Dual infrared limits of 6d [formula omitted] theory

Olaf Lechtenfeld,Alexander D Popov

doi:10.1016/j.physletb.2019.02.051

Abstract

Compactifying type AN−1 6d N=(2,0) supersymmetric CFT on a product manifold M4×Σ2=M3×S˜1×S1×I either over S1 or over S˜1 leads to maximally supersymmetric 5d gauge theories on M4×I or on M3×Σ2, respectively. Choosing the radii of S1 and S˜1 inversely proportional to each other, these 5d gauge theories are dual to one another since their coupling constants e2 and e˜2 are proportional to those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU(N) Yang–Mills theories on M4×I and on M3×Σ2, where M4⊃M3=Rt×Tp2 with time t and a punctured 2-torus, and I⊂Σ2 is an interval. In the first case, shrinking I to a point reduces to Yang–Mills theory or to the Skyrme model on M4, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on M3 and employing the adiabatic method, we derive in the infrared limit a non-linear SU(N) sigma model with a baby-Skyrme-type term on Σ2, which can be reduced further to AN−1 Toda theory.

Highlights

Introduction and summaryThe famous Alday-Gaiotto-Tachikawa (AGT) 2d-4d correspondence [1] relates Liouville field theory on a punctured Riemann surface Σ2 and SU(2) super-Yang–Mills (SYM) theory on a fourdimensional manifold M 4
This correspondence was quickly extended to 2d AN−1 Toda field theory and 4d SU(N ) SYM [3]
One way to interpret these correspondences is to start from 6d N =(2,0) supersymmetric conformal field theory (CFT) on M n ×M 6−n

Summary

Introduction and summary

The famous Alday-Gaiotto-Tachikawa (AGT) 2d-4d correspondence [1] relates Liouville field theory on a punctured Riemann surface Σ2 and SU(2) super-Yang–Mills (SYM) theory on a fourdimensional manifold M 4 (see e.g. [2] for a nice review and references). [2] for a nice review and references) This correspondence was quickly extended to 2d AN−1 Toda field theory and 4d SU(N ) SYM [3]. Depending on the choice of reduction – translational invariance along IR or adiabatic approach [14, 15, 16, 17, 18, 4] – one obtains (N ≥ 2 extended) Yang–Mills theory or the Skyrme model on the manifold MR40. We propose a geometric background for establishing 4d-2d AGT correspondences between field theories on M 4 and Σ2 We argue that these correspondences depend on the topology of M 4 and Σ2 and on the method employed for deriving the low-energy effective field theories in the infrared (using symmetries, adiabatic limit, constraints etc.)

Two dual ways from 6d to 5d Yang–Mills theory

Moduli space of YM vacua on M3