Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction

Abstract

An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the combinatorial and analytic properties of the asymptotic eigenfunctions of the ordinary Macdonald operator of $A$-type, and the Euler transformation formula for Kajihara and Noumi's multiple basic hypergeometric series. That factorization formula provides us with a reasonable algebraic description of the 5D (K-theoretic) Alday-Gaiotto-Tachikawa (AGT) correspondence, and the interpretation of the invariance under the preferred direction from the point of view of the $SL(2,\mathbb{Z})$ duality of the DIM algebra.

arXiv comment54 pages, 2 figures

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• A slow review of the AGT correspondence
• Quantum toroidal algebras and solvable structures in gauge/string theory
• ABCD of qq-characters
• Non-stationary difference equation for q-Virasoro conformal blocks
• MacMahon KZ equation for Ding-Iohara-Miki algebra
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• Intertwining operator and integrable hierarchies from topological strings
• Proof of 5D $A_n$ AGT conjecture at $\beta =1$
• Direct Sum Structure of the Super Virasoro Algebra and a Fermion Algebra Arising from the Quantum Toroidal ${\mathfrak{gl}}_2$