On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

By Vasyl Alba, V. Fateev, A. Litvinov, G. Tarnopolskiy. 2010

286 citations recorded by the source corpus.

In their recent paper, Alday et al. (Lett Math Phys 91:167–197, 2010) proposed a relation between $${\mathcal{N}=2}$$ four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired by the exact form of the instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra $${\mathcal{A}={\sf Vir} \otimes\mathcal{H}}$$ which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of $${\mathcal{A}}$$. The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called $${Z_{{\sf bif}}}$$ appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in Alday et al. (2010). We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin–Ono integrable hierarchy.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c0d1cd15f0db6562d71318b753be1fcf366f4570

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Citations in the corpus

How instanton combinatorics solves Painlevé VI, V and IIIs (2013, methodology): “as a linear combination of Virasoro conformal blocks. Being purely representation-theoretic quantities, these CFT special functions can be computed in several ways. In particular, the recently proven [1] AGT conjecture [2] relates them to instanton partition functions in N= 2 SUSY 4D Yang-Mills theo”
Quantum Geometry and Quiver Gauge Theories (2013, background): “The tensor product of n level 1 modules of gl1 affine Yangian relates to the ordinary W(sln)-algebra, the symmetry of the 2d conformal sln-Toda field theory; and since in the rational limit equivariant K-theory contracts to the equivariant cohomology, the ordinary W(sln) algebra relates to the equiv”
Instanton counting with a surface operator and the chain-saw quiver (2011, background): “For N > 1 with no surface operator, or equivalently when L = SU(N) and P = SL(N), VP is the Verma module of the WN algebra and F is the Fock space of a free boson [5, 6, 61, 62].”
Semiclassical Virasoro blocks from AdS3 gravity (2015, methodology): “The coefficients have also been computed using combinatorial methods inspired by the AGT correspondence [25,26].”
Virasoro conformal blocks in closed form (2015, methodology): “In [11], a formula was derived for the coefficient of an arbitrary term in a series expansion in the conformal cross-ratio.”