nLab
AGT correspondence

Context

Duality in string theory

Functorial Quantum field theory

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

String theory

Contents

Idea

The AGT correspondence (AGT 09) is a relation between

  1. the instanton-partition function of SU(2) n+3g3SU(2)^{n+3g-3}-N=2 D=4 super Yang-Mills theory (Nekrasov’s partition function, e.g. Szabo 15 (2.1))

  2. the conformal blocks of Liouville theory on an nn-punctured Riemann surface C g,nC_{g,n} of genus gg

Here the idea is that C g,nC_{g,n} is the super Yang-Mills theory obtained by compactifying the worldvolume 6d (2,0)-supersymmetric QFT of two M5-branes, see at N=2 D=4 super Yang-Mills theory, the section Construction by compactification).

In particular, the N=2 D=4 super Yang-Mills theory is a quiver gauge theory and the correspondence matches the shape of its quiver-diagram to the genus and punctures of the Riemann surface:

More generally, this construction yields something like a decomposition of the 6d (2,0)-superconformal QFT into a 2d SCFT “with values in 4d SYM field theory” (e.g. Tachikawa 10, slide 25 (33 of 54)). Hence composition with any kind of suitable invariant of the 4d field theories yields an actual 2d SCFT, for instance taking the superconformal index in 4d yields a 2d TQFT (GPRR 10). In this picture of “4d-SYM field theory-valued 2d SCFT” one has the following correspondences:

Wrapping the M5-brane on a 3-manifold instead yields: 3d-3d correspondence.

References

A 2d SCFT argued to describe the KK-compactification of the M5-brane on a 4-manifold (specifically: a complex surface) originates with

The origin of the AGT correspondence is:

The 2d TQFT obtained from this by forming the 4d index is discussed in

Relation of the AGT-correspondence to the D=6 N=(2,0) SCFT

and to the 3d-3d correspondence:

Brief surveys include

More detailed review is in

See also

The AGT correspondence is treated with the help of a Riemann-Hilbert problem in

category: physics