Exact Results in D=2 Supersymmetric Gauge Theories (1206.2606v4)

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.

• The paper demonstrates dual partition function representations via Coulomb integration and vortex/anti-vortex summation on the Higgs branch.
• The paper employs localization techniques that precisely quantify vortex and anti-vortex contributions in 2D N=(2,2) gauge theories.
• The paper establishes a novel correspondence between gauge theory partition functions and Toda CFT correlators, supporting Seiberg-like dualities.

Overview of "Exact Results in $D=2$ Supersymmetric Gauge Theories"

This paper presents an in-depth examination of two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories on the sphere ($S^2$), focusing on the computation of their partition functions. The authors demonstrate that these partition functions possess dual descriptions — either as integrals over the Coulomb branch or as sums over vortex and anti-vortex excitations on the Higgs branches. Their findings elucidate a novel connection with correlation functions in two-dimensional Liouville/Toda conformal field theories (CFTs).

Key Findings

1. Dual Representations of Partition Functions:
   • Coulomb Branch: The partition function can be expressed as an integral over vector multiplet field configurations, representing the Coulomb phase of the theory. This representation is characterized by parameters $a$ (Coulomb branch parameter), $B$ (quantized flux on $S^2$), and gauge theory parameters $\tau$.
   • Higgs Branch: The partition function admits a representation as a discrete sum over Higgs branches involving vortex and anti-vortex contributions. Such representations highlight the theory's modular properties and are interpreted as a manifestation of Seiberg duality — an equivalence between seemingly different quantum field theories.
2. Vortex/Anti-vortex Contributions:
   • The vortex and anti-vortex configurations localize at the north and south poles of the sphere, respectively.
   • These contributions are captured by the vortex partition function in an $\Omega$-background, providing a link to the modular measures found in Toda CFT.
3. Connection to Toda CFT:
   • The partition functions of certain two-dimensional gauge theories on $S^2$ correlate with Toda CFT correlators involving degenerate vertex operators. For example, in the framework of $\mathcal{N}=(2,2)$ SQCD with $U(N)$ gauge group, the correspondence with Toda CFT provides insights into modular transformations and offers a geometric realization of phenomena such as topology-changing transitions (e.g., flop transitions).
4. Implications for Seiberg Duality:
   • The paper provides evidence for Seiberg-like dualities in two dimensions, analogous to those in higher dimensions. It demonstrates that partition functions of conjectured dual pairs are equivalent under certain conditions, supporting the idea of analytic continuation and crossing symmetries in moduli space.

Theoretical and Practical Implications

• Modular Invariance and Dualities: The work underscores the role of modular properties in understanding dualities and topological transitions in quantum field theories. The dual representations enrich our comprehension of quantum field theories at a fundamental level.
• Future Prospects for Gauge Theory-CFT Correspondences: The established correspondence with Toda CFT opens avenues for exploring other CFT analogs and dualities, potentially extending the landscape of known gauge theory-CFT correspondences akin to the AGT conjecture.
• Applications in String Theory: Given that some gauge theories flow to Calabi-Yau sigma models in the infrared, these exact results contribute to the comprehension of string theory on non-perturbative backgrounds and the computation of worldsheet instantons.

The paper sheds light on intricate aspects of $\mathcal{N}=(2,2)$ supersymmetric gauge theories and their deep connections with integrable systems and CFTs, paving the way for further explorations in quantum field theory and string theory landscapes.