Four dimensional N = 2 Argyres-Douglas theories have been recently conjectured to be described by N = 1 Lagrangian theories. Such models, once reduced to 3d, should be mirror dual to Lagrangian N = 4 theories. This has been numerically checked through the matching of the partition functions on the three sphere. In this article, we provide an analytic derivation for this result in the A 2n−1 case via hyperbolic hypergeometric integrals. We study the D 4 case as well, commenting on some open questions and possible resolutions. In the second part of the paper we discuss other integral identities leading to the matching of the partition functions in 3d dual pairs involving higher monopole superpotentials.

We construct a quantisation of the Teichmüller spaces of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential. Figure 5. The pentagon equation. Quantum Teichmüller theoryQuantisation of the Teichmüller theory of punctured Riemann surfaces was developed by Kashaev in [11] and independently by Fock and Chekhov in [15,17]. We will associate a

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