Partition functions of N=(2,2) gauge theories on S^2 and vortices (1206.2356v3)

Abstract: We apply localization techniques to compute the partition function of a two-dimensional N=(2,2) R-symmetric theory of vector and chiral multiplets on S^2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet-Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. As applications we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

• The paper introduces a novel localization method that reduces the path integral to a finite-dimensional matrix integral over the Cartan subalgebra.
• It details the computation of partition functions via both Coulomb and Higgs branch localization, emphasizing vortex contributions and magnetic flux summation.
• The work confirms dualities in U(N) gauge theories and lays the groundwork for further explorations in mirror symmetry and non-perturbative dynamics.

An Exploration of the Partition Functions in $N=(2,2)$ Gauge Theories on $S^2$ through Localization

This paper by Francesco Benini and Stefano Cremonesi ventures into the rigorous computation of partition functions of two-dimensional $N=(2,2)$ supersymmetric gauge theories on the sphere $S^2$ utilizing the powerful toolkit of localization. These theories, intrinsic to understanding key theoretical structures, include vector and chiral multiplets, and are formulated in the curved background of a round sphere. The authors explore intricacies involving the path integral's reduction into a finite-dimensional integral defined over the gauge group's Cartan subalgebra, thus simplifying traditionally challenging evaluations.

Initially, the paper provides a solid foundation by discussing the importance of $N=(2,2)$ supersymmetric theories as laboratories for exact computations in quantum field theory. One crucial highlight is the development of localization techniques, enabling researchers to bypass the intricacies of non-abelian interactions by focusing on semiclassical limits in path integrals—a strategy stemming from Witten’s earlier work.

Benini and Cremonesi effectively summarize the mathematical groundwork required for localization on $S^2$, detailing the involvement of Killing spinors in the complex S$^2$ setup. The complexification of conformal transformations on curved backgrounds emerges as significant, invoking deep connections between $SU(2|1)$ superalgebras and exact superconformal methods, which enhance analytical tractability.

The authors advance Renormalization Group (RG) indepen-dent calculations and explore fundamental configurations such as vector zero-modes. Importantly, they categorize contributions of determinants into topological sectors—equivalent to summing over GNO-quantized magnetic fluxes. Through sophisticated matrix integrals, they deliver an expression reflecting a gauge group’s partition function, reinforcing the method's versatility and previous conjectural outcomes by Hori-Tong.

A key accomplishment is their exploration of alternative paths in localization: while the conventional matrix integral emerges on the Coulomb branch, the paper also pioneers computing analogous expressions through localization on the Higgs branch. The core technique involves refining $Q$-exact deformations by manipulating the path's integration contour of real fields, presenting a rich tapestry connecting discrete vortex contributions at spherical poles to vortex partition functions, weighted with semiclassical terms in finite sums.

This research provides robust analytical tools for calculating sphere partition functions, covering a gamut of $U(N)$ gauge theories with flavors in fundamental, antifundamental, and adjoint representations. It confirms the conjectured equivalences of partition functions for various dual theories, extending the groundwork laid by Hori and Tong.

Practically and theoretically, the implications are manifold. Dedicated sections elaborate on special cases and conjectural dualities, propelling understandings such as Seiberg-like dualities for lower-dimensional theories while ensuring congruity across different mathematical approaches. Future directions allude to leveraging the findings to decode mirror symmetry and aid in deciphering non-perturbative aspects of $N=(2,2)$ theories.

This work holds rich potential for future avenues, potentially bridging calculations between two and three-dimensional theories through dimensional reduction insights, underpinning emergent techniques such as index formulations and litmus tests for dualities. Crucially, it intersects advanced topics like supersymmetry breaking and vortex dynamics, crucial for understanding correlators and more sophisticated phenomena in mathematical physics and beyond. Not only does this paper enrich computational methodologies, but it lays theoretical bricks towards understanding the tapestry of gauge/string direct correspondence, matrix models, and the search for exact results in supergravity and geometry, all haLLMarks resonating through the field of high-energy physics.