Exact Results in D=2 Supersymmetric Gauge Theories

TL;DR

The paper provides exact results for 2d $\mathcal{N}=(2,2)$ gauge theories on $S^2$ via localization, revealing two dual representations of the partition function: a Coulomb-branch integral over the vector multiplet and a Higgs-branch sum over vortices. It demonstrates a gauge-theory/Toda CFT correspondence, where partition functions match Liouville/Toda correlators with degenerate insertions, and shows that flop transitions correspond to crossing symmetry in the CFT. The work also furnishes quantitative evidence for Seiberg-like dualities in two dimensions by matching partition functions of dual pairs in appropriate limits. Together, these results illuminate modular properties, nonperturbative structures, and phase transitions in 2d supersymmetric gauge theories, with implications for Calabi–Yau sigma models and string worldsheet dynamics.

Abstract

We compute exactly the partition function of two dimensional 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 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.

Figures (5)

• Figure 1: Higgs vacua. Vortices and anti-vortices on these vacua contribute to $Z_{\text{Higgs}}(m, \tau)$
• Figure 2: Vortex and anti-vortex configurations in the Higgs branch
• Figure 3: SQED partition function as Toda CFT correlator
• Figure 4: Toda CFT $s$-channel conformal block which reproduces the vortex partition function of SQED for a Higgs vacuum matching the choice of channel $1\leq p\leq N_F$. The full four point correlator is equal to the SQED partition function on $S^2$. The momenta $\alpha_1$ and $\alpha_2$ encode the masses of fundamental and anti-fundamental chiral multiplets, with $SU(N_F)\times SU(N_F)$ flavour symmetry, while ${\rm \hat{m}}$ captures the remaining $U(1)_{\text{diag}}$ flavour symmetry.
• Figure 5: Decoupling limit of the AGT correspondence with a simple surface operator.