It was shown in [1] for 2d topological conformal field theory (TCFT) [2,3] and more recently in [4]-[12] for the noncritical string theory [13]-[17] that several models of these two types can be exactly solved using their connection with the Frobenius manifold (FM) structure introduced by Dubrovin [18]. More precisely, these models are connected with a special case of FMs, the so-called Saito Frobenius manifolds (SFMs), originally called the flat structures [19], which arise on the space of the versal deformations of the isolated singularities together with the flat coordinate systems after a suitable so-called primitive forms are chosen. In this paper, we explore the connection of the models of TCFT and noncritical string theory with the SFM. The crucial point for obtaining an explicit expression for the correlators is finding the flat coordinates of SFMs as functions of the parameters of the deformed singularity. We propose a new direct way to find the flat coordinates using the integral representation for the solutions of the GaussManin system connected with the corresponding SFM. We illustrate our approach in the A n singularity case. We also address the possible generalization of our approach for the models investigated in [21], which are SU (N ) k /(SU (N − 1) k+1 × U (1)) Kazama-Suzuki theories [22]. We prove a theorem that the potential connected with these models is an isolated singularity, which is a condition for the FM structure to emerge on its deformation manifold. This fact allows using the DVV approach to solve such Kazama-Suzuki models.

Let X and X ! be a pair of symplectic varieties dual with respect to 3D-mirror symmetry. The K-theoretic limit of the elliptic duality interface is an equivariant K-theory class m ∈ K(X × X ! ). We show that this class provides correspondencesmapping the K-theoretic stable envelopes to the K-theoretic stable envelopes. This construction allows us to extend the action of various representation theoretic objects on K(X), such as action of quantum groups, quantum Weyl groups, R-matrices etc., to their action on K(X ! ). In particular, we relate the wall R-matrices of X to the Rmatrices of the dual variety X ! .As an example, we apply our results to X = Hilb n (C 2 ) -the Hilbert scheme of n points in the complex plane. In this case we arrive at the conjectures of E.Gorsky and A.Negut from [13].

No abstract

We study generalized Deligne categories and related tensor envelopes for the universal two-dimensional cobordism theories described by rational functions, recently defined by Sazdanovic and one of the authors.

No abstract