(A^{\mathrm{stab}}(\mathscr{W}^k(\mathfrak{g},f))\), for an appropriately chosen current \(h\), is isomorphic to the finite \(W\)-algebra \(U(\mathfrak{l},f)\) associated with the minimal Levi subalgebra \(\mathfrak{l}\) containing \(f\), and the stable Zhu algebra \(A^{\mathrm{stab}}(\mathscr{W}_k(\mathfrak{g},f))\) of the simple quotient \(\mathscr{W}_k(\mathfrak{g},f)\) of \(\mathscr{W}^k(\mathfrak{g},f)\) is a quotient of \(U(\mathfrak{l},f)\). Similarly, \(X^h_{W_k(\mathfrak{g},f)}\) is the closed subvariety of the Slodowy slice \(f+\mathfrak{l}^e\) at \(f\) in \(\mathfrak{l}\). We thus show that all quasi-lisse \(W\)-algebras are stably quasi-lisse, and all quasi-lisse admissible \(W\)-algebras of standard Levi type are stably rational as well, see Theorem 20 and Theorem 36 below. This represents a large extension of Zhu’s modularity results [1] to quasi-lisse vertex algebras.

The appearance of modular and Jacobi forms in representation theory has a long history. In [6] Kac and Peterson determined modular properties of characters of integrable representations of affine Kac-Moody algebras by expressing them in terms of theta functions, and deduced consequences such as asymptotic growth of representations. In [7] Feigin and Fuchs developed analogous character formulas for minimal representations of the Virasoro algebra. Further impetus came from the role of the moonshine module vertex algebra \(V^\natural\) in monstrous moonshine [8], [9].

These phenomena are naturally explained by Zhu’s theorem. Furthermore Zhu’s theorem plays an important role in conformal field theory, and the resulting representation of \(SL_2(\mathbb{Z})\) on the space \(\mathcal{C}^\nabla(V,\omega)\) of global flat sections, appears in Huang’s proof of the Verlinde formula [10]. Modularity also has important structural applications for vertex algebras. For example, [11], [12] use properties of weight \(2\) cusp forms, including Deligne’s bound on Fourier coefficients [13], to obtain estimates on the weight-one spaces of orbifolds of vertex algebras. These estimates were used in [14] to show that every holomorphic vertex algebra of central charge \(24\) with nontrivial weight-one space is an orbifold of the Leech lattice vertex algebra.

Jacobi invariance is similarly important. Schellekens used it to constrain the weight-one Lie algebra of a holomorphic conformal vertex algebra of central charge \(24\), leading to his list of \(71\) possibilities [15] [16]; see also [17]. For lisse rational vertex algebras the Jacobi property was established in [18], [19] by expanding 2 as a Taylor series in \(\alpha\), identifying the coefficients with \(n\)-point functions, and applying Zhu’s results. Closer in spirit to the present work is [20], where supertrace functions of topological \(N=2\) vertex algebras were shown to be flat sections of superconformal blocks, see also [21]. The lisse condition is assumed in all these works.

It has long been expected that analogues of Zhu’s theorem should hold under somewhat weaker hypotheses. Indeed the space \(\mathcal{C}(E_\tau,V)\) of ordinary conformal blocks is finite dimensional not only for lisse vertex algebras, but for all quasi-lisse vertex algebras [2]. This class includes the simple affine vertex algebras \(L_k(\mathfrak{g})\) at admissible levels [22]. Although the representation category of admissible \(L_k(\mathfrak{g})\) is very large, it was shown in [23] for \(\mathfrak{g}=\mathfrak{sl}_2\) and in [24] in general that the \(L_k(\mathfrak{g})\)-modules in category \(\mathcal{O}\) for \(\widehat\mathfrak{g}\) are precisely the admissible modules of Kac and Wakimoto, who had shown that the characters of the latter are modular invariant [5], [25].

The connection \(\nabla\) on genus-one conformal blocks gives rise to modular linear differential equations (MLDE) satisfied by vertex algebra characters. These equations have been studied since the early days of conformal field theory [26] [27], and later in the context of vertex algebras, providing useful constraints, even without assumptions of rationality [2], [28]–[30].

The 4d/2d duality of [31] relating \(4\)-dimensional \(N=2\) SUSY theories with vertex algebras provides further motivation. Modular properties of Schur indices were observed in [32], and in view of the 4d/2d duality, would be natural consequences of modularity for the corresponding vertex algebras. The associated MLDEs have been investigated in detail in [33], and flavoured analogues have been studied in [34]. Here the inclusion of “flavour” roughly corresponds to the passage from trace functions of the form 1 to that of 2 . In particular, the vertex algebras associated with \(\mathcal{S}\)-class theories [31], [33], [35] in genus \(0\) have been constructed in [36] as chiral quantisations of Moore-Tachikawa symplectic varieties [37]. These vertex algebras are known to be quasi-lisse in type \(A\) and believed to be so in general, and their characters are conjecturally quasi-modular and have interesting relations to multiple \(q\)-zeta values [38]. Many non-admissible affine vertex algebras are also quasi-lisse [39].

Thus motivated, it is reasonable to seek a description of conformal blocks in terms of trace functions on appropriate modules for the whole class of quasi-lisse vertex algebras. By including moduli of line bundles in our approach, trace functions on a sufficiently large category of modules are rendered convergent and, under a reasonable rationality condition, span the corresponding spaces of conformal blocks.

There are many generalisations of Zhu’s theorem in the literature. Dong, Li and Mason proved a generalisation to twisted modules [40], a result fundamental to orbifold constructions of holomorphic vertex algebras [16], [41]–[43]. Generalisations to vertex superalgebras with “good” and “bad” statistics were obtained in [44] and [45], respectively. A version for vertex superalgebras graded by rational conformal weights was established in [46], and used in [47] to lift the Kac–Wakimoto modular transformation formulas for admissible affine vertex algebras from characters to trace functions; see also [48], [49]. Modularity of trace functions involving intertwining operators has been applied by Huang to rigidity of tensor categories [50], [51]. There has also been substantial interest in modularity for lisse but non-rational vertex algebras [52]. Miyamoto showed that modularity is restored by adjoining suitable “pseudotrace functions”, and the analysis of these functions was a key ingredient in McRae’s proof that semisimplicity of Zhu’s algebra implies rationality for lisse vertex algebras [53], [54]. We note that while relatively few non-rational lisse vertex algebras are known [55]–[57], there is by contrast a rich supply of non-rational quasi-lisse examples [36], [39], [58]. We expect extensions of our results in all these directions to be possible.

Finally, our results describe charged conformal blocks in terms of stable \(V\)-modules, characterised among weight modules by a support condition. Other categories of modules, such as spectral flows of stable modules, arise by modifying this condition. It would be interesting to relate charged conformal blocks associated with these different categories, perhaps in connection with the proposal of Creutzig and Ridout to recover Verlinde-type formulas from distributional characters [59], see also [60], [61].

Acknowledgements: The authors would like to thank R. Heluani, Y.-Z. Huang, V. Kac and H. Movasati for discussions. JvE gratefully acknowledges RIMS in Kyoto for excellent working conditions during his stay as visiting professor in 2024, during which time work on this project began. TA was partially supported by JSPS KAKENHI Grant Numbers 21H04993, 25K21659 and 26H01997. HL was supported by JSPS KAKENHI Grant Number 21H04993. JvE was supported by grants Serra-2023-0001, CNPq 306498/2023-5 and FAPERJ 204.322/2025.

2 Elliptic curves, line bundles, and modular forms↩︎

In this section we recall background material on modular forms and elliptic functions, and line bundles over elliptic curves. Throughout the paper \(\mathcal{H}\) denotes the upper half complex plane, and we reserve the symbols \(\tau \in \mathcal{H}\) for a complex variable and \(q = e^{2\pi i \tau}\), and \(\alpha\in \mathbb{C}\) and \(y = e^{2\pi i \alpha}\).

2.1 Modular forms and elliptic functions↩︎

For each integer \(k \geq 1\) the Eisenstein series is defined as \[\begin{align} \label{eq:Eisenstein46def} \widetilde{G}_{2k}(q) = (2\pi i)^{2k} \left[ -\frac{B_{2k}}{(2k)!} + \frac{2}{(2k-1)!} \sum_{n=1}^\infty n^{2k-1} \frac{q^n}{1-q^n} \right], \end{align}\tag{3}\] where the \(B_{2k}\) are the Bernoulli numbers: \(B_2 = 1/6\), \(B_4 = -1/30\), \(B_6 = 1/42\), etc. This series is convergent for \(|q|<1\), and we set \(G_{2k}(\tau) = \widetilde{G}_{2k}(q)\) for \(\tau \in \mathcal{H}\). The holomorphic function \(G_{2k}(\tau)\) is also referred to as an Eisenstein series.

For all \(k \geq 1\) the Eisenstein series satisfies \[\begin{align} G_{2k}\left(\frac{a\tau+b}{c\tau+d}\right) = (c\tau+d)^{2k} G_{2k}(\tau) - 2\pi i c (c\tau+d) \delta_{k,1} \qquad \text{for all \left(\begin{array}{cc}a&b\\c&d\\ \end{array} \right) \in SL_2(\mathbb{Z}),} \end{align}\] i.e., \(G_{2k}(\tau)\) is a modular form of weight \(2k\) for \(k \geq 2\). The Eisenstein series \(G_2(\tau)\) is sometimes referred to as a quasimodular form.

For each integer \(k \geq 1\) we consider the Laurent series \[\begin{align} \label{eq:wp46def} \widetilde{\wp}_k(z, q) = z^{-k} + \pi i \delta_{k,1} + (-1)^k \sum_{n=0}^\infty \binom{2n+1}{k-1} \widetilde{G}_{2n+2}(q) z^{2n+2-k}. \end{align}\tag{4}\] At \(q = e^{2\pi i \tau}\) the series is convergent for \(0 < |z| < r\) where \(r\) is the minimal norm of a nonzero point in the lattice \(\Lambda = \mathbb{Z}+ \mathbb{Z}\tau \subset \mathbb{C}\). In this domain the limit coincides with a meromorphic function, defined on all of \(\mathbb{C}\) with poles on \(\Lambda\), which we denote \(\wp_k(z, \tau)\).

We remark that 4 differs slightly from the convention used by Zhu in [1]. Indeed \(\wp_1(z, \tau) = \wp_1^{\text{Zhu}}(z, \tau) + \pi i - G_2(\tau)z\) and \(\wp_2(z, \tau) = \wp_2^{\text{Zhu}}(z, \tau) + G_2(\tau)\), while \(\wp_k(z, \tau) = \wp_k^{\text{Zhu}}(z, \tau)\) for \(k \geq 3\).

We consider, for each integer \(k \geq 1\), the series \[\begin{align} \label{eq:zhu463469} P^+_k(w, q) = \frac{(2\pi i)^k}{(k-1)!} \sum_{n=1}^\infty \left[ \frac{n^{k-1} w^n}{1-q^n} + (-1)^k \frac{n^{k-1} w^{-n}q^n}{1-q^n} \right], \end{align}\tag{5}\] convergent in the domain \(|q| < |w| < 1\). Notice that \(2\pi i w \partial_w P^+_k(w, q) = k P^+_{k+1}(w, q)\). This is precisely the series \(P_k(w,q)\) introduced in [1].

If we denote the limit of \(P^+_k(w, q)\) also by the same symbol, then we have \(P^+_k(e^{2\pi i z}, e^{2\pi i \tau}) = (-1)^k {\wp}_k(z, \tau)\) [1]. Let us be careful to note, however, that substitution of \(w = e^{2\pi i z}\) into the formal series defining \(P^+_k(w, q)\) does not yield the formal series defining \((-1)^k \widetilde{\wp}_k(z, q)\). Indeed 5 involves the summation \(w + w^2 + w^3 + w^4 + \cdots\), and the substitution \(w = e^{2\pi i z}\) here is ill defined.

It will also be convenient for us to have the following variant: \[\begin{align} \label{eq:zhu46346946minus} P^-_k(w, q) = \frac{(2\pi i)^k}{(k-1)!} \sum_{n=1}^\infty \left[ \frac{n^{k-1} w^n q^n}{1-q^n} + (-1)^k \frac{n^{k-1} w^{-n}}{1-q^n} \right], \end{align}\tag{6}\] a series in \(\mathbb{C}((w^{-1}))[[q]]\) convergent in the domain \(|q| < |w^{-1}| < 1\). As before we have \(2\pi i w \partial_w P^-_k(w, q) = k P^-_{k+1}(w, q)\), and \(P^-_k(e^{2\pi i z}, e^{2\pi i \tau}) = (-1)^k {\wp}_k(z, \tau)\) when the left hand side converges.

We recall the theta function [62] \[\begin{align} \theta(z, \tau) = \theta_{11}(z, \tau) = i \sum_{n \in \mathbb{Z}} (-1)^n q^{(n+1/2)^2/2} e^{2\pi i (n+1/2) z}, \end{align}\] which is holomorphic on \(z \in \mathbb{C}\) and vanishes only at \(z \in \Lambda\). It satisfies the relations (see the table at [62]) \[\begin{align} \label{eq:theta46xforms} \theta(z+1, \tau) &= -\theta(z, \tau), \\ \theta(z+\tau, \tau) &= -q^{-1/2} e^{-2\pi i z} \theta(z, \tau). \end{align}\tag{7}\] It is known that \(\theta'(0, \tau) = -2\pi \eta(\tau)^3\), where \(\theta'(z, \tau)\) denotes the derivative in \(z\) of \(\theta\), and \(\eta(\tau)\) is the Dedekind eta function \[\eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n).\]

Now we define \[\begin{align} \label{eq:psi46def} \psi(z, \alpha, \tau) = \frac{\theta'(0, \tau)\theta(z+\alpha, \tau)}{\theta(z, \tau)\theta(\alpha, \tau)} \end{align}\tag{8}\] (see [20]) which we see satisfies \[\begin{align} \label{eq:psi.transf} \psi(z+1, \alpha, \tau) &= \psi(z, \alpha, \tau), \\ \psi(z+\tau, \alpha, \tau) &= e^{-2\pi i\alpha} \psi(z, \alpha, \tau). \end{align}\tag{9}\]

We now consider the Laurent series expansion of \(\psi(z, \alpha, \tau)\) in powers of \(z\). We define \[\begin{align} \label{eq:Ppm46def} \begin{aligned} P_1^+(z, y, q) &= \sum_{n \geq 0} y^{-n} - \frac{e^{2\pi i z}}{e^{2\pi i z}-1} + \sum_{n = 1}^\infty \left( \frac{y^{-1}q^n}{1-y^{-1}q^n} e^{2\pi i n z} - \frac{y q^n}{1-yq^n} e^{-2\pi i n z} \right), \\ P_1^-(z, y, q) &= -\sum_{n \geq 1} y^{n} - \frac{e^{2\pi i z}}{e^{2\pi i z}-1} + \sum_{n = 1}^\infty \left( \frac{y^{-1}q^n}{1-y^{-1}q^n} e^{2\pi i n z} - \frac{y q^n}{1-yq^n} e^{-2\pi i n z} \right). \end{aligned} \end{align}\tag{10}\] These series converge in the region \(|q|<|y^{\mp 1}|<1\). The only difference between \(P_1^+\) and \(P_1^-\) is the first summation, and in both cases the summation is an expansion of the same rational function \(y/(y-1)\). Now [49] asserts that \[\psi(z; -\alpha, \tau) = -2\pi i P_1^{\pm}(z; e^{2\pi i \alpha}, e^{2\pi i \tau}).\] On the other hand we write \[\label{eq:P1pm-Laurent} P_1^\pm(z,y,q) = -\frac{1}{2\pi i z} + \sum_{m= 1}^\infty E_m^\pm(y,q)\,(2\pi i z)^{m-1}.\tag{11}\] The series \(P_1^\pm\) differ only in the constant coefficient, so we have \[E_m(y,q) := E_m^+(y,q) = E_m^-(y,q) \in \mathbb{C}[y^{\pm 1}][[q]],\qquad m\ge 2,\] while \[E_1^\pm(y,q) = -\frac{1}{2\pi i}P_1^\pm(y,q)-\frac{1}{2}.\]

A more general class of twisted Weierstrass functions has been considered in [49] (see also [40]) \[P_k\left[\begin{smallmatrix} \theta \\ \phi \end{smallmatrix}\right](z, \tau) = \frac{(-2\pi i)^k}{(k-1)!} \sum_{n \in \mathbb{Z} + \lambda}\!\!\!{}'\,\,\, \frac{n^{k-1} e^{n z}}{1 - \theta^{-1} q^n},\] where \(\lambda\in \mathbb{R}\) is chosen so \(e^{2\pi i \lambda} = \phi\), and the prime means omit the summand \(n=0\) if \((\theta, \phi) = (1, 1)\). Then, our \(P_1^\pm(z,y, q)\) are particular expansions of \(P_1\left[\begin{smallmatrix} y \\ 1 \end{smallmatrix}\right](z, \tau)\). These have series expansions \[\begin{align} P_k\left[\begin{smallmatrix} \theta \\ \phi \end{smallmatrix}\right](z, \tau) = z^{-k} + (-1)^k \sum_{m=k}^\infty \binom{m-1}{k-1} G_m\left[\begin{smallmatrix} \theta \\ \phi \end{smallmatrix}\right](\tau) z^{m-k}, \end{align}\] in terms of twisted Eisenstein series \(G_m\left[\begin{smallmatrix} \theta \\ \phi \end{smallmatrix}\right](\tau)\). The functions \(E_m\) above are now recovered as \(E_m(y,q) = G_m\left[\begin{smallmatrix} y \\ 1 \end{smallmatrix}\right](\tau)\). The results of [29] assert that the ring generated by twisted Eisenstein series is Noetherian. In Section 4 below we will require the following special case:

In fact, according to [63] the functions \(E_m^\pm(y,q)\) are polynomials in the functions \(P_m^\pm(y,q)\), \(m \leq 3\).

2.2 Elliptic curves and line bundles↩︎

This section establishes the connection between the analytic functions in Section 2.1 and the algebro-geometric framework of elliptic curves and degree-zero line bundles, which form the foundation for charged conformal blocks.

We introduce a family \(T \rightarrow B\) of pairs \((E, \mathcal{L})\) consisting of a smooth elliptic curve \(E\) and a holomorphic line bundle \(\mathcal{L}\) of degree \(0\) over \(E\). The base of the family will be \(B = \mathbb{C}\times \mathcal{H}\), points of which we will denote \((\alpha, \tau)\). We now describe the fibres. We consider the following group action of \(\mathbb{Z}^2\) on \(B \times \mathbb{C}^2\): \[\begin{align} \label{eq:elliptic46family} (m, n) \cdot (\alpha, \tau, v, z) = (\alpha, \tau, e^{-2\pi i m \alpha} v, z + m\tau + n). \end{align}\tag{12}\] The quotient \(T = B \times \mathbb{C}^2 / \mathbb{Z}^2\) is the total space of our family, and in the fibre over the point \((\alpha, \tau)\) the projection \(\mathcal{L}\rightarrow E\) is \((v, z) \mapsto z\).

As above we write \(q = e^{2\pi i \tau}\), let us here and throughout also write \(y = e^{2\pi i \alpha}\). The fibre \((E, \mathcal{L})_{(\alpha, \tau)}\) can also be presented as the quotient of \(\mathbb{C}\times \mathbb{C}^\times\) by the following action of \(\mathbb{Z}\): \[m \cdot (v, w) = (y^{-m} v, q^m w).\]

We now describe meromorphic sections of \(\mathcal{L}_\alpha\) and relate them to the functions introduced in Section 2.1 above.

Let \(X\) be a smooth complex curve of any genus \(g\). The exponential exact sequence of sheaves \[0 \rightarrow 2\pi i \underline{\mathbb{Z}} \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X^\times \rightarrow 0,\] yields a long exact sequence in cohomology and in particular \[\mathop{\mathrm{Pic}}(X) = H^1(X, \mathcal{O}_X^\times) \rightarrow H^2(X, \underline{\mathbb{Z}}) \cong \mathbb{Z}\] takes a line bundle \(\mathcal{L}\in \mathop{\mathrm{Pic}}(X)\) to its degree. The kernel of the degree map is isomorphic to the quotient \[\mathop{\mathrm{Jac}}(X) = H^1(X, \mathcal{O}_X) / H^1(X, \underline{\mathbb{Z}}) \cong \mathbb{C}^g / \mathbb{Z}^g\] i.e., a complex \(g\)-dimensional torus. The other characterisation of degree is in terms of meromorphic sections. Namely a section \(\sigma\) of \(\mathcal{L}\) defines a divisor \(\sum_{p \in X} s_p [p]\) where \(s_p\) is the degree of vanishing of \(\sigma\) at \(p\), and \(\deg(\mathcal{L})\) is \(\sum_{p \in X} s_p\). From this description of the degree, it follows that for \(\deg(\mathcal{L}) < 0\) one has \(H^0(X, \mathcal{L}) = 0\).

Let \(\mathcal{L}\) be a line bundle on \(X\) of degree \(d\). We write \(h^0(X, \mathcal{L})\) for the dimension of \(H^0(X, \mathcal{L})\) the vector space of global sections. The Riemann-Roch theorem asserts that \[h^0(X, \mathcal{L}) - h^0(X, \mathcal{L}^{-1} \otimes \omega_X) = d + 1 - g,\] where \(\omega_X\) is the canonical bundle, i.e., the bundle of \(1\)-forms since \(X\) is a curve. For a point \(p \in X\) and \(k \in \mathbb{Z}\) we can consider the line bundle \(\mathcal{O}_X(k p)\) associated to the divisor \(k p\), characterised by its space \(H^0(X, \mathcal{O}_X(kp))\) of global sections being the set of meromorphic functions on \(X\), regular outside \(p\), and with pole at \(p\) of order not greater than \(k\). The degree of \(\mathcal{O}_X(kp)\) is \(k\). We can similarly form \(\mathcal{L}(kp) \cong \mathcal{L}\otimes \mathcal{O}_X(kp)\). Note also that \(\mathcal{L}(kp)^{-1} \cong \mathcal{L}^{-1}(-kp)\).

If \(E\) is a curve of genus \(1\) and \(\mathcal{L}\) a line bundle on \(E\), then \(\omega_E \cong \mathcal{O}_E\) and we obtain \[h^0(E, \mathcal{L}(kp)) - h^0(E, \mathcal{L}^{-1}(-kp)) = k.\] For \(k > 0\) the second term on the LHS vanishes so \(h^0(E, \mathcal{L}(kp)) = k\). For \(k=0\) all we can conclude is that \(h^0(E, \mathcal{L}) = h^0(E, \mathcal{L}^{-1})\). But if \(\mathcal{L}= \mathcal{O}_E\) we know both sides here are \(1\). If \(\mathcal{L}\) is nontrivial then, since it has degree \(0\), any nonzero global meromorphic section of \(\mathcal{L}\) must have as many poles as it has zeroes. So if such a nonzero section has no poles, it has no zeroes either, which is impossible for \(\mathcal{L}\) nontrivial.

In short, for a smooth elliptic curve \(E\) the dimensions \(h^0(E, \mathcal{L}(kp))\) are given for \(k \geq 0\) by \[\begin{align} 1, 1, 2, 3, 4, \ldots \quad \text{if \mathcal{L}\cong \mathcal{O}_E} \\ 0, 1, 2, 3, 4, \ldots \quad \text{if \mathcal{L}\not\cong \mathcal{O}_E}. \end{align}\] These spaces of sections are described in terms of Weierstrass elliptic functions and theta functions. For \(\mathcal{L}= \mathcal{O}_E\) the sum of all \(H^0(E, \mathcal{L}(kp))\) is spanned by the functions \(1\) and \(\wp_k(z, \tau)\) for \(k \geq 2\). Let \((E, \mathcal{L})\) be the elliptic curve and line bundle corresponding to \((\alpha, \tau)\) with \(\alpha\notin \Lambda\) now. If we define \[\begin{align} \label{eq:varphi46def} \varphi(z) = \frac{\theta(z-\alpha, \tau)}{\theta(z, \tau)}, \end{align}\tag{13}\] then from 7 we see that \[\begin{align} \varphi(z+1) &= \varphi(z), \\ \varphi(z+\tau) &= e^{2\pi i\alpha} \varphi(z). \end{align}\] Hence \(\varphi(z) v\) defines a meromorphic section of the line bundle \(\mathcal{L}_\alpha\) with a simple pole at \(z=0\) (and a simple zero at \(z=\alpha\)).

Note that the function \(\psi(z, -\alpha, \tau)\), defined in 8 above, is just \(\varphi(z)\) up to a normalisation which depends on \(\alpha\) and \(\tau\) but not \(z\). The normalisation \(\psi\) is more convenient for our purposes than \(\varphi\), because the coefficients \(E_m(y, q)\) of its expansion 11 lie in a Noetherian ring, Proposition 4.

3 Vertex algebras and chiral algebras↩︎

We refer to [65] for background on the theory of vertex algebras. For the material on chiral algebras constructed from vertex algebras via associated bundle constructions we follow [4]. The definitive reference on chiral algebras is [3].

This section has two parts. Section 3.1 fixes notation and conventions for vertex algebras and conformal vertex algebras used throughout the paper. Section 3.2 recalls the associated-bundle construction of chiral algebras and discusses the notion of charged conformal block on curves, providing the geometric framework used in later sections.

We now recall a few pieces of notation. For a Laurent series \(f(z) = \sum_n f_n z^n \in \mathbb{C}(\!(z)\!)\) the formal residue \(\mathop{\mathrm{Res}}_z f(z)\) is just the coefficient \(f_{-1}\). If \(\rho(w) = \sum_{n=1}^{\infty}a_nw^n\in \mathbb{C}[\![w]\!]\) is an invertible power series, i.e., \(a_1\neq 0\), the following formal change of coordinates formula holds: \[\begin{align} \label{change95of95variable} \mathop{\mathrm{Res}}_{z}f(z) = \mathop{\mathrm{Res}}_{w}\left( f(\rho(w)) \rho'(w) \right). \end{align}\tag{14}\] One also has the formal integration by parts formula: \[\begin{align} \mathop{\mathrm{Res}}_z (f(z) g'(z)) = -\mathop{\mathrm{Res}}_z(f'(z) g(z)). \end{align}\] When we write \((1+t)^n\) for \(n\in \mathbb{Z}\) in general, we mean \[\begin{align} \label{binom} (1+t)^n=\sum_{j\geq 0}\binom{n}{j} t^j, \quad \text{where} \quad \binom{n}{j}=\frac{n(n-1)\cdots (n-j+1)}{j!}. \end{align}\tag{15}\]

3.1 Vertex algebras↩︎

The translation operator \(T\in \text{End}(V)\) is defined by \(T(a)=a_{(-2)}|0\rangle\) for all \(a \in V\), and 16 implies that \([T, Y(a, z)] = Y(Ta,z)=\partial_zY(a,z)\), for all \(a\in V\). A useful special case of the Borcherds identity is the commutator formula \[\begin{align} \label{eq:va46comm46fla} [a_{(m)}, b_{(n)}] = \sum_{j \geq 0} \binom{m}{j} (a_{(j)}b)_{(m+n-j)}. \end{align}\tag{17}\]

If \(a \in V\) is an eigenvector of \(L_0\) then we refer to the eigenvalue as its conformal weight and we denote it by \(\Delta_a\). Given \(a,b\in V\) and \(n\in \mathbb{Z}\), we have \[\Delta_{a_{(n)}b}=\Delta_{a}+\Delta_{b}-n-1.\] We denote by \(V_\Delta\) the vector subspace of vectors of conformal weight \(\Delta\). It is useful to introduce the following notation for \(a\in V_{\Delta}\): \[\begin{align} \label{eq:conf46modes} Y(a,z)=\sum_{n\in \mathbb{Z}} a_nz^{-n-\Delta}. \end{align}\tag{19}\] In particular we have \(a_{(n)}=a_{n-\Delta+1}\), and \(a_0 : V_{\Delta} \rightarrow V_\Delta\) for all \(a \in V\) and \(\Delta\in \mathbb{Z}\).

A current is a vector \(h \in V_1\) of conformal weight \(1\), such that \(h_0 = h_{(0)}\) acts semisimply on \(V\) and \[\begin{align} h_{n}h = 0 \quad \text{for n \geq 2} \quad \text{and} \quad h_1h = 2\kappa \left|0\right\rangle \end{align}\] for some constant \(\kappa \in \mathbb{C}\) which we call the index. In OPE notation \[h(z) h(w) \sim \frac{2\kappa \left|0\right\rangle}{(z-w)^2}.\] We call the eigenvalues of \(h_{0}\) the charge; we denote by \(V^{(k)}\) the space of vectors in \(V\) with charge \(k\).

Let \(\rho(t) = \sum_{n=1}^\infty \rho_n t^n = \exp(\sum_{n \geq 1} a_n t^{n+1} \partial_t) (a_1)^{t\partial_t} t\) be an invertible power series, i.e., \(\rho_1 = a_0 \neq 0\). Huang established the following formula [66] \[\begin{align} \label{eq:huang46fla} Y(a, z) = R(\rho) Y(R(\rho_z)^{-1}a, \rho(z)) R(\rho)^{-1}, \end{align}\tag{20}\] where \(\rho_z\) is the series defined by \(\rho_z(t) = \rho(z+t)-\rho(z)\), and \(R(\rho) \in \mathop{\mathrm{End}}(V)\) is a linear automorphism defined by the following procedure: \[\begin{align} \label{eq:R40rho4146def} R(\rho) = \exp\left( -\sum_{n \geq 1} a_n L_n \right) a_0^{-L_0}. \end{align}\tag{21}\] This formula for \(R(\rho)\) makes sense because \(L_0\) acts on \(V\) with integer eigenvalues, and the expression is well-defined because each \(L_{>0}\) acts locally nilpotently.

We recall Zhu’s modified vertex algebra structure, which is useful in formulating the modular transformation properties of trace functions. For all \(a \in V\) one defines [1] \[\begin{align} Y[a,z] = Y(e^{2\pi i z L_0} a, \phi(z)), \end{align}\] where \(\phi(t) = e^{2\pi i t}-1\). If we write \(R = R(\phi)\) then \(R(\phi_z) = R(\phi) e^{-2\pi i z L_0}\) since \(\phi_z(t) = e^{2\pi i z}\phi(t)\). Now we have \[\begin{align} Y[a, z] = R^{-1} Y(Ra, z) R \end{align}\] because of 20 . So in fact \(R : (V, Y[-,z]) \rightarrow (V, Y(-,z))\) is an isomorphism of vertex algebras.

The vertex algebra \((V, Y[-,z])\) is conformal with conformal vector \(\widetilde{\omega} = R^{-1}\omega = (2\pi i)^2(\omega - \tfrac{c}{24}\left|0\right\rangle)\). We write \(L[z] = Y[\widetilde{\omega}, z]\). One also checks that \(R^{-1}(h) = \widetilde{h} = 2\pi i (h + \frac{1}{2}L_1 h)\). Let’s assume \(h\) is primary for now. We write \(h[t] = Y[\widetilde{h}, t] = \sum_n h_{[n]} t^{-n-1}\).

We note that \(h_{[1]}\) is related to the notation \(I(h)\) used in [47]. Specifically [47] \[(2\pi i)^2 h_{([1])} = I(h) = \sum_{j \geq 1} \frac{(-1)^j}{j} h_j.\]

3.2 Associated chiral algebras↩︎

In this section we follow closely the presentation in [4]. It is possible to view invertible power series as continuous automorphisms of the topological algebra \(\mathcal{O}= \mathbb{C}[[t]]\). Let \(V\) be a conformal vertex algebra. Then the group \(\mathop{\mathrm{Aut}}(\mathcal{O})\) of all continuous automorphisms of \(\mathcal{O}\) has a representation on \(V\) via the construction \(\rho \mapsto R(\rho)\) defined in 21 .

If \(X\) is a smooth complex curve and \(p \in X\) a point, the local ring \(\mathcal{O}_p\) is naturally an \(\mathop{\mathrm{Aut}}(\mathcal{O})\)-torsor, i.e., carries a simple transitive right \(\mathop{\mathrm{Aut}}(\mathcal{O})\)-action. Informally elements of \(\mathcal{O}_p\) are choices of local coordinate in a neighbourhood of \(p\), more precisely in the formal disc \(D_p = \mathop{\mathrm{Spec}}{\mathcal{O}_p}\), and \(\mathop{\mathrm{Aut}}(\mathcal{O})\) is the group of all changes of coordinate. Through an associated bundle construction one obtains an infinite dimensional vector bundle \(\mathcal{V}\) over \(X\) whose fibre over \(p\) is \(\mathcal{O}_p \times_{\mathop{\mathrm{Aut}}(\mathcal{O})} V\). The bundle \(\mathcal{A}= \mathcal{V}\otimes \omega_X\) carries the structure of a chiral algebra [3]. We do not explain precisely what a chiral algebra is here, except to say that there is an action \[\mu_p : \Gamma(D_p^\times, \mathcal{A}) \times \mathcal{V}_p \rightarrow \mathcal{V}_p,\] where \(D_p^\times\) is the punctured formal disc, and that a choice of coordinate \(z \in \mathcal{O}_p\) at the point \(p\) induces identifications \(\mathcal{V}_p \rightarrow V\) and \(\Gamma(D_p^\times, \mathcal{A}) \rightarrow V[\![z]\!]\) upon which the action becomes \[\begin{align} \label{eq:chiral46action46explicit} \mu_p(f(z) a, b) = \mathop{\mathrm{Res}}_z f(z) Y(a, z)b. \end{align}\tag{22}\] The associated bundle construction and Huang’s formula 20 together guarantee that this action is well-defined, independently of the choice of coordinate \(z\).

Within this context we have the definition of the space of conformal blocks on \(X\) at \(p\):

Although we will not require it in this work, the construction can be adapted to include multiple points \(p\) on \(X\) and the insertion of nontrivial \(V\)-modules at the points instead of \(V\).

We now describe an enhancement of these constructions in which the geometric input consists of an algebraic curve \(X\) as well as a holomorphic line bundle \(\mathcal{L}\) over \(X\). We shall be brief, as the theory we require is treated in [4] in greater generality.

Let \(\mathbb{G}\) denote the algebraic group characterised by \(\mathbb{G}(R) = R^\times\). In particular \(\mathbb{G}(\mathcal{O})\) may be identified with formal power series \(a_0 + a_1 t + \cdots\) in which \(a_0 \in \mathbb{C}^\times\) and \(a_n \in \mathbb{C}\) for \(n \geq 1\). The group \(\mathop{\mathrm{Aut}}(\mathcal{O})\) of continuous automorphisms of \(\mathcal{O}\) acts on \(\mathbb{G}(\mathcal{O})\) and we may form the semidirect product \(G = \mathop{\mathrm{Aut}}(\mathcal{O}) \ltimes \mathbb{G}(\mathcal{O})\).

Let \(\mathcal{L}\) be a holomorphic line bundle over \(X\) and \(p \in X\), then \(\mathbb{G}(\mathcal{O})\) acts on the set of trivialisations of \(\mathcal{L}|_{D_p}\) via multiplication. This action combines with the action of \(\mathop{\mathrm{Aut}}(\mathcal{O})\) on \(\mathcal{O}_p\), so that the semidirect product \(G\) acts on the set \(\mathcal{P}_p\) of pairs \((s, z)\) consisting of \(z \in \mathcal{O}_p\) a coordinate on \(D_p\) and \(s\) a trivialisation of the restriction \(\mathcal{L}|_{D_p}\). In this way \(\mathcal{P}_p\) acquires the structure of a \(G\)-torsor.

At the level of Lie algebras \(\mathop{\mathrm{\mathbf{Lie}}}\mathop{\mathrm{Aut}}(\mathcal{O})\) is spanned by \(L_m = -z^{m+1} \partial_z\) for \(m \geq 0\). These elements, of course, satisfy the commutator relations of the Virasoro algebra 18 . The Lie algebra \(\mathop{\mathrm{\mathbf{Lie}}}\mathbb{G}(\mathcal{O})\) is abelian, spanned by \(h_n\) for \(n \geq 0\), where (after a choice of local coordinate \(z\)) \(h_n\) corresponds to multiplication by \(z^n\). In the semidirect product \(\mathop{\mathrm{\mathbf{Lie}}}(G)\) we have the Lie bracket \[\begin{align} \label{eq:L46h46rel} [L_m, h_n] = -n h_{m+n}. \end{align}\tag{23}\]

Let \((V, \omega, h)\) be a charged conformal vertex algebra. Since \(h\) is quasi-primary, the commutator formula 17 gives the same relation 23 as above. Thus \(V\) carries a representation of \(\mathop{\mathrm{\mathbf{Lie}}}(G)\). Because of the integral charge condition, this representation exponentiates to a representation of \(G\). After a choice of local coordinate \(z\), the elements \(e^a\) and \(e^{a z}\) of \(\mathbb{G}(\mathcal{O})\) are mapped to \(e^{a h_0}\) and \(e^{a h_1}\) in \(\mathop{\mathrm{End}}(V)\), respectively.

Let \((V, \omega, h)\) be a charged conformal vertex algebra, \(X\) a smooth complex curve and \(\mathcal{L}\) a holomorphic line bundle over \(X\). From these data we obtain, as above, a \(G\)-action on \(V\) and the \(G\)-torsors \(\mathcal{P}_p\) for each \(p \in X\). Through an associated bundle construction one obtains an infinite dimensional vector bundle \(\mathcal{V}\) over \(X\) whose fibre over \(p\) is \(\mathcal{P}_p \times_{G} V\). As before \(\Gamma(D_p^\times, \mathcal{V}\otimes \omega_X)\) acts on \(\mathcal{V}_p\), and each choice of local coordinate \(z\) at \(p\) and trivialising section \(s\) of \(\mathcal{L}|_{D_p}\) yields an isomorphism which identifies the action with the residue 22.

Through the formalism of Virasoro localisation, the spaces of conformal blocks assemble to form a quasicoherent sheaf, in fact a \(\mathcal{D}\)-module, as we vary the moduli parameters of the curve \(X\) [67] [4]. In the following, we shall restrict attention to the case of \(X = E_\tau\) smooth elliptic curves and write \(\mathcal{C}(E_\tau, V)\) generally in place of \(\mathcal{C}(E_\tau, 0, V)\). The connection on the sheaf of conformal blocks comes from an instance of the notion of Ward identity in physics, and is interpreted as a Gauss-Manin connection in the chiral algebra picture [3] and is given explicitly in the following definition, which is essentially the definition of \(1\)-point function in [1].

In a similar way, we introduce a connection on charged conformal blocks as follows.

Using the material on elliptic functions in Section 2 we now cast the definition of charged conformal blocks more explicitly. First we briefly explain for the case of ordinary conformal blocks.

We consider the elliptic curve \(E_\tau = \mathbb{C}/ \Lambda\) and the local coordinate at the point \(0\) induced by the standard coordinate \(z\) on \(\mathbb{C}\). We denote this local coordinate also by \(z\). Using this choice the fibre \(\mathcal{V}_0\) is identified with \(V\). The bundle \(\mathcal{V}\) is trivial, and elements of \(\Gamma(E_\tau \backslash 0, \mathcal{V}\otimes \omega_E)\) are linear combinations of \(a \, dz\) and \(\wp_k(z, \tau) a \, dz\) for \(k \geq 2\) and \(a \in V\). The space \(\mathcal{C}(E_\tau, V)\) is thus identified with the space of linear functionals \(\Phi : V \rightarrow \mathbb{C}\) such that \[\begin{align} \label{eq:CB46analytic} \begin{aligned} \Phi(a_{(0)}b) &= 0 \quad \text{for all a, b \in V}, \\ \text{and} \quad \Phi(\mathop{\mathrm{Res}}_t \wp_2(t, \tau) Y(a, t) b) &= 0 \quad \text{for all a, b \in V}. \end{aligned} \end{align}\tag{26}\] We note that \[\Phi(\mathop{\mathrm{Res}}_t \wp_k(t, \tau) Y(a, t) b) = 0\] for all \(k \geq 2\) follows from the case \(k=2\) by substituting \(Ta\) in place of \(a\) and transfering the derivative to the elliptic function using the integration by parts formula.

We now describe charged conformal blocks. It is instructive to work over \(\mathbb{C}\) and then pass to the quotient as in 12 . Consider the map \(f : \mathbb{C}\rightarrow \mathbb{C}\) defined by \(f(z) = z + \tau\). Denote by \(z_p\) the local coordinate at \(p\) induced by the global coordinate \(z\). Let \(a \in V^{(k)}\) and consider the local sections \((v, z_p, a)\) and \((v, z_{f(p)}, a)\). The quotient by the \(\mathbb{Z}^2\)-action identifies \((v, p)\) with \((e^{-2\pi i \alpha}v, f(p))\), so \[f^*((v, z_{f(p)}, a)) = (e^{-2\pi i \alpha}v, z_p, a) = (v, z_p, e^{-2\pi i \alpha h_0} a) = (v, z_p, e^{-2\pi i k \alpha} a).\] If \(k\alpha\notin \Lambda\) then the function \(\psi(z, -k\alpha, \tau)\) is well-defined and meromorphic in \(z\) and, because of the transformation formula 9 , we obtain a section \[\psi(z, -k\alpha, \tau) (v, z, a) \, dz \in \Gamma(E \backslash 0, \mathcal{V}\otimes \omega_E).\] In fact, this section has a simple pole at \(0\). We now have a characterisation of charged conformal blocks similar to (26 ): \[\begin{align} \label{eq:twisted46CB46analytic} \begin{aligned} \Phi(a_{(0)}b) &= 0 \quad \text{for b \in V and a \in V^{(k)} if k\alpha\in \Lambda}, \\ \Phi(\mathop{\mathrm{Res}}_t \wp_2(t, \tau) Y(a, t) b) &= 0 \quad \text{for b \in V and a \in V^{(k)} if k\alpha\in \Lambda}, \\ \text{and} \quad \Phi(\mathop{\mathrm{Res}}_t \psi(t, -k\alpha, \tau) Y(a, t) b) &= 0 \quad \text{for b \in V and a \in V^{(k)} if k\alpha\notin \Lambda}. \end{aligned} \end{align}\tag{27}\]

4 Finiteness of charged conformal blocks↩︎

In this section we prove finite dimensionality of spaces of charged conformal blocks under appropriate hypotheses on the vertex algebra \(V\). In order to do this, it is convenient to first formulate a notion of formal coinvariants inspired by 27 , but expressed purely in terms of series in \(q\) and \(y\).

We say that a conformal vertex algebra \(V\) is strongly generated by a subset \(U \subset V\) if \(V\) is spanned by vectors of the form \(u^{(1)}_{(-n_1)} \cdots u^{(r)}_{(-n_r)} \left|0\right\rangle\) with \(u^{(j)} \in U\) and \(n_j \geq 1\). We call \(V\) finitely strongly generated if it is strongly generated by a finite set \(\{u_1,\ldots,u_r\}\).

Inspired by 27 , we formulate a definition of formal coinvariants.

Technically this definition of the space of formal coinvariants depends on the auxiliary datum of the choice of strong generators. However this choice is not essential, nor is the value of \(N\) appearing in the definition, so long as it is finite.

We recall that the vector space quotient \(R_V = V / V_{(-2)}V\) carries the structure of a Poisson algebra, when endowed with commutative product \(a \cdot b = a_{(-1)}b\) and Poisson bracket \(\{a, b\} = a_{(0)}b\). In [1] Zhu introduced this algebra, as well as the finiteness condition \(\dim(R_V) < \infty\) which serves as a sufficient condition guaranteeing \(\dim \mathcal{C}(E_\tau, V) < \infty\) [1]. In fact the much weaker condition \(\dim R_V / \{R_V, R_V\} < \infty\) (implied in particular by the quasi-lisse condition) is already sufficient [2].

We form an analogous definition in the charged case.

It is easy to see, but not completely obvious, that \(R_V^h\) is a Poisson algebra. It is clear that \(R_V^h\) is naturally a quotient of \(V^{(0)}\), and any element of \(I \cap R_V^{(0)}\) is expressed as a linear combination of terms of the form \(a \cdot b\) where \(a \in R_V^{(c)}\) and \(b \in R_V^{(-c)}\) for some \(c \neq 0\). But then for any \(x \in R_V^{(0)}\) we have \(\{a \cdot b, x\} = a \cdot \{b, x\} + b \cdot \{a, x\} \in I\).

For convenience let us denote \[C(V) = V^{(0)}_{(-2)}V^{(0)} + \sum_{c \neq 0} V^{(c)}_{(-1)}V^{(-c)},\] so that \(R_V^h = V^{(0)} / C(V)\).

In particular if \(V\) is stably quasi-lisse then \(\dim R_V^h / \{R_V^h, R_V^h\} < \infty\) [2], [68]. Notice that if we take \(h = 0\) then the charge grading collapses to \(V = V^{(0)}\), and \(R_V^h = R_V\) so that we recover the usual quasi-lisse condition. On the other hand if \(V\) is quasi-lisse, and \(h \neq 0\), it is not clear if \(V\) is stably quasi-lisse. The issue is that \(I \subset R_V\) is not necessarily an ideal for \(\{\cdot, \cdot\}\). Nevertheless, we shall examine several classes of examples below, and in these cases the stable quasi-lisse condition indeed follows from the quasi-lisse condition.

In the following corollary, and in many places throughout the paper, we will require that our vertex algebra \(V\) be finitely strongly generated, and the number \(N\) defined as in the statement of Proposition 17 will play a role. On the understanding that \(N\) will always be as in Proposition 17, we shall not repeat its definition in full. The following result is analogous to [69].

We will strengthen this result in Corolloary 47 below: in fact \(\mathcal{C}((E, \mathcal{L})_{(\alpha, \tau)}, V)\) is finite dimensional whenever \(N\alpha\notin \mathbb{Z}+ \mathbb{Z}\tau\).

In the remainder of this section we establish the stable quasi-lisse condition for several classes of affine vertex algebras and \(W\)-algebras. The following result and its proof are closely related to [47].

Proposition 19 applies, in particular, to all simple affine vertex algebras \(L_k(\mathfrak{g})\) of admissible level \(k\), since these are quasi-lisse [22]. We recall briefly that an admissible level for the simple Lie algebra \(\mathfrak{g}\) [5] is a rational number of the form \(k = -h^\vee + p/u\) where \(p, u \in \mathbb{Z}_{\geq 1}\) are coprime, and \(p \geq h^\vee\) if \((r^\vee, u) = 1\) in which case \(k\) is said to be a principal admissible level, and \(p \geq h\) if \((r^\vee, u) = r^\vee\) in which case \(k\) is said to be a coprincipal admissible level.

The first examples of quasi-lisse \(L_k(\mathfrak{g})\) of non-admissible level \(k\) were discovered in [39] in connection with the Deligne exceptional series. In particular \(L_k(\mathfrak{g})\) is quasi-lisse for \(\mathfrak{g}= D_4, E_6, E_7, E_8\) and \(k\) integral in the range \(-h^\vee/6-1 \leq k \leq -1\).

We next discuss the stable quasi-lisse condition for affine \(W\)-algebras. Let \(\mathscr{W}^k(\mathfrak{g},f)=H_{DS,f}^0(V^k(\mathfrak{g}))\) be the universal affine \(W\)-algebra associated with the simple Lie algebra \(\mathfrak{g}\), nilpotent element \(f \in \mathfrak{g}\), and level \(k\in \mathbb{C}\). Here \(H_{DS,f}(?)\) denotes the functor of quantized Drinfeld-Sokolov reduction associated with \(f\) [70], [71]. We also denote by \(\mathscr{W}_k(g,f)\) the unique simple graded quotient of \(\mathscr{W}^k(\mathfrak{g},f)\).

Let \(\mathfrak{s}=\{e,h,f\}\) be an \(\mathfrak{sl}_2\)-triple in \(\mathfrak{g}\) associated with \(f\). The adjoint action of \(h\) induces a grading \(\mathfrak{g}=\bigoplus_{j\in \frac{1}{2}\mathbb{Z}}\mathfrak{g}_j\) of \(\mathfrak{g}\) by eigenvalues, i.e., \(\mathfrak{g}_j=\{x\in \mathfrak{g}\mid [h,x]=2j x\}\). If we denote by \(\mathfrak{g}^f\) the centraliser of \(f\) in \(\mathfrak{g}\), and \(\mathfrak{g}_j^f=\mathfrak{g}^f\cap \mathfrak{g}_j\), then we have the induced grading \(\mathfrak{g}^f=\bigoplus_{j\in \frac{1}{2}\mathbb{Z}_{\leq 0}} \mathfrak{g}^f_{j}\). According to [72] the associated variety of \(\mathscr{W}^k(\mathfrak{g},f)\) is the Slodowy slice at \(f\), that is \[\begin{align} \label{eq:DSK46iso} R_{\mathscr{W}^k(\mathfrak{g},f)} \cong \mathbb{C}[S_f]\cong \mathbb{C}[\mathfrak{g}^f], \end{align}\tag{29}\] where \(S_f=f+\mathfrak{g}^e\) is the Slodowy slice.

We denote by \(\mathfrak{g}^{\natural}=\mathfrak{g}^f_0\) the centralizer of \(\mathfrak{s}\) in \(\mathfrak{g}\). Let \(\mathfrak{t}\) be a maximal toral subalgebra of \(\mathfrak{g}^\natural\) and \(\mathfrak{h}\) a Cartan subalgebra of \(\mathfrak{g}\) containing \(h\) and \(\mathfrak{t}\). Note that \(\mathfrak{t}\) is also a maximal toral subalgebra of \(\mathfrak{g}^f\). We denote by \(\Delta \subset \mathfrak{h}^*\) the root system of \(\mathfrak{g}\) with respect to \(\mathfrak{h}\), and by \(\Delta^f \subset \mathfrak{t}^*\) the restricted root system, that is, the set of nonzero weights of \(\mathfrak{t}\) acting on \(\mathfrak{g}^f\) [73], [74]. The centralizer of \(\mathfrak{t}\) in \(\mathfrak{g}\) is a minimal Levi subalgebra \(\mathfrak{l}\) of \(\mathfrak{g}\) containing \(f\), and we clearly have the following restricted root space decomposition \[\begin{align} \mathfrak{g}^f = \mathfrak{l}^f \oplus \bigoplus_{\alpha \in \Delta^f} \mathfrak{g}^f_{\alpha} \end{align}\] Here the zero weight space \(\mathfrak{l}^f = \mathfrak{l} \cap \mathfrak{g}^f\) is the centralizer of \(f\) in \(\mathfrak{l}\). Let \(\Pi^f\) be a base of the restricted root system \(\Delta^f\), so that any element of \(\Delta^f\) can be written as \(\sum_{\alpha\in \Delta^f} a_\alpha \alpha\) with either all \(a_{\alpha}\in \mathbb{Z}_{\geq 0}\) or all \(a_{\alpha}\in \mathbb{Z}_{\leq 0}\). Finally set \[\begin{align} \label{eq:rho95f} \rho_f=\frac{1}{2}\sum_{\alpha\in \Delta^f} \alpha, \end{align}\tag{30}\]

The weight-one space of \(\mathscr{W}^k(\mathfrak{g},f)\) generates the universal affine vertex algebra associated with \(\mathfrak{g}^{\natural}\) and we have a vertex algebra embedding \[\begin{align} \label{eq:weight-one-subalgenra} V^{k^{\natural}}(\mathfrak{g}^\natural)\hookrightarrow \mathscr{W}^k(\mathfrak{g},f), \end{align}\tag{31}\] where \(k^{\natural}\) is some complex number that depends on \(k\).

Let \(\{h_1,\dots, h_{\dim \mathfrak{t}}, x_1,\dots, x_r\}\) be a basis of \(\mathfrak{l}^f\) such that \(\{h_1,\dots, h_{\dim \mathfrak{t}}\}\) is a basis of \(\mathfrak{t}\). Let us also choose a basis \(\{y_{\alpha,i}\}\) of each restricted weight space \(\mathfrak{g}_{\alpha}^f\), so that \(\mathbb{C}[\mathfrak{g}^f] = \mathbb{C}[h_i, x_i, y_{\alpha,i}]\). Considering the surjection \(\mathscr{W}^k(\mathfrak{g},f)\twoheadrightarrow R_{\mathscr{W}^k(\mathfrak{g},f)}\cong \mathbb{C}[\mathfrak{g}^f]\), we may choose lifts of the listed generators to a set of strong generators of \(\mathscr{W}^k(\mathfrak{g},f)\), whose associated quantum fields we denote \[\begin{align} h_i(z), \quad x_i(z),\quad \text{and} \quad y_{\alpha,i}(z), \end{align}\] respectively, with the properties that \(\{h_i(z)\}\) generates the image of the Heisenberg algebra associated with \(\mathfrak{t}\) by 31 , \[\begin{align} h_i(z)x_j(w)\sim 0, \quad \text{and} \quad h_i(z)y_{\alpha,j}(w)\sim \frac{\alpha(h_i)}{z-w}y_{\alpha,j}(w). \end{align}\] We denote the element in the image of 31 corresponding to the element \(\rho_f\) also by \({\rho_f}\).

Notice that the charge grading of \(\mathscr{W}^k(\mathfrak{g},f)\) by \(\rho_f\) runs over \(\mathbb{Z}_{\geq 0}\) and so, from Definition 15, we have \(R_{\mathscr{W}^k(\mathfrak{g},f)}^{\rho_f} = R_{\mathscr{W}^k(\mathfrak{g},f)}^{(0)} = \mathbb{C}[h_i, x_i]\), and the isomorphism 29 yields \[\begin{align} R_{\mathscr{W}^k(\mathfrak{g},f)}^{\rho_f} \cong \mathbb{C}[S^{\mathfrak{l}}_f] \end{align}\] at the level of Poisson algebras, where \(S^{\mathfrak{l}}_f\) is the Slodowy slice \(f+\mathfrak{l}^e\) at \(f\) in the Levi subalgebra \(\mathfrak{l}\) (cf. [74]).

5 Trace functions and charged conformal blocks↩︎

Let \((V, \omega, h)\) be a charged conformal vertex algebra. In this section we prove that certain trace functions on an appropriate category of \(V\)-modules, which we call stable \(V\)-modules, are flat sections of charged conformal blocks in a formal sense. The precise statement is Proposition 22 below.

In short, the proposition identifies trace functions on stable \(V\)-modules as charged conformal blocks relative to Zhu’s vertex algebra structure \(Y[-, z]\) on \(V\).

In the course of the proof of Proposition 22 we proved the following useful identity.

We now pass from unnormalised to normalised trace functions.

6 Holonomicity↩︎

Let us denote \(X = \{(y, q) \in \mathbb{C}^2 \mid 0 \leq |q| < |y^{-1}|^N < 1 \}\) and let \(U\) be a connected and simply connected open subset of \(\{(y, q) \in X \mid q \neq 0\}\). In this section we show that the differential equations 24 and 25 equip the sheaf \(\mathcal{C}|_U\) with the structure of an integrable connection \(\nabla\). Furthermore the equations have a regular singularity at \(q=0\).

6.1 Integrability of the connection↩︎

For notational convenience we shall introduce operators \[\begin{align} [R_L S](a, y, q) &= -S(\mathop{\mathrm{Res}}_z \widetilde{\wp}_1(z, q) L[z] a, y, q), \\ \text{and} \quad [R_h S](a, y, q) &= -S(\mathop{\mathrm{Res}}_z \widetilde{\wp}_1(z, q) h[z] a, y, q). \end{align}\] (The minus signs are introduced to make the commutation relations of \(R_L\) and \(R_h\) more cleanly reflect those of \(L\) and \(h\), as in the dual action of a Lie algebra, see below.) Then the differential equations 24 and 25 are expressed as \[\begin{align} \left( y\frac{\partial}{\partial y} + \frac{1}{2\pi i} R_h \right)S &= 0, \\ \left( q\frac{\partial}{\partial q} + \frac{1}{(2\pi i)^2} R_L \right) S &= 0. \end{align}\] Explicitly the connection \(\nabla : \mathcal{C}\rightarrow \mathcal{C}\otimes \Omega_U\) becomes \[\begin{align} \label{eq:ward-connection} \nabla(S) = \left(\frac{\partial}{\partial q} + \frac{1}{(2\pi i)^2 q} R_L\right)S \otimes dq + \left(\frac{\partial}{\partial y} + \frac{1}{2\pi i y} R_h\right)S \otimes dy. \end{align}\tag{39}\]

6.2 Regularity↩︎

We continue in the setup of Section 6.1. We make use of the following standard result, in this case paraphrased from [76].

In particular, the restriction \(\mathcal{C}|_U\) is a vector bundle of finite rank \(r\) with connection \(\nabla\). Since we assume \(U\) simply connected, \(\mathcal{C}|_U\) is trivial and we may choose a trivialisation \(\mathcal{C}|_U \rightarrow U \times \mathbb{C}^r\) with frame \(\mathbf{e}=(e_1,\dots,e_r)\). Let \(u = u(y, q)\) be the coordinate column vector of a horizontal section \(S \in \Gamma(U,\mathcal{C})\) with respect to \(e\). Then \(u\) satisfies a first-order system \[\label{eq:system-AB} q\frac{\partial}{\partial q} u = B(y,q)\,u,\qquad y\frac{\partial}{\partial y} u = A(y,q)\,u\tag{40}\] for some matrices \(A,B \in \mathop{\mathrm{Mat}}_r(\mathcal{O}_X)\). Indeed the entries of the matrices \(A, B\) are expressed as elements of the ring \(R^{(N)} \subset \mathbb{C}(\!(y^{-1})\!)[\![q]\!]\) and these series are convergent in \(X\).

The intersection of \(X\) with the locus \(q=0\) is the set \(L = \{y \in \mathbb{C}\mid |y| > 1\}\). The matrices \(A\) and \(B\) have series expansions \[A(y,q) = \sum_{n=0}^\infty A_n(y)\,q^n \quad \text{and} \quad B(y,q)=\sum_{n=0}^\infty B_n(y)\,q^n\] where \(A_n(y), B_n(y) \in \mathbb{C}(\!(y^{-1})\!)\) are convergent on \(L\).

We wish to use regularity of the connection at \(q=0\) to prove that \(u(y,q)\) has a series expansion of Frobenius type, in analogy with the classical theory of ordinary differential equations with regular singular point. To be able to do this, we need the following simple but key isospectrality result.

7 Stable rationality and the stable Zhu algebra↩︎

Let \((V, \omega, h)\) be a charged conformal vertex algebra, and let \(M = \bigoplus_{\Delta, k} M_\Delta^{(k)}\) be a nonzero stable \(V\)-module. If \(\Delta_0\) is the minimal value of \(\Delta\) for which \(M_\Delta= \bigoplus_k M_\Delta^{(k)}\) is nonzero, then we denote \(M_{\Delta_0}\) by \(M_{\operatorname{top}}\). If \(k_0\) is the maximal value of \(k\) for which \(M_{\Delta_0}^{(k_0)}\) is nonzero, then we denote \(M_{\Delta_0}^{(k_0)}\) by \(M_{\operatorname{peak}}\).

7.1 The stable Zhu algebra↩︎

We begin by recalling the definition of the Zhu algebra \(A(V)\) of a conformal vertex algebra \((V, \omega)\) [1]. Let \(J \subset V\) denote the vector subspace spanned by elements of the form \[\mathop{\mathrm{Res}}_w w^{-2}(1+w)^{\Delta_a}Y(a,w)b \, dw, \quad \text{a, b \in V}.\] By the Borcherds identity 16 we have \[\begin{align} \bigl(\mathop{\mathrm{Res}}_w w^{-2}(1+w)^{\Delta_a}Y(a,w)b \bigr)_0 &= \sum_{j\in \mathbb{Z}_+}\binom{\Delta(a)}{j}(a_{(-2+j)}b)_0 \\ &= \sum_{j\in \mathbb{Z}_+}(-1)^j\bigl(a_{-1-j}b_{1+j}-b_{-1-j}a_{1+j}\bigr). \end{align}\] Hence, if \(M\) is a positive-energy \(V\)-module and \(M_{\operatorname{top}}\) denotes its lowest conformal-weight subspace, then the assignment \[V \to \mathop{\mathrm{End}}(M_{\operatorname{top}}), \qquad a \mapsto a_0,\] induces a linear map \[V/J \to \rm{End}(M_{\operatorname{top}}).\] The remarkable observation of Zhu is that \(V/J\) acquires the structure of an associative algebra with the product \[a * b = \mathop{\mathrm{Res}}_w w^{-1}(1+w)^{\Delta(a)}Y(a,w)b \, dw, \quad \text{a, b \in V},\] unital with unit \(\left|0\right\rangle\), and the map above is a homomorphism of associative algebras. The algebra \((V/J, *)\) is denoted \(A(V)\). For any irreducible left \(A(V)\)-module \(N\), there exists an irreducible positive energy \(V\)-module \(M\) such that \(M_{\operatorname{top}} = N\) as \(A(V)\)-modules.

We record the following useful result [1].

Assume now that \((V, \omega, h)\) is a charged conformal vertex algebra. Then \(A(V)\) inherits a \(\mathbb{Z}\)-grading by charge from that on \(V\), we write \(A(V)=\bigoplus_{c\in\mathbb{Z}} A(V)^{(c)}\).

Let \(M\) be a stable \(V\)-module, so that \(M_{\operatorname{top}}\) is a left \(A(V)\)-module. Then since \(a_0\) raises the \(h_0\)-eigenvalue by \(c\) for \(a\in V^{(c)}\), the space \(M_{\operatorname{peak}}\) is annihilated by \(a_0\) for all \(a\in V^{(c)}\) with \(c>0\). This motivates us to define:

By construction, the action of \(A(V)\) on \(M_{\operatorname{top}}\) induces an action of \(A^{\operatorname{stab}}(V)\) on \(M_{\operatorname{peak}}\).

In the rest of this section we discuss stable rationality of affine vertex algebras and \(W\)-algebras.

Recall that a nilpotent element \(f\) of \(\mathfrak{g}\) is said to be of standard Levi type if it is a principal nilpotent element in some Levi subalgebra of \(\mathfrak{g}\). Note that if \(\mathfrak{g}\) is of type \(A\) then all nilpotent elements are of standard Levi type.

Theorem 36 provides a large supply of examples to which our main results apply, beyond admissible level affine vertex algebras.

7.2 Exhaustion by trace functions↩︎

In this subsection we complete the proof of Theorem 2. Let \(S\) be a genus-one charged conformal block, i.e.a flat section of the connection \(\nabla\) on \(\mathcal{C}\) in the sense of Section 6. Our aim is to show that, when \(A^{\operatorname{stab}}(V)\) is finite-dimensional and semisimple, \(S\) is a finite linear combination of trace functions on irreducible stable \(V\)-modules. The argument follows from Zhu’s exhaustion method [1], applied first to the \(y\)-expansion of each \(q\)-sector and then to powers of \(\log(q)\). If \(V\) is stably rational, the hypothesis on \(A^{\operatorname{stab}}(V)\) holds by Proposition 34.

By Proposition 29, applied to a local trivialisation of \(\mathcal{C}\), the block \(S(a,y,q)\) admits a Frobenius expansion. We use the conventions of ?? –?? . Thus \(\alpha\) indexes a \(q\)-sector \(q^{r_\alpha}\) (\(r_\alpha\) constant, no two differing by an integer), \(j \in \mathbb{Z}_{\geq 0}\) is an extra power of \(q\), \(\mu \leq M_{\alpha,j}\) is the power of \(\log(q)\) in the slice \(q^{r_\alpha+j}\), \(\beta\) indexes a \(y\)-sector \(y^{h_\beta}\), and \(\ell\) indexes the tail \(\sum_\ell y^{-\ell}\) at \(y=\infty\). We have \[\begin{align} S(a,y,q) &= \sum_{\alpha=1}^K \sum_{j=0}^\infty \sum_{\mu=0}^{M_{\alpha,j}} q^{r_\alpha} \log(q)^\mu\, q^j\, S_{\alpha,\mu,j}(a,y), \tag{46} \\ \text{in which} \quad S_{\alpha,\mu,0}(a,y) &= \sum_{\beta=1}^L \sum_{\nu=0}^{N_{\alpha,\mu,\beta}} y^{h_\beta} \log(y)^\nu \sum_{\ell=0}^\infty C_{\alpha,\beta,\mu,\nu,0,\ell}(a)\, y^{-\ell}. \tag{47} \end{align}\] Each \(C_{\alpha,\beta,\mu,\nu,0,\ell} : V \to \mathbb{C}\) is linear, as in ?? . The subscripts \((\alpha,\mu,0)\) on \(S_{\alpha,\mu,0}\) and the fourth entry \(0\) in \(C_{\alpha,\beta,\mu,\nu,0,\ell}\) record the \(q\)-index \(j=0\) in 46 . When \(\ell=0\) we set \(C_{\alpha,\beta,\mu,\nu,0} := C_{\alpha,\beta,\mu,\nu,0,0}\) for the head coefficient of \(y^{h_\beta}\log(y)^\nu\). For \(j \geq 1\), the coefficient \(S_{\alpha,\mu,j}\) has the same \(y\)-expansion with \(0\) replaced by \(j\). The \(y\)-direction arguments below use the slice \(j=0\); a uniform bound on the integers \(M_{\alpha,j}\) is proved later in Lemma 8.

For \(a,b \in V^{(0)}\) we write \(a *_q b = \mathop{\mathrm{Res}}_z \widetilde{\wp}_1(z,q)\,Y[a,z]b\,dz\). The Ward identities 24 and 25 become \[\begin{align} S(h *_q a,y,q) &= y\frac{d}{dy}S(a,y,q), \tag{48} \\ S((\omega-\tfrac{c}{24}\left|0\right\rangle) *_q a,y,q) &= q\frac{d}{dq}S(a,y,q). \tag{49} \end{align}\]

The first step is to show that the head coefficients descend to linear functionals on the stable Zhu algebra.

We next verify two head-coefficient identities used in the exhaustion arguments below (in the \(j=0\) slice of 46 ; here \(N_{\alpha,\mu,\beta}\) is the top \(\log(y)\)-power in the sector \(y^{h_\beta}\) of \(S_{\alpha,\mu,0}\), and \(M_{\alpha,0}\) is the top \(\log(q)\)-power in \(q^{r_\alpha}\)): \[\begin{align} C_{\alpha,\beta,\mu,\nu,0}\bigl((h-h_{\beta})*a\bigr) &= 0 && \text{for \nu=N_{\alpha,\mu,\beta},} \tag{53} \\ C_{\alpha,\beta,M_{\alpha,0},\nu,0}\bigl((\omega-\tfrac{c}{24}-r_{\alpha})*a\bigr) &= 0 && \text{for each \nu \geq 0.} \tag{54} \end{align}\] Identity 54 is the \(q\)-identity of [1]; 53 is the analogous \(y\)-identity from the current \(h\) and 48 . At \(q=0\), \[\widetilde{\wp}_1(z,q=0) = 2\pi i\, \frac{e^{2\pi i z}}{e^{2\pi i z}-1},\] so the constant term in \(q\) of \(h*_q a\) is \(h*a\).

Proof of 53 . Fix \(\alpha\), \(\mu\), and \(\beta\), and set \(\nu=N_{\alpha,\mu,\beta}\). Then \(C_{\alpha,\beta,\mu,\nu+1,0}\equiv 0\) on the sector \(y^{h_\beta}\), by maximality of \(\nu\). Since 48 involves only \(y\partial_y\), it restricts to each slice \(S_{\alpha,\mu,0}\) in 47 . Substituting into 48 and comparing the head coefficient of \(y^{h_\beta}(\log y)^\nu\), the left-hand side gives \(C_{\alpha,\beta,\mu,\nu,0}(h*a)\). On the right-hand side, only \((\log y)^\nu\) and \((\log y)^{\nu+1}\) contribute, since \(y\partial_y(\log y)^\nu\) contains \((\log y)^{\nu-1}\) and therefore lies in the comparison at \(\nu-1\); the \((\log y)^{\nu+1}\) term vanishes by maximality of \(\nu\). Thus \[C_{\alpha,\beta,\mu,\nu,0}(h*a)=h_\beta\,C_{\alpha,\beta,\mu,\nu,0}(a),\] which is 53 .

Proof of 54 . Fix \(\alpha\), \(\beta\), and \(\nu \geq 0\). Since 49 involves only \(q\partial_q\), it restricts to the sector \(q^{r_\alpha}\) in the \(j=0\) slice, and \(\nu\) may be arbitrary in 47 . Applying \(q\partial_q\) to this sector as in 44 , only the top power \(\log(q)^{M_{\alpha,0}}\) contributes to the coefficient of \(q^{r_\alpha}(\log q)^{M_{\alpha,0}}\) on the right of 49 ; comparing with the left-hand side yields 54 .

The following lemma is the common exhaustion step; it will be applied separately in the \(y\)- and \(q\)-directions.

We now treat the \(y\)-expansion of the leading \(q\)-coefficient \(S_{\alpha,\mu,0}\) in 46 . Fix \(\alpha\) and \(\mu\) and decompose \[\label{eq:T46logy46decomp} S_{\alpha,\mu,0}(a,y)=\sum_{\nu\geq 0}\log(y)^\nu T_{\alpha,\mu,\nu}(a,y).\tag{55}\] Here \(T_{\alpha,\mu,\nu}\) is the coefficient of \((\log y)^\nu\) in the slice \(S_{\alpha,\mu,0}\) (the subscript \(\nu\) indexes \(\log y\), whereas the subscript \(0\) on \(S_{\alpha,\mu,0}\) records \(j=0\) in 46 ). Set \[\nu_{\max} := \max\bigl\{\nu \in \mathbb{Z}_{\geq 0} \mid \text{some head coefficient C_{\alpha,\beta,\mu,\nu,0} is nonzero}\bigr\},\] with the convention that if no such \(\nu\) exists, then \(\nu_{\max}=0\) and \(T_{\alpha,\mu,\nu_{\max}}=0\).

We turn to the \(q\)-expansion. Write \[S(a,y,q)=\sum_{\mu=0}^\infty \log(q)^\mu S^{(q)}_\mu(a,y,q),\] and for an irreducible stable \(V\)-module \(W\) set \[S_W(a,y,q):=\mathop{\mathrm{Tr}}_W a_0 y^{h_0} q^{L_0-c/24}.\]

8 Modularity of the connection↩︎

In Section 2.2 we introduced a family \(T \rightarrow B\) of pairs \((E, \mathcal{L})_{(\alpha, \tau)}\) consisting of \(E = E_\tau\) a smooth elliptic curve and \(\mathcal{L}= \mathcal{L}_\alpha\) a holomorphic line bundle over \(E\) of degree \(0\). In Section 3.2 we described the bundle \(\mathcal{C}\) of charged conformal blocks over \(B\), associated with a charged conformal vertex algebra \((V, \omega, h)\) and the connection \(\nabla\) on \(\mathcal{C}\) corresponding to the differential equations 24 and 25 .

In Sections 4 and 6 we saw that, for suitable \(V\), the pair \((\mathcal{C}, \nabla)\) is an integrable connection in the domain \[\begin{align} X = \{(\alpha, \tau) \in B \mid N \alpha\notin \mathbb{Z}+ \mathbb{Z}\tau\} \subset B \end{align}\] (where \(N\) is a positive integer which depends on \(V\)), and the trace function \(S_{M}(u, \alpha, \tau)\) associated with a stable \(V\)-module is convergent for \((\alpha, \tau)\) in a certain domain \(Y \subset X\), and converges in \(Y\) to a flat section of \((\mathcal{C}, \nabla)\). Explicitly \[\begin{align} Y = \{(\alpha, \tau) \in B \mid -\mathop{\mathrm{Im}}(\tau) < N \mathop{\mathrm{Im}}(\alpha) < 0 \} \subset X. \end{align}\] Furthermore, if \(V\) is stably rational, the finite set of trace functions \(S_M\) associated with irreducible stable \(V\)-modules forms a basis of the space of flat sections of \((\mathcal{C}, \nabla)\) on \(Y\).

Although the connection \(\nabla\) develops singularities on the lattice \(N \alpha\in \mathbb{Z}+ \mathbb{Z}\tau\), we now prove that these singularities are regular. The proof is straightforward, and just requires some careful book keeping with respect to pole orders.

The spaces \(B\) and \(T\) carry compatible actions of the groups \(SL_2(\mathbb{Z})\) and \(\mathbb{Z}^2\), and thus of their semidirect product the Jacobi group \(J = SL_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\). The fibres over points in the same \(J\)-orbit are isomorphic. These actions are well known and are as follows. For \(A = \left(\begin{smallmatrix} a & b \\ c & d \\ \end{smallmatrix}\right) \in SL_2(\mathbb{Z})\) and \((\alpha, \tau) \in B\) we have \[A \cdot (\alpha, \tau) = \left( \frac{\alpha}{c\tau+d}, \frac{a\tau+b}{c\tau+d} \right).\] Writing \((\alpha', \tau') = A \cdot (\alpha, \tau)\), the action of \(A\) on \(T\) is \[A \cdot (\alpha, \tau, v, z) = \left( \alpha', \tau', e^{2\pi i \frac{c}{c\tau+d} \alpha z} v, \frac{z}{c\tau+d} \right).\] For \((m, n) \in \mathbb{Z}^2\) the action is \[\begin{align} (m, n) \cdot (\alpha, \tau, v, z) = (\alpha+ m\tau + n, \tau, e^{-2\pi i m z} v, z). \end{align}\] It is straightforward to verify that \[(E, \mathcal{L})_{A \cdot (\alpha, \tau)} \cong (E, \mathcal{L})_{(\alpha, \tau)} \quad \text{and} \quad (E, \mathcal{L})_{(m, n) \cdot (\alpha, \tau)} \cong (E, \mathcal{L})_{(\alpha, \tau)},\] for all \(A \in SL_2(\mathbb{Z})\) and \((m, n) \in \mathbb{Z}^2\).

The main result of this section is that the connection \(\nabla\) is \(J\)-equivariant, and that flat sections of \((\mathcal{C}, \nabla)\) transform as Jacobi forms of a certain index \(\kappa\) which is determined by the charged conformal vertex algebra \((V, \omega, h)\). In particular if \(u \in V\) is primary of conformal weight \(k\), then the flat sections \(S_M(u, \alpha, \tau)\) of \(\mathcal{C}\) transform as a vector-valued Jacobi form of weight \(k\) and \(\kappa\) is the index defined as in Section 3.1, namely \(h_{1}h = 2\kappa \left|0\right\rangle\). The precise statement is now as follows.

The proof of this proposition is essentially by direct calculation, given below in Appendices 10.1 and 10.2. We now indicate the idea behind the calculations. Let \(A \in SL_2(\mathbb{Z})\) and let us consider the isomorphism \(f : (E, \mathcal{L})_{(\alpha, \tau)} \rightarrow (E, \mathcal{L})_{A \cdot (\alpha, \tau)}\) given by \[f(v, z) = \left( e^{2\pi i \frac{c}{c\tau+d} \alpha z} v, \frac{z}{c\tau+d} \right).\] This isomorphism induces a nontrivial identification between the fibres \(\mathcal{V}_0\) on the two curves \(E_{\tau}\) and \(E_{A \cdot \tau}\), as per the associated bundle construction described in Section 3.2. Explicitly, for each \(u \in V\), we have in \(\mathcal{V}_0\) the equality \[\left( e^{2\pi i \frac{c\alpha}{c\tau+d} z} v, \frac{z}{c\tau+d}, u \right) = \left( v, z, e^{2\pi i \frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} u \right).\]

Let us now consider a flat local section \(S(u, \alpha, \tau)\) of \((\mathcal{C}, \nabla)\), more precisely \(S((v, z, u), \alpha, \tau)\). We define a new section \(S'((v, z, u), \alpha, \tau)\) by \[\begin{align} \label{eq:mod46xform} \begin{aligned} S'((v', z', u), \alpha', \tau') &= e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v', z', u), \alpha, \tau) \\ &= e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}}(c\tau+d)^{L_{[0]}} u), \alpha, \tau), \end{aligned} \end{align}\tag{58}\] where \((\alpha', \tau', v', z') = A \cdot (\alpha, \tau, v, z)\). The proof of Proposition [prop:connec46is46modular] consists in using 24 and 25 to prove the following: \[\begin{align} \frac{\partial}{\partial\alpha'} S((v', z', u), \alpha', \tau') = S((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau) h[t] u), \alpha', \tau' ) \tag{59} \\ \text{and} \quad 2\pi i \frac{\partial}{\partial\tau'} S((v', z', u), \alpha', \tau') = S((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau) L[t] u), \alpha', \tau' ). \tag{60} \end{align}\] The verification is straightforward but lengthy and we relegate it to Appendix 10.1.

Similarly, for a flat local section \(S((v, z, u), \alpha, \tau)\) of \((\mathcal{C}, \nabla)\) and \((m, n) \in \mathbb{Z}^2\), we define a new section \(S'((v, z, u), \alpha, \tau)\) by \[\begin{align} \label{eq:ell46xform} \begin{aligned} S'((v', z', u), \alpha', \tau') &= e^{ -2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v', z', u), \alpha, \tau) \\ &= e^{ -2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, e^{-2\pi i m h_{[1]}} u), \alpha, \tau) \end{aligned} \end{align}\tag{61}\] where \((\alpha', \tau', v', z') = (m, n) \cdot (\alpha, \tau, v, z)\). The proof of Proposition [prop:connec46is46elliptic] consists again in using 24 and 25 to prove 59 and 60 . The computations are significantly easier in this case, and are given in Appendix 10.2.

Using the action of \(SL_2(\mathbb{Z})\) on \(X\), we now show that the trace functions can be analytically continued along paths from \(Y\) to any point of \(X\).

In fact for almost all points we can take \(k=1\), and for the remainder \(k=2\) suffices.

Let \((E, \mathcal{L})\) be the fibre of \(T \rightarrow B\) over a point \((\alpha, \tau) \in X\). Since, by Lemma 9, \((\alpha, \tau)\) has an image \((\alpha', \tau') \in Y\), we can conclude that charged conformal blocks on \((E, \mathcal{L})\) are expressible in terms of trace functions on stable \(V\)-modules. More precisely we have the following corollary.

We now show the consequences of these results for the example of simple affine vertex algebras at admissible level. As before we consider the charged conformal vertex algebra \(V = (L_k(\mathfrak{g}), \omega, \rho)\), as in Proposition 19 where we showed that \(V\) is stably quasi-lisse, and Proposition 35, where we showed that \(V\) is stably rational also. By Proposition 35 and the main result of [24], we see that the category of stable \(V\)-modules is semisimple, with simple objects precisely the admissible \(\widehat\mathfrak{g}\)-modules \(L_k(\lambda)\) as \(\lambda\) ranges over \(\mathrm{Prin}^k\) the set of principal/coprincipal admissible weights of level \(k\).

We now obtain the following as a consequence of our main results. We write \(L_k(\mathfrak{g})\) for \((L_k(\mathfrak{g}), \omega, \rho)\).

Here \(P\) and \(\check{P}\) are the weight and coweight lattices of \(\mathfrak{g}\), and \(p,u\) are as in the definition of admissible level \(k=-h^\vee+p/u\) above. We recall that \(\nu\in P\) is dominant integral if \(\left<\nu,\alpha^\vee\right>\in\mathbb{Z}_{\geq 0}\) for all simple roots \(\alpha\), and is regular if \(\left<\lambda,\alpha^\vee\right>\neq 0\) for all such \(\alpha\). With this convention we set \[\begin{align} P_+^{p}:=\{\nu \in P\mid{}& \text{\nu is dominant integral and (\nu,\theta)=p}\}. \end{align}\] and \(P_+^{p,\operatorname{reg}}\) its subset of regular weights. If \(k\) is coprincipal, we write \({}^{L}\check{P}_+^{p,\operatorname{reg}}\) for the analogous set of regular dominant integral coweights for the Langlands dual root system \({}^{L}\Delta\). The parametrisation of admissible weights [5] yields bijections \(\operatorname{Adm}^k\cong (\check{P}/u\check{P})\times P_+^{p,\operatorname{reg}}\) in the principal case [84], and \(\operatorname{Adm}^k\cong (\check{P}/uP)\times {}^{L}\check{P}_+^{p,\operatorname{reg}}\) in the coprincipal case [24].

In particular the index is generally fractional and can be either positive or negative, according to the sign of the level \(k\).

9 Examples of MLDEs↩︎

We have discussed a number of examples already throughout the text, in particular simple affine vertex algebras at admissible levels, and quasi-lisse \(W\)-algebras obtained as quantum Drinfeld-Sokolov reductions. We have not explicitly discussed modular linear differential equations (MLDEs), though there is interest in determining them explicitly in examples. In this section we briefly examine the explicit form of the connection \(\nabla\) on \(\mathcal{C}\) and the associated MLDEs in two simple cases.

9.1 The rational vertex algebra \(L_1(\mathfrak{sl}_2)\)↩︎

We take \(V = L_1(\mathfrak{sl}_2)\) with \(\omega\) the Sugawara conformal vector and \(h = \rho = \alpha/2\) the Weyl vector. In particular \([\rho, E] = E\) and \([\rho, F] = -F\). The central charge of \((V, \omega, \rho)\) is \(c=1\) and the index is \(\kappa = 1/4\). The vertex algebra \(V\) is rational and has two irreducible modules \(W^0 = V\) and another which we denote \(W^1\). The normalised characters of these \(V\)-modules, defined by \[\chi_i(\alpha, \tau) = \mathop{\mathrm{Tr}}_{W^i} y^{h_0} q^{L_0-c/24},\] are given by the formulas \(\chi_i(\alpha, \tau) = \theta_{i0}(\alpha, \tau) / \eta(\tau)\) where [62] \[\begin{align} \theta_{i0}(\alpha, \tau) = \sum_{n \in \mathbb{Z}} q^{(n+i/2)^2/2} e^{2\pi i (n+i/2)z}. \end{align}\] These characters are the restrictions to \(u=\left|0\right\rangle\) of two charged conformal blocks \(S_{W^i}(u, y, q)\).

On the other hand let \(S(u, y, q)\) be a charged conformal block for \((V, \omega, h)\), and write \(S_{0}\) and \(S_1\) for its restrictions to \(u = \left|0\right\rangle\) and \(u = h\), respectively. In this example \(V^{(0)} / C(V)\) is \(2\) dimensional, spanned by \(\left|0\right\rangle\) and \(h\). The equations 24 and 25 therefore reduce to a \(2 \times 2\) holonomic system on \(S_{0}\) and \(S_1\). We find in fact that \[\begin{align} \label{eq:sl246y46eq} y \frac{\partial}{\partial y} \left[ \begin{array}{c} S_0 \\ S_1 \\ \end{array} \right] = \left[ \begin{array}{cc} 0 & 1 \\ \tfrac{1}{16}(4Q_2-4Q_1^2-E_2) & Q_1 \\ \end{array} \right] \left[ \begin{array}{c} S_0 \\ S_1 \\ \end{array} \right] \end{align}\tag{63}\] and \[\begin{align} \label{eq:sl246q46eq} q \frac{\partial}{\partial q} \left[ \begin{array}{c} S_0 \\ S_1 \\ \end{array} \right] = \left[ \begin{array}{cc} \tfrac{1}{48}(12Q_2-12Q_1^2-E_2) & -Q_1 \\ \tfrac{1}{16}(4Q_1^3-12Q_1Q_2+8Q_3-E_2Q_1) & \tfrac{1}{48}(36Q_1^2-36Q_2+E_2) \\ \end{array} \right] \left[ \begin{array}{c} S_0 \\ S_1 \\ \end{array} \right] \end{align}\tag{64}\] To avoid cumbersome factors of \(2\pi i\) in these equations, we have set \[Q_k(y, q) = \frac{1}{2}\delta_{k,1} + (2\pi i)^{-k} P^+_k(y, q),\] where \(P^+_k(y,q)\) is as in 5 with \(w=y\) (a series in \(\mathbb{C}[y][[q]]\), not the three-variable series \(P_1^+(z,y,q)\) of 10 ). Here \(E_k(q) = -(2\pi i)^{-k}(k!/B_k)\) is the Eisenstein series normalised with constant coefficient \(1\).

In particular we deduce from 63 the following MLDE in \(y\) for \(u(y, q) = S_0(\alpha, \tau)\): \[\begin{align} \label{eq:MLDE46level461} \begin{aligned} \left(y\frac{\partial}{\partial y}\right)^2 u - Q_1 \left(y\frac{\partial}{\partial y}\right) u &-\left(\frac{1}{4} Q_2 - \frac{1}{4} Q_1^2 - \frac{1}{16} E_2 \right) u = 0. \end{aligned} \end{align}\tag{65}\]

9.2 The admissible level vertex algebra \(L_{-4/3}(\mathfrak{sl}_2)\)↩︎

We take \(V = L_{-4/3}(\mathfrak{sl}_2)\) with \(\omega\) and \(h\) as above. The central charge of \((V, \omega, \rho)\) is \(c=-6\) and the index is \(\kappa = -1/3\). This vertex algebra is stably rational with three stable modules \(L_k(\lambda)\), where \(\lambda\) takes the values \(0\), \(- \frac{2}{3}\varpi\) and \(- \frac{4}{3}\varpi\), where \(\varpi = \alpha/2\) is the fundamental weight. (In fact \(\rho = \varpi\) since \(\mathfrak{g}= \mathfrak{sl}_2\).)

The characters of these modules are well known from the work of Kac and Wakimoto [5], [85]. To express them easily we may introduce the theta functions \[\begin{align} \theta_j(y, q) = \sum_{n \in \mathbb{Z}} (-1)^n y^{n+1/2-j/3} q^{(n+1/2-j/3)^2/2}, \qquad j=0,1,2. \end{align}\] Then the three normalised characters are \(\chi_i(y,q) = {\theta_i(y, q^3)}/{\theta_0(y,q)}\).

In this example \(V^{(0)} / C(V)\) is \(3\)-dimensional, spanned by the images of the vectors \(\left|0\right\rangle\), \(h\) and \(h_{(-1)}h\). If we write \(S_i(\alpha, \tau)\) for \(i=0,1,2\) for the evaluations of a charged conformal block \(S(u, \alpha, \tau)\) on \(\left|0\right\rangle\), \(h\) and \(h_{(-1)}h\), respectively, then the equations 24 and 25 reduce to a \(3 \times 3\) holonomic system on \((S_0, S_1, S_2)\). The explicit form for this system is quite tedious to write out in full, but we obtain from it the following MLDE in \(y\) for \(u(y, q) = S_0(\alpha, \tau)\): \[\begin{align} \label{eq:MLDE46level46-4473} \begin{aligned} \left(y\frac{\partial}{\partial y}\right)^3 u - 2Q_1 \left(y\frac{\partial}{\partial y}\right)^2 u &-\left(\frac{10}{3} Q_2 - \frac{4}{3} Q_1^2 + \frac{1}{9} E_2 \right) \left(y\frac{\partial}{\partial y}\right) u \\ &+ \left(\frac{2}{27} E_2 Q_1 - \frac{52}{27} Q_3 + \frac{20}{9} Q_1 Q_2 - \frac{8}{27} Q_1^3 \right) u = 0. \end{aligned} \end{align}\tag{66}\] The three normalised characters \(\chi_i(\alpha, \tau)\) satisfy this MLDE.

10 Verification of invariance under the Jacobi group↩︎

Before beginning the calculations, we record some commutator relations that will be used.

Applying Lemma 10 to \(A = h_{[1]}\) and \(B = L_{([m])}\) and using the relations \[\begin{align} \mathop{\mathrm{ad}}_{h_{[1]}}(L_{([m])}) &= h_m \\ \mathop{\mathrm{ad}}_{h_{[1]}}^2(L_{([m])}) &= \mathop{\mathrm{ad}}_{h_{[1]}}(h_{[m]}) = \delta_{m,-1} \left<h, h\right> = \delta_{m,-1} \left<h, h\right> \\ \mathop{\mathrm{ad}}_{h_{[1]}}^{\geq 3}(L_{([m])}) &= 0, \end{align}\] yields \[\begin{align} L_{([m])} h_{[1]}^k = h_{[1]}^k L_{([m])} - k h_{[1]}^{k-1} h_{[m]} + \delta_{m,-1} \binom{k}{2} \left<h, h\right> h_{[1]}^{k-2}. \end{align}\] From this, in turn, we obtain \[\begin{align} \sum_{k \in \mathbb{Z}_+} \frac{\lambda^k}{k!} L_{([m])} h_{[1]}^k &= \sum_{k \in \mathbb{Z}_+} \frac{\lambda^k}{k!} h_{[1]}^k L_{([m])} - \sum_{k \in \mathbb{Z}_+} \frac{\lambda^k}{k!} k h_{[1]}^{k-1} h_{[m]} + \delta_{m,-1} \sum_{k \in \mathbb{Z}_+} \frac{\lambda^k}{k!} \binom{k}{2} \left<h, h\right> h_{[1]}^{k-2} \\ &= e^{\lambda h_{[1]}} L_{([m])} - \lambda e^{\lambda h_{[1]}} h_{[m]} + \delta_{m,-1} \lambda^2\frac{\left<h, h\right>}{2} e^{\lambda h_{[1]}}. \end{align}\] Rewriting in terms of the field \(L[t] = \sum_m L_{([m])} t^{-m-1}\) we get \[\begin{align} L[t] e^{\lambda h_{[1]}} &= \sum_{m \in \mathbb{Z}} L_{([m])} e^{\lambda h_{[1]}} t^{-m-1} \nonumber \\ &= e^{\lambda h_{[1]}} \sum_{m \in \mathbb{Z}} \left( L_{([m])} - \lambda h_{[m]} + \delta_{m,-1} \lambda^2\frac{\left<h, h\right> }{2} \right) t^{-m-1} \nonumber \\ &= e^{\lambda h_{[1]}} L[t] - \lambda e^{\lambda h_{[1]}} h[t] + \lambda^2\frac{\left<h, h\right> }{2}e^{\lambda h_{[1]}}. \label{eq:L[t].exp.comm} \end{align}\tag{67}\]

Applying Lemma 10 to \(A = h_{[1]}\) and \(B = h_{[m]}\) and using the relations \[\begin{align} \mathop{\mathrm{ad}}_{h_{[1]}}(h_{[m]}) &= \delta_{m,-1} \left<h, h\right> \\ \mathop{\mathrm{ad}}_{h_{[1]}}^{\geq 2}(h_{[m]}) &= 0, \end{align}\] yields \[\begin{align} h_{[m]} h_{[1]}^k = h_{[1]}^k h_{[m]} - \delta_{m,-1} k \left<h, h\right> h_{[1]}^{k-1}. \end{align}\] we can say that \[\begin{align} h[t] e^{ \lambda h_{[1]} } &= e^{ \lambda h_{[1]} } h[t] - \left<h, h\right> \sum_{k=0}^\infty \frac{1}{(k-1)!} \lambda^{k} h_{[1]}^{k-1} \nonumber \\ &= e^{ \lambda h_{[1]} } h[t] - \lambda\left<h, h\right> e^{ \lambda h_{[1]} }. \label{eq:h[t].exp.comm} \end{align}\tag{68}\]

10.1 Verification of invariance under modular transformations↩︎

Let \(A \in SL_2(\mathbb{Z})\) and \((\alpha', \tau') = A \cdot (\alpha, \tau)\) as in Section 8. The partial derivatives in \(\tau\) and \(\alpha\) transform as follows: If \((\alpha', \tau') = A(\alpha, \tau)\) then \[\begin{align} \frac{\partial}{\partial \alpha'} &= (c\tau+d) \frac{\partial}{\partial \alpha} \\ \frac{\partial}{\partial \tau'} &= (c\tau+d)^2 \frac{\partial}{\partial \tau} + c\alpha(c\tau+d) \frac{\partial}{\partial \alpha} \end{align}\]

10.1.1 Modular invariance of the \(y\)-equation↩︎

Let \(S((v, z, u), \alpha, \tau)\) be a flat section of \(\mathcal{C}\) and let \(S'((v', z', u), \alpha', \tau')\) be given by equation 58 . We need to show 59 . We compute \[\begin{align} & \frac{\partial}{\partial\alpha'}S'((v', z', u), \alpha', \tau') \\ = {} & (c\tau+d) \frac{\partial}{\partial\alpha}S'((v', z', u), \alpha', \tau') \\ = {} & (c\tau+d) \frac{\partial}{\partial\alpha}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}}(c\tau+d)^{L_{[0]}} u), \alpha, \tau)\right) \\ = {} & (c\tau+d) \frac{\partial}{\partial\alpha}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u} \sum_{j=0}^\infty \frac{1}{j!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j S((v, z, h_{[1]}^j u), \alpha, \tau)\right) \\ = {} & (c\tau+d) \sum_{j=0}^\infty \frac{1}{j!} \frac{\partial}{\partial\alpha}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j \right) S((v, z, h_{[1]}^j u), \alpha, \tau) \tag{69} \\ & + e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+1} \sum_{j=0}^\infty \frac{1}{j!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j \frac{\partial}{\partial\alpha} S((v, z, h_{[1]}^j u), \alpha, \tau). \tag{70} \end{align}\] We simplify 69 first and then return to 70 later. We have \[\begin{align} \text{\eqref{eqn:y46D}} = {} & (c\tau+d)^{\nabla_u+1} \sum_{j=0}^\infty \frac{1}{j!} \frac{d}{d\alpha}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j \right) S((v, z, h_{[1]}^j u), \alpha, \tau) \\ = {} & (c\tau+d)^{\nabla_u+1} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{j!} \frac{d}{d\alpha}\left( 2\pi i\frac{c\alpha}{c\tau+d} \right)^j S((v, z, h_{[1]}^j u), \alpha, \tau) \tag{71} \\ &+ (c\tau+d)^{\nabla_u+1} \frac{d}{d\alpha} (e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}) S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]} } u), \alpha, \tau). \tag{72} \end{align}\] We have \[\begin{align} \text{\eqref{eqn:y46D2}} = {} & (c\tau+d) 2\pi i \kappa \frac{c}{c\tau+d} (2\alpha) e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]} } (c\tau+d)^{L_{[0]}} u), \alpha, \tau) \\ = {} & 2\pi i \kappa {c} (2\alpha) S'((v', z', u), \alpha', \tau'). \end{align}\] And we have \[\begin{align} \text{\eqref{eqn:y46D1}} = {} & (c\tau+d)^{\nabla_u+1} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{j!} \frac{2\pi i c}{c\tau+d} j (2\pi i\frac{c\alpha}{c\tau+d})^{j-1} S((v, z, h_{[1]}^j u), \alpha, \tau) \\ = {} & {2\pi i c} (c\tau+d)^{\nabla_u} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{(j-1)!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^{j-1} S((v, z, h_{[1]}^j u), \alpha, \tau) \\ = {} & {2\pi i c} (c\tau+d)^{\nabla_u} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]} } h_{[1]} u), \alpha, \tau) \\ = {} & {2\pi i c} (c\tau+d) e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]} }(c\tau+d)^{L_{[0]}} h_{[1]} u), \alpha, \tau) \\ = {} & {2\pi i c} (c\tau+d) S'((v', z', h_{[1]} u), \alpha', \tau'). \end{align}\]

Now we turn to 70 , rewriting it using 24 as \[\begin{align} \text{\eqref{eqn:y46C}} = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+1} \sum_{j=0}^\infty \frac{1}{j!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) h[t] h_{[1]}^j u), \alpha, \tau) \\ = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+1} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) h[t] e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} u), \alpha, \tau) \end{align}\] To proceed further we commute \(h[t]\) with \(e^{h_{[1]}}\) using equation 68 (with \(\lambda= 2\pi i\frac{c\alpha}{c\tau+d}\)). We have \[\begin{align} \text{\eqref{eqn:y46C}} = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+1} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) h[t] e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} u), \alpha, \tau) \\ = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+1} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h[t] u), \alpha, \tau) \tag{73} \\ &- 2\pi i\frac{c\alpha\left<h, h\right>}{c\tau+d} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+1} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau)e^{ 2\pi i\frac{c\alpha}{c\tau+d} h_{([1])}} u), \alpha, \tau). \tag{74} \end{align}\] We continue \[\begin{align} \text{\eqref{eqn:y46C2}} = {} & -2\pi i{c\alpha\left<h, h\right>} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u} S((v, z, e^{ 2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} u), \alpha, \tau) \\ = {} & -2\pi i{c\alpha\left<h, h\right>} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{ 2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} u), \alpha, \tau) \\ = {} & -2\pi i{c\alpha\left<h, h\right>} S'((v', z', u), \alpha', \tau'). \end{align}\] Before simplifying 73 we recall that \(\wp_1(t, \tau) = \sum_{i=0}^\infty P^{i+1}(\tau) t^{i}\), where for all \(k\) we have \[P^k(\tau') = (c\tau+d)^k P^k(\tau),\] except when \(k=2\) in which case \(P^2(\tau) = -G_2(\tau)\) and \[G_2(\tau') = (c\tau+d)^2 G_2(\tau) - 2\pi i c (c\tau+d).\] We apply this in the following simplifications: \[\begin{align} \text{\eqref{eqn:y46C1}} = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u+1} \sum_{i} P^{i+1}(\tau) S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h_{[i]}u), \alpha, \tau) \\ = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u+1} \sum_{i} (c\tau+d)^{-i-1} P^{i+1}(\tau') S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h_{[i]}u), \alpha, \tau) \\ &- e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u+1} 2\pi i c (c\tau+d)^{-1} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h_{[1]}u), \alpha, \tau) \\ = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{i} P^{i+1}(\tau') S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} h_{[i]}u), \alpha, \tau) \\ &- 2\pi i c (c\tau+d) e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} h_{[1]}u), \alpha, \tau) \\ = {} & \sum_{i} P^{i+1}(\tau') S'((v', z', h_{[i]}u), \alpha', \tau') - 2\pi i c (c\tau+d) S'((v', z', h_{[1]}u), \alpha', \tau') \\ = {} & S'((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau') h[t] u), \alpha', \tau') \tag{75} \\ &- 2\pi i c (c\tau+d) S'((v', z', h_{[1]}u), \alpha', \tau'). \tag{76} \end{align}\] It remains to check that 76 + 74 + 71 + 72 = 0. But 76 + 71 = 0, and once we set \[\kappa = \frac{\left<h, h\right>}{2}\] we get 74 + 72 = 0.

10.1.2 Modular invariance of the \(q\)-equation↩︎

Let \(S((v, z, u), \alpha, \tau)\) be a flat section of \(\mathcal{C}\) and let \(S'((v', z', u), \alpha', \tau')\) be given by equation 58 . We need to show 60 . We compute \[\begin{align} 2\pi i\frac{\partial}{\partial\tau'}S'((v', z', u), \alpha', \tau') = {} & 2\pi i(c\tau+d)^2 \frac{\partial}{\partial\tau}S'((v', z', u), \alpha', \tau') \tag{77} \\ &+ 2\pi i c\alpha(c\tau+d) \frac{\partial}{\partial\alpha} S'((v', z', u), \alpha', \tau'). \tag{78} \end{align}\] Taking advantage of the calculations done above in Section 10.1.1, we have \[\begin{align} \text{\eqref{eqn:I2}} = 2\pi i c\alpha S'((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau') h[t] u), \alpha', \tau'). \end{align}\] We now analyse 77 . \[\begin{align} \text{\eqref{eqn:I1}} = {} & 2\pi i(c\tau+d)^2 \frac{\partial}{\partial\tau}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}}(c\tau+d)^{L_{[0]}} u), \alpha, \tau)\right) \\ = {} & (2\pi i)(c\tau+d)^2 \frac{\partial}{\partial\tau}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u} \sum_{j=0}^\infty \frac{1}{j!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j S((v, z, h_{[1]}^j u), \alpha, \tau)\right) \\ = {} & (2\pi i)(c\tau+d)^2 \sum_{j=0}^\infty \frac{1}{j!} \frac{\partial}{\partial\tau}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j \right) S((v, z, h_{[1]}^j u), \alpha, \tau) \tag{79} \\ &+ (2\pi i)e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+2} \sum_{j=0}^\infty \frac{1}{j!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j \frac{\partial}{\partial\tau} S((v, z, h_{[1]}^j u), \alpha, \tau). \tag{80} \end{align}\] We leave 80 until later, focusing for now on 79 . We have \[\begin{align} \text{\eqref{eqn:D}} = {} & 2\pi i (c\tau+d)^2 \frac{\partial}{\partial \tau}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \right) \sum_{j=0}^\infty \frac{1}{j!} (c\tau+d)^{\nabla_u} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j S((v, z, h_{[1]}^j u), \alpha, \tau) \tag{81} \\ &+ 2\pi i (c\tau+d)^2 \frac{d}{d\tau}\left( (c\tau+d)^{\nabla_u} \right) e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{j!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j S((v, z, h_{[1]}^j u), \alpha, \tau) \tag{82} \\ &+ 2\pi i (c\tau+d)^{\nabla_u+2} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{j!} \frac{\partial}{\partial\tau} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j S((v, z, h_{[1]}^j u), \alpha, \tau). \tag{83} \end{align}\] We simplify each of these three terms in turn. We have \[\begin{align} \text{\eqref{eqn:D1}} = {} & 2\pi i (c\tau+d)^2 \frac{\partial}{\partial\tau}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \right) (c\tau+d)^{\nabla_u} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{([1])} } u), \alpha, \tau) \\ = {} & 2\pi i (c\tau+d)^2 \frac{\partial}{\partial\tau}\left(e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \right) S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{([1])} } (c\tau+d)^{L_{[0]}} u), \alpha, \tau) \\ = {} & -2\pi i (c\tau+d)^2 (2\pi i) \kappa \frac{c^2}{(c\tau+d)^2}\alpha^2 e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{([1])} } (c\tau+d)^{L_{[0]}} u), \alpha, \tau) \\ = {} & -2\pi i (c\tau+d)^2 (2\pi i) \kappa \frac{c^2}{(c\tau+d)^2}\alpha^2 S'((v', z', u), \alpha', \tau') \\ = {} & -(2\pi i c \alpha)^2 \kappa S'((v', z', u), \alpha', \tau'). \end{align}\] Next \[\begin{align} \text{\eqref{eqn:D2}} = {} & 2\pi i (c\tau+d)^2 \frac{d}{d\tau}\left( (c\tau+d)^{\nabla_u} \right) e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{([1])} } u), \alpha, \tau) \\ = {} & 2\pi i c \nabla_u (c\tau+d)^{\nabla_u+1} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{([1])} } u), \alpha, \tau) \\ = {} & 2\pi i c \nabla_u (c\tau+d) e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{([1])} }(c\tau+d)^{L_{[0]}} u), \alpha, \tau) \\ = {} & 2\pi i c \nabla_u (c\tau+d) S'((v', z', u), \alpha', \tau'). \end{align}\] And \[\begin{align} \text{\eqref{eqn:D3}} = {} & 2\pi i (c\tau+d)^{\nabla_u+2} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{j!} \frac{\partial}{\partial\tau}\left( \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j \right) S((v, z, h_{[1]}^j u), \alpha, \tau) \\ = {} & 2\pi i (c\tau+d)^{\nabla_u+2} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{j!} \frac{-2\pi i c^2\alpha}{(c\tau+d)^2} j \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^{j-1} S((v, z, h_{[1]}^j u), \alpha, \tau) \\ = {} & -(2\pi i)^2 c^2\alpha(c\tau+d)^{\nabla_u} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_{j=0}^\infty \frac{1}{(j-1)!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^{j-1} S((v, z, h_{[1]}^j u), \alpha, \tau) \\ = {} & -(2\pi i)^2 c^2\alpha(c\tau+d)^{\nabla_u} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d}h_{[1]}} h_{([1])} u), \alpha, \tau) \\ = {} & -(2\pi i)^2 c^2\alpha(c\tau+d) e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d}h_{[1]}} (c\tau+d)^{L_{[0]}} h_{([1])} u), \alpha, \tau) \\ = {} & -(2\pi i)^2 c^2\alpha(c\tau+d) S'((v', z', h_{[1]} u), \alpha', \tau'). \end{align}\]

We use the DE to simplify 80 , obtaining \[\begin{align} \text{\eqref{eqn:C}} = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+2} \sum_{j=0}^\infty \frac{1}{j!} \left(2\pi i\frac{c\alpha}{c\tau+d}\right)^j S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) L[t] h_{[1]}^j u), \alpha, \tau) \\ = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+2} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) L[t] e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} u), \alpha, \tau). \end{align}\] To proceed further we need to commute \(L[t]\) with \(e^{h_{[1]}}\), using equation 67 . The term (C) equals \[\begin{align} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+2} S\left(\left(v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{\lambda h_{[1]}} \left( L[t] - \lambda h[t] + \lambda^2\frac{\left<h, h\right> }{2} \right) u\right), \alpha, \tau \right) \end{align}\] (where \(\lambda= 2\pi i\frac{c\alpha}{c\tau+d}\)). Let’s identify the three terms with \(L[t]\), \(- \lambda h[t]\) and \(\lambda^2 \frac{\left<h, h\right>}{2}\) as (C1), (C2) and (C3), respectively. First let’s focus on \[\begin{align} \text{(C1)} = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+2} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} L[t] u), \alpha, \tau) \\ = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u+2} \sum_i P^{i+2}(\tau) S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} L_{[i]}u), \alpha, \tau) \\ = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}\sum_i (c\tau+d)^{\nabla_u-i} P^{i+2}(\tau') S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} L_{[i]}u), \alpha, \tau) \tag{84} \\ &- 2\pi i c e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u+1} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} L_{[0]}u), \alpha, \tau). \tag{85} \end{align}\] Notice that \[\begin{align} \text{\eqref{eqn:C139}} = {} & - 2\pi i c \nabla_u e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d)^{\nabla_u+1} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} u), \alpha, \tau) \\ = {} & - 2\pi i c \nabla_u e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d) S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}}(c\tau+d)^{L_{[0]}} u), \alpha, \tau) \\ = {} & - 2\pi i c \nabla_u (c\tau+d) S'((v', z', u), \alpha', \tau'). \end{align}\] The main term 84 continues as \[\begin{align} \text{\eqref{eqn:C10}} = {} & e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}\sum_i P^{i+2}(\tau') S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} L_{[i]}u), \alpha, \tau) \\ = {} & \sum_i P^{i+2}(\tau') S'((v', z', L_{[i]}u), \alpha', \tau') \\ = {} & S'((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau') L[t] u), \alpha', \tau'). \end{align}\]

Next is \[\begin{align} \text{(C2)} = {} & -2\pi i\frac{c\alpha}{c\tau+d} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+2} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h[t] u), \alpha, \tau) \\ = {} & -2\pi i {c\alpha} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+1} \sum_i P^{i+1}(\tau) S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h_{[i]} u), \alpha, \tau) \\ = {} & -2\pi i {c\alpha} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_i (c\tau+d)^{\nabla_u-i} P^{i+1}(\tau') S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h_{[i]} u), \alpha, \tau) \tag{86} \\ &+ (2\pi i c)^2\alpha e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h_{[1]} u), \alpha, \tau). \tag{87} \end{align}\] Note that \[\begin{align} \text{\eqref{eqn:C239}} = {} & (2\pi i c)^2\alpha e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} (c\tau+d) S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} h_{[1]} u), \alpha, \tau) \\ = {} & (2\pi i c)^2\alpha(c\tau+d) S'((v', z', h_{[1]} u), \alpha', \tau'). \end{align}\] The main term 86 continues as \[\begin{align} \text{\eqref{eqn:C20}} = {} & -2\pi i {c\alpha} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} \sum_i (c\tau+d)^{\nabla_u-i} P^{i+1}(\tau') S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} h_{[i]} u), \alpha, \tau) \\ = {} & -2\pi i {c\alpha} e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau') e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} h[t] u), \alpha, \tau) \\ = {} & -2\pi i {c\alpha} S'((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau') h[t] u), \alpha', \tau'). \end{align}\] Next is (C3) which simplifies as: \[\begin{align} \text{(C3)} = {} & \frac{\left<h,h\right>}{2} (2\pi i\frac{c\alpha}{c\tau+d})^2 e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2}(c\tau+d)^{\nabla_u+2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} u), \alpha, \tau) \\ = {} & \frac{\left<h,h\right>}{2} (2\pi i {c\alpha})^2 e^{2\pi i \kappa \frac{c}{c\tau+d} \alpha^2} S((v, z, e^{2\pi i\frac{c\alpha}{c\tau+d} h_{[1]}} (c\tau+d)^{L_{[0]}} u), \alpha, \tau) \\ = {} & \frac{\left<h,h\right>}{2} (2\pi i {c\alpha})^2 S'((v', z', u), \alpha', \tau'). \end{align}\] Next we note that 78 + (C20) = 0, (D1) + (C3) = 0 (assuming \(\kappa = \left<h,h\right>/2\)), (D2) + (C1’) = 0, (D3) + (C2’) = 0. The only remaining term is (C10), so we have proved 60 at last.

10.2 Verification of invariance under elliptic transformations↩︎

Let \((m, n) \in \mathbb{Z}^2\) and \((\alpha', \tau') = (m, n) \cdot (\alpha, \tau)\) as in Section 8. The partial derivatives in \(\tau\) and \(\alpha\) transform as follows: \[\begin{align} \frac{\partial}{\partial \alpha'} &= \frac{\partial}{\partial \alpha} \\ \frac{\partial}{\partial \tau'} &= \frac{\partial}{\partial \tau} - m \frac{\partial}{\partial \alpha}. \end{align}\]

10.2.1 Elliptic invariance of the \(y\)-equation↩︎

Let \(S((v, z, u), \alpha, \tau)\) be a flat section of \(\mathcal{C}\) and let \(S'((v', z', u), \alpha', \tau')\) be given by equation 61 . We need to show 59 . We compute \[\begin{align} \frac{\partial}{\partial\alpha'}S'((v', z', u), \alpha', \tau') &= \frac{\partial}{\partial\alpha}S'((v', z', u), \alpha', \tau') \\ = {} & \frac{\partial}{\partial\alpha}\left( e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S( ( v, z, e^{2\pi i m h_{[1]}} u), \alpha, \tau) \right) \\ = {} & -2\pi i (2m \kappa) e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S( ( v, z, e^{-2\pi i m h_{[1]}} u), \alpha, \tau) \tag{88} \\ & + e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } \frac{\partial}{\partial\alpha} S( ( v, z, e^{-2\pi i m h_{[1]}} u), \alpha, \tau). \tag{89} \end{align}\]

We simplify 89 , rewriting it using 24 as \[\begin{align} \text{\eqref{eqn:y46C46ell}} = {} & e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) h[t] e^{-2\pi i m h_{[1]}} u), \alpha, \tau). \end{align}\] To proceed further we commute \(h[t]\) with \(e^{h_{[1]}}\) using equation 68 (with \(\lambda= -2\pi i m\)). We have \[\begin{align} \text{\eqref{eqn:y46C46ell}} = {} & e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{-2\pi i m h_{[1]}} h[t] u), \alpha, \tau) \tag{90} \\ &+ 2\pi i{m \left<h, h\right>} e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau)e^{-2\pi i m h_{[1]}} u), \alpha, \tau). \tag{91} \end{align}\] We continue \[\begin{align} \text{\eqref{eqn:y46C246ell}} = {} & 2\pi i{m \left<h, h\right>} e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, e^{-2\pi i m h_{[1]}} u), \alpha, \tau) \\ = {} & 2\pi i{m \left<h, h\right>} S'((v', z', u), \alpha', \tau'). \end{align}\] Since \(\tau' = \tau\) the transformation of \(\wp_1(t, \tau)\) is trivial and we have \[\begin{align} \text{\eqref{eqn:y46C146ell}} = {} & e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{-2\pi i m h_{[1]}} h[t] u), \alpha, \tau) \\ = {} & e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau') h[t] u), \alpha', \tau') \end{align}\] as required.

10.2.2 Elliptic invariance of the \(q\)-equation↩︎

Let \(S((v, z, u), \alpha, \tau)\) be a flat section of \(\mathcal{C}\) and let \(S'((v', z', u), \alpha', \tau')\) be given by equation 61 . We need to show 60 . We compute \[\begin{align} 2\pi i\frac{\partial}{\partial\tau'}S'((v', z', u), \alpha', \tau') = {} & 2\pi i \frac{\partial}{\partial\tau}S((v', z', u), \alpha', \tau') \tag{92} \\ &- 2\pi i m \frac{\partial}{\partial\alpha} S'((v', z', u), \alpha', \tau'). \tag{93} \end{align}\] Taking advantage of the calculations done above in Section 10.2.1, we have \[\begin{align} \text{\eqref{eqn:I246ell}} = -2\pi i m S'((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau') h[t] u), \tau', \alpha' ). \end{align}\] We now analyse 92 .

We compute \[\begin{align} \frac{\partial}{\partial\tau}S'((v', z', u), \alpha', \tau') = {} & \frac{\partial}{\partial\tau}\left( e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S( ( v, z, e^{2\pi i m h_{[1]}} u), \alpha, \tau) \right) \\ = {} & -2\pi i (m^2 \kappa) e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S( ( v, z, e^{-2\pi i m h_{[1]}} u), \alpha, \tau) \tag{94} \\ & + e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } \frac{\partial}{\partial\tau} S( ( v, z, e^{-2\pi i m h_{[1]}} u), \alpha, \tau). \tag{95} \end{align}\]

We simplify 89 , rewriting it using 24 as \[\begin{align} \text{\eqref{eqn:y46C46ell}} = {} & e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) h[t] e^{-2\pi i m h_{[1]}} u), \alpha, \tau). \end{align}\] To proceed further we commute \(L[t]\) with \(e^{h_{[1]}}\) using equation 67 (with \(\lambda= -2\pi i m\)). The term (C) equals \[\begin{align} e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{-2\pi i m h_{[1]}} \left(L[t] + (2\pi i m) h[t] + (2\pi i m)^2\frac{\left<h,h\right>}{2} \right) u), \alpha, \tau). \end{align}\] Let’s identify the three terms with \(L[t]\), \((2\pi i m) h[t]\) and \((2\pi i m)^2\frac{\left<h,h\right>}{2}\) as (C1), (C2) and (C3), respectively. First we have \[\begin{align} \text{(C1)} = {} & e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S((v, z, \mathop{\mathrm{Res}}_t \wp_1(t, \tau) e^{-2\pi i m h_{[1]}} L[t] u), \alpha, \tau) \\ = {} & e^{- 2\pi i \kappa (m^2 \tau + 2m \alpha) } S'((v', z', \mathop{\mathrm{Res}}_t \wp_1(t, \tau') L[t] u), \alpha', \tau'). \end{align}\] Now (C2) cancels (I2) and (C3) cancels (D).

T.A.:
Okinawa Institute of Science and Technology (OIST)
Onna, Kunigami District, Okinawa 904-0495, Japan
tomoyuki.arakawa@oist.jp

J.vE.:
Instituto de Matemática Pura e Aplicada (IMPA)
Jardim Botânico, Rio de Janeiro, RJ 22.460-320, Brazil
jethro@impa.br

H.L:
Okinawa Institute of Science and Technology (OIST)
Onna, Kunigami District, Okinawa 904-0495, Japan
hao-li@oist.jp

References↩︎

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1. Okinawa Institute of Science and Technology (OIST), tomoyuki.arakawa@oist.jp↩︎

2. Instituto de Matemática Pura e Aplicada (IMPA), jethro@impa.br↩︎

3. Okinawa Institute of Science and Technology (OIST), hao-li@oist.jp↩︎