Gauge/Liouville Triality

Abstract

Conformal blocks of Liouville theory have a Coulomb-gas representation as Dotsenko-Fateev (DF) integrals over the positions of screening charges. For q-deformed Liouville, the conformal blocks on a sphere with an arbitrary number of punctures are manifestly the same, when written in DF representation, as the partition functions of a class of 3d U(N) gauge theories with N=2 supersymmetry, in the Omega-background. Coupling the 3d gauge theory to a flavor in fundamental representation corresponds to inserting a Liouville vertex operator; the two real mass parameters determine the momentum and position of the puncture. The DF integrals can be computed by residues. The result is the instanton sum of a five dimensional N=1 gauge theory. The positions of the poles are labeled by tuples of partitions, the residues of the integrand are the Nekrasov summands.

50 pages, 3 figures. v2: Typos corrected, aspects of 3d gauge theory clarified, references added

References in corpus (9)

• Toda Theories, Matrix Models, Topological Strings, and N=2 Gauge Systems
• Generalized Superconformal Index for Three Dimensional Field Theories
• On Three Dimensional Quiver Gauge Theories and Integrability
• Seiberg-Witten Theory and Random Partitions
• A direct proof of AGT conjecture at beta = 1
• M-theory and a Topological String Duality
• On "Dotsenko-Fateev" representation of the toric conformal blocks
• Holomorphic Blocks for 3d Non-abelian Partition Functions
• Refined Chern-Simons Theory and Knot Homology

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• Quantum q-Langlands Correspondence
• A geometric recipe for twisted superpotentials
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