Abstract
We study five dimensional AGT correspondence by means of the q-deformed beta-ensemble technique. We provide a special basis of states in the q-deformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop equations for the beta-ensemble and obtain the factorization formulas for the corresponding matrix elements. We prove the spectral duality for Nekrasov functions and discuss its meaning for conformal blocks. We also clarify the relation between topological strings and q-Liouville vertex operators.
Highlights
To the expansion of four-point conformal block in terms of a certain complete system of basis vectors |A, B, α, labelled by pairs of partitions, which can be written schematically as
We provide a special basis of states in the q-deformed CFT Hilbert space consisting of generalized Macdonald polynomials, derive the loop equations for the beta-ensemble and obtain the factorization formulas for the corresponding matrix elements
We prove the spectral duality for SU(2) Nekrasov functions and discuss its meaning for conformal blocks
Summary
Generalized Macdonald polynomials are symmetric polynomials in two sets of variables xi and xi labelled by pairs of partitions Y1, Y2 and depending on an extra parameter Q. There is a convenient way to normalize generalized Macdonald polynomials: MY(q1,tY)2(Q|p, p) = mY 1(p)mY 2(p)+ W =Y cY W mW 1(p)mW 2(p). Notice that in this normalization generalized Macdonald polynomials depend polynomially on the parameters Q, q and t. Generalized Macdonald polynomials form a complete basis in the space of symmetric functions in two sets of variables: CY CY. The completeness (2.11) of the generalized Macdonald polynomials can be employed in the last line of eq (3.1) and gives the following expansion. Observe that a new parameter a has appeared in the last lines of eq (3.2) This is an extra parameter of generalized Macdonald polynomials, which should be tuned to conform with the parameters of the Selberg average according to eqs. We develop the loop equations for the q-deformed beta-ensemble (3.4) in order to check eq (3.3)
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