The elliptic version of the (q,t)(q,t)-Knizhnik–Zamolodchikov equation (KZE) satisfied by torus correlators of Ding–Iohara–Miki (DIM) algebra intertwiners has solutions built from the double elliptic gamma function. For the one-point torus correlator the modular transformation of the solution is diagonal (a scalar prefactor), but for higher-point correlators (n≥1n\geq1 extra insertions) the solutions mix under modular transformations through a non-diagonal matrix that has never been computed. The open problem is to compute these modular-transformation matrices exactly and to identify the group they generate; the source paper suggests, from the structure of the double elliptic gamma function, that the modular group acting on the solution space may be enhanced from SL(2,Z)SL(2,\mathbb{Z}) to SL(4,Z)SL(4,\mathbb{Z}).

Background

The Knizhnik–Zamolodchikov equation (KZE) is the system of differential equations satisfied by correlation functions (conformal blocks) of the Wess–Zumino–Witten model of two-dimensional conformal field theory; it expresses how the correlators change when the insertion points move, with the variation governed by the action of an affine Lie algebra. Its qq-deformed analogue (qKZ) is a difference equation associated with quantum affine algebras, and the elliptic (KZB) version involves correlation functions on a torus, where the modular group SL(2,Z)SL(2,\mathbb{Z}) of transformations of the torus modulus acts on the space of solutions.

The Ding–Iohara–Miki (DIM) algebra Uq,t(gl^^1)U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1), also called the quantum toroidal algebra of gl1\mathfrak{gl}_1, is a two-parameter deformation of (a central extension of) the algebra of functions on a double-loop space; it is the symmetry algebra underlying the refined topological string (a two-parameter deformation of the topological string resolving the usual Omega-background into two independent equivariant parameters) and the AGT-type correspondences (after Alday–Gaiotto–Tachikawa, relating supersymmetric gauge theories to conformal field theories and integrable systems) between gauge theories and integrable systems. Its intertwiners — operators mapping between representations, the algebraic counterpart of the refined topological vertex used in refined topological string amplitudes — can be composed into network correlators. Awata, Kanno, Mironov, Morozov, Morozov, Ohkubo and Zenkevich (2017) showed that correlators of DIM intertwiners satisfy a (q,t)(q,t)-deformed KZ equation: a difference equation in the spectral parameters of the intertwiners in which the braiding transformations are performed by the RR-matrix of the DIM algebra (the operator encoding the quasitriangular structure of the algebra). Its solutions are identified, via spectral duality (a duality exchanging the gauge-theory Coulomb-branch and instanton-counting descriptions, under which KZ-type building blocks become Nekrasov functions and vice versa), with building blocks of the Nekrasov partition functions — the exact instanton-counting functions of supersymmetric gauge theories in an Omega-background (a supersymmetry-preserving deformation of flat space that renders the instanton sum finite).

The same authors also constructed an elliptic version of the (q,t)(q,t)-KZ equation, relevant to six-dimensional gauge theories compactified on a torus and to the partition functions of M-strings (BPS strings arising from M2-branes suspended between parallel M5-branes). Its basic solutions are expressed through the double elliptic gamma function G2(x∣q0,q1,q2)G_2(x|q_0,q_1,q_2), the third member of the hierarchy of multiple elliptic gamma functions GrG_r that begins with the Jacobi theta function (G0G_0) and the elliptic gamma function (G1G_1). For the one-point torus correlator (n=0n=0), the modular transformation of the solution is diagonal: it is multiplication by a simple explicit factor, and exact modular invariance can be restored by an antiholomorphic term satisfying a holomorphic anomaly equation (the analogue, in topological string theory, of the statement that holomorphic amplitudes are only quasimodular and become modular after antiholomorphic completion). For higher-point correlators (n≥1n\geq1), however, the Φ\Phi and Φˉ\bar{\Phi} propagators building the solutions are not modular invariant, and the modular transformation mixes the solutions through a non-trivial matrix whose exact form is unknown.

Two precedents motivate the conjecture. First, Felder and Varchenko showed that the elliptic gamma function G1G_1 transforms under a group isomorphic to SL(3,Z)⋉Z3SL(3,\mathbb{Z})\ltimes\mathbb{Z}^3 acting on its three nome parameters, and studied modular and monodromy properties of solutions to the elliptic qKZB equations together with Tarasov. Second, in those elliptic qKZB cases the modular group acting on the solution space is enlarged from SL(2,Z)SL(2,\mathbb{Z}) to SL(3,Z)SL(3,\mathbb{Z}). Since the DIM solutions at n=0n=0 involve G2G_2, whose natural modular group is SL(4,Z)SL(4,\mathbb{Z}) (acting on the four parameters q0,q1,q2q_0,q_1,q_2 and the argument), Awata et al. conjectured that the modular group acting on the solutions of the elliptic (q,t)(q,t)-KZ equation is enhanced beyond SL(2,Z)SL(2,\mathbb{Z}), plausibly to SL(4,Z)SL(4,\mathbb{Z}), and stated that computing the non-diagonal modular matrices for n≥1n\geq1 exactly is an open and very interesting problem.

Problem Statement

For the elliptic (q,t)(q,t)-Knizhnik–Zamolodchikov equation of the Ding–Iohara–Miki algebra Uq,t(gl^^1)U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1), as formulated by Awata, Kanno, Mironov, Morozov, Morozov, Ohkubo and Zenkevich (Phys. Rev. D 96 (2017) 026021, arXiv:1703.06084), compute exactly the modular-transformation matrices that act on the space of solutions for higher-point torus correlators (those beyond the one-point torus block, i.e. containing the Φ\Phi and Φˉ\bar{\Phi} propagators that are not modular invariant; here nn counts the extra insertions beyond the one-point block, so the one-point torus block is n=0n=0 and higher-point correlators are n≥1n\geq1, the one-point torus block being n=1n=1 in Section 4 of the source paper's indexing by the number of intertwiner pairs), where the transformation is known to be non-diagonal and "no longer given by a simple overall factor". The open problem is to give closed-form expressions for these matrices in an explicitly specified basis of the solution space, in terms of the parameters of the theory (the deformation parameters qq and tt, the elliptic nome Q=e2πiτQ=e^{2\pi i\tau} with τ\tau the torus modulus, and the spectral parameters and representation labels of the inserted intertwiners), and to identify the group generated by these matrices, in relation to the enhancement from SL(2,Z)SL(2,\mathbb{Z}) to SL(4,Z)SL(4,\mathbb{Z}) suggested in the source by the double elliptic gamma function structure of the solutions; the n=0n=0 case must reproduce the known diagonal modular transformation of the one-point torus block.

References

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