Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 077, 55 pages      arXiv:1906.00600

Twisted Representations of Algebra of $q$-Difference Operators, Twisted $q$-$W$ Algebras and Conformal Blocks

Mikhail Bershtein abcde and Roman Gonin bc
a) Landau Institute for Theoretical Physics, Chernogolovka, Russia
b) Center for Advanced Studies, Skolkovo Institute of Science and Technology, Moscow, Russia
c) National Research University Higher School of Economics, Moscow, Russia
d) Institute for Information Transmission Problems, Moscow, Russia
e) Independent University of Moscow, Moscow, Russia

Received November 22, 2019, in final form August 01, 2020; Published online August 16, 2020

We study certain representations of quantum toroidal $\mathfrak{gl}_1$ algebra for $q=t$. We construct explicit bosonization of the Fock modules $\mathcal{F}_u^{(n',n)}$ with a nontrivial slope $n'/n$. As a vector space, it is naturally identified with the basic level 1 representation of affine $\mathfrak{gl}_n$. We also study twisted $W$-algebras of $\mathfrak{sl}_n$ acting on these Fock modules. As an application, we prove the relation on $q$-deformed conformal blocks which was conjectured in the study of $q$-deformation of isomonodromy/CFT correspondence.

Key words: quantum algebras; toroidal algebras; $W$-algebras; conformal blocks; Nekrasov partition function; Whittaker vector.

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