Abstract

In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory T2. The partition functions exhibit jumps when passing from one chamber in the parameter space to another. Such jumps can be attributed to a change of the integration contour in the free field representation of Liouville conformal blocks. We compare the partition functions of the T2 theories representing trifundamental half hypermultiplets in N = 2, d = 4 field theories to the partition functions associated to bifundamental hypermultiplets. We find that both are related to the same Liouville conformal blocks up to inessential factors. In order to establish this picture we combine and compare results obtained using topological vertex techniques, matrix models and topological recursion. We furthermore check that the partition functions obtained by gluing two T2 vertices can be represented in terms of a four point Liouville conformal block. Our results indicate that the T2 vertex offers a useful starting point for developing an analog of the instanton calculus for SUSY gauge theories with trifundamental hypermultiplets.

Highlights

  • The discovery of relations between N = 2, d = 4 supersymmetric gauge theories and conformal field theory by Alday, Gaiotto and Tachikawa [1] has stimulated a large amount of work

  • In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory T2

  • We find that both are related to the same Liouville conformal blocks up to inessential factors

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Summary

Introduction

Analogs of the AGT-correspondence have been found relating instanton partition functions in five-dimensional supersymmetric gauge theories to q-deformations of conformal blocks for the WN -algbra. Geometric engineering one may naturally associate the strip diagrams with fundamentalor bi-fundamental hypermultiplets It is well-known that the strip partition functions represent the basic building blocks for the instanton partition functions in linear or circular quiver gauge theories. For this case one has a clear understanding of the relation between the instanton partition functions and conformal blocks This state of affairs naturally raises two basic questions:. For the reader’s (and our) convenience we have collected some of the key formulae in appendix D

The topological vertex computation
The computation
Resummation into a product formula
Dependence on the choice of preferred direction
T2 web with non-empty external Young tableaux
The 4D limit
A useful factorisation
The limit of the regular part
Renormalising the singular part
Extending the domain of definition of the T2 partition functions
Integral representation of the T2 partition function
Imposing the specialisation condition to the topological strings
The matrix integral as a sum of residues
The free field representation for q-Liouville
A simple example
The matrix integral
The matrix integral as a Selberg integral
Comparison with the strip vertex
Topological vertex computation
Comparison
Relation between mirror curves of strip and T2
Geometric transitions and matrix models
Topological recursion
TN versus conformal blocks
Mapping to Virasoro four point conformal blocks

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