SU(N) q-Toda equations from mass deformed ABJM theory

It is known that the partition functions of the U(N)k × U(N + M)−k ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painlevé III3 equation. In this paper we have suggested that a similar bilinear relation holds for the ABJM theory with $$ \mathcal{N} $$ N = 6 preserving mass deformation for an arbitrary complex value of mass parameter, to which we have provided several non-trivial checks by using the exact values of the partition function for various N, k, M and the mass parameter. For particular choices of the mass parameters labeled by integers ν, a as m1 = m2 = −πi(ν − 2a)/ν, the bilinear relation corresponds to the q-deformation of the affine SU(ν) Toda equation in τ-form.

Universal correlators and novel cosets in 2d RCFT

The two-character level-1 WZW models corresponding to Lie algebras in the Cvitanović-Deligne series A1, A2, G2, D4, F4, E6, E7 have been argued to form coset pairs with respect to the meromorphic E8,1 CFT. Evidence for this has taken the form of holomorphic bilinear relations between the characters. We propose that suitable 4-point functions of primaries in these models also obey bilinear relations that combine them into current correlators for E8,1, and provide strong evidence that these relations hold in each case. Different cases work out due to special identities involving tensor invariants of the algebra or hypergeometric functions. In particular these results verify previous calculations of correlators for exceptional WZW models, which have rather subtle features. We also find evidence that the intermediate vertex operator algebras A0.5 and E7.5, as well as the three-character A4,1 theory, also appear to satisfy the novel coset relation.

Homogeneity of cohomology classes associated with Koszul matrix factorizations

In this work we prove the so-called dimension property for the cohomological field theory associated with a homogeneous polynomial $W$ with an isolated singularity, in the algebraic framework of [A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, J. Reine Angew. Math. 714 (2016), 1–122]. This amounts to showing that some cohomology classes on the Deligne–Mumford moduli spaces of stable curves, constructed using Fourier–Mukai-type functors associated with matrix factorizations, live in prescribed dimension. The proof is based on a homogeneity result established in [A. Polishchuk and A. Vaintrob, Algebraic construction of Witten’s top Chern class, in Advances in algebraic geometry motivated by physics (Lowell, MA, 2000) (American Mathematical Society, Providence, RI, 2001), 229–249] for certain characteristic classes of Koszul matrix factorizations of $0$. To reduce to this result, we use the theory of Fourier–Mukai-type functors involving matrix factorizations and the natural rational lattices in the relevant Hochschild homology spaces, as well as a version of Hodge–Riemann bilinear relations for Hochschild homology of matrix factorizations. Our approach also gives a proof of the dimension property for the cohomological field theories associated with some quasihomogeneous polynomials with an isolated singularity.

FROM KP/UC HIERARCHIES TO PAINLEVÉ EQUATIONS

We study the underlying relationship between Painlevé equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painlevé equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g. Lax formalism, Hirota bilinear relations for τ-functions, Weyl group symmetry and algebraic solutions in terms of the character polynomials, i.e. the Schur function and the universal character.