Abstract
Abstract We compute exactly the partition function of two dimensional $ \mathcal{N} $ = (2, 2) gauge theories on S 2 and show that it admits two dual descriptions: either as an integral over the Coulomb branch or as a sum over vortex and anti-vortex excitations on the Higgs branches of the theory. We further demonstrate that correlation functions in two dimensional Liouville/Toda CFT compute the S 2 partition function for a class of $ \mathcal{N} $ = (2, 2) gauge theories, thereby uncovering novel modular properties in two dimensional gauge theories. Some of these gauge theories flow in the infrared to Calabi-Yau sigma models — such as the conifold — and the topology changing flop transition is realized as crossing symmetry in Liouville/Toda CFT. Evidence for Seiberg duality in two dimensions is exhibited by demonstrating that the partition function of conjectured Seiberg dual pairs are the same.
Highlights
It has long been recognized that many of the dynamical and quantum properties of four dimensional gauge theories are mirrored in two dimensional quantum field theories
We demonstrate that the partition function of certain two dimensional N = (2, 2) gauge theories on S2 admits a dual description in terms of correlation functions in two dimensional Liouville/Toda CFT
By explicitly evaluating the integral representation in the Coulomb branch, we find exact agreement with the Higgs branch representation of the partition function
Summary
It has long been recognized that many of the dynamical and quantum properties of four dimensional gauge theories are mirrored in two dimensional quantum field theories. Constructing a supersymmetric Lagrangian on S2 requires finding supersymmetry transformations on the vector and chiral multiplet fields that represent the SU(2|1) algebra We construct these by restricting the N = (2, 2) superconformal transformations to those corresponding to the SU(2|1) subalgebra. SU(2|1) supersymmetry (see equations (2.17) and (2.18)) implies that the mass parameters are given by a constant background expectation value for the scalar field σ2 in the vector multiplet for GF This can be taken in the Cartan subalgebra of the flavour symmetry group GF.
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