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Genus Two Generalization of $A_1$ spherical DAHA

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arxiv 1704.02947 v1 pith:UEUPBRDO submitted 2017-04-10 math.QA hep-thmath.RT

classification math.QAhep-thmath.RT
keywords algebragenusoperatorsautomorphismsdahadifferencegeneralizationspherical
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abstract

We consider a system of three commuting difference operators in three variables $x_{12},x_{13},x_{23}$ with two generic complex parameters $q,t$. This system and its eigenfunctions generalize the trigonometric $A_1$ Ruijsenaars-Schneider model and $A_1$ Macdonald polynomials, respectively. The principal object of study in this paper is the algebra generated by these difference operators together with operators of multiplication by $x_{ij} + x_{ij}^{-1}$. We represent the Dehn twists by outer automorphisms of this algebra and prove that these automorphisms satisfy all relations of the mapping class group of the closed genus two surface. Therefore we argue from topological perspective this algebra is a genus two generalization of $A_1$ spherical DAHA.

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  1. Non-invertible twisted compactification of class $\mathcal S$ theory and $(B,B,B)$ branes

    hep-th 2024-12 conditional novelty 6.0 of 10

    Non-invertible twisted compactification of class S theories on S^1 produces 3d N=4 sigma models whose target spaces are fixed-point sets of mapping class group actions on Hitchin moduli space, i.e. (B,B,B) branes.

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