Abstract
We compute the light asymptotic limit of A n−1 Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of A 2 Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions.
Highlights
The heavy and light asymptotic limits were reconsidered in [10] for complex solutions of the analytically continued Liouville theory
The essential point here is the fact that there are explicit combinatorial formulas for the Nekrasov partition function [20, 21], which can be successfully applied in 2d CFT
This could be anticipated since in the light asymptotic limit the infinite Virasoro symmetry reduces to SL(2) algebra whose representations are classified with one row Young tableaux
Summary
Due to AGT duality, this partition function is directly related to specific four point conformal block in 2d An−1 Toda field theory. Before describing this relation let us briefly recall few facts about Toda theory. In what follows it would be convenient to represent the roots, weights and Cartan elements of An−1 as n-component vectors with the usual Kronecker scalar product, subject to the condition that sum of components is zero This is equivalent to more conventional representation of these quantities as diagonal traceless n×n matrices with the pairing given by trace.
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