Comments on twisted indices in 3d supersymmetric gauge theories (1605.06531v2)

Abstract: We study three-dimensional ${\mathcal N}=2$ supersymmetric gauge theories on ${\Sigma_g \times S^1}$ with a topological twist along $\Sigma_g$, a genus-$g$ Riemann surface. The twisted supersymmetric index at genus $g$ and the correlation functions of half-BPS loop operators on $S^1$ can be computed exactly by supersymmetric localization. For $g=1$, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe equations for the theory on ${\mathbb R}^2 \times S^1$. This also provides a powerful and simple tool to study 3d ${\mathcal N}=2$ Seiberg dualities. Finally, we study A- and B-twisted indices for ${\mathcal N}=4$ supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the $S^2 \times S^1$ twisted indices and the Hilbert series of ${\mathcal N}=4$ moduli spaces.

• The paper computes twisted indices in 3d N=2 supersymmetric gauge theories on Sigma_g x S^1 using supersymmetric localization and Jeffrey-Kirwan residues.
• It applies these indices to study the quantum algebra of Wilson loops and confirm 3d Seiberg dualities, including the Aharony duality.
• The research also explores A/B-twisted indices for N=4 mirror symmetry and discusses broader implications for dualities and integrable models.

An Expert Overview of Twisted Indices in 3d Supersymmetric Gauge Theories

The research paper "Comments on twisted indices in 3d supersymmetric gauge theories" presents a detailed examination of twisted indices in three-dimensional $N=2$ supersymmetric gauge theories. The authors, Cyril Closset and Heeyeon Kim, explore the computation of these indices through supersymmetric localization and analyze their applications in the paper of gauge theories and related dualities.

Twisted Supersymmetric Indices and Correlation Functions

The paper investigates three-dimensional $N=2$ supersymmetric gauge theories formulated on a product space ${\Sigma_g \times S^1}$, where $\Sigma_g$ denotes a genus-$g$ Riemann surface. The authors utilize a topological twist, enabling the preservation of supersymmetry, and compute the twisted indices using supersymmetric localization techniques. This approach provides precise calculations of correlation functions involving half-BPS loop operators on $S^1$.

A critical aspect of calculating these twisted indices involves using Jeffrey-Kirwan (JK) residues of a holomorphic form over a complexified Coulomb branch. These residues consider various configurations including matter singularities and monopole operator singularities. The paper explains the handling of these residues in detail, highlighting nuances related to the choice of auxiliary vectors and ensuring the integrals avoid non-projective singularities.

Applications in Gauge Theories and Seiberg Dualities

The paper elaborates on how twisted indices can offer insights into the quantum algebra of supersymmetric Wilson loops across different gauge theories. For a Yang-Mills-Chern-Simons matter theory, the twisted indices reveal the quantum relations these Wilson loops satisfy, providing a finite-dimensional algebraic description.

One of the significant implications of this research is its contribution to understanding Seiberg dualities in three-dimensional gauge theories. The authors apply the twisted index framework to substantiate various dualities, including the well-known Aharony duality. By comparing twisted indices across dual descriptions, they confirm that dual theories share identical partition functions when accounting for specific CS contact terms and matter content alterations.

Special Cases and Mirror Symmetry

The researchers also present findings related to mirror symmetry in $N=4$ supersymmetric theories by exploring the $A$- and $B$-twisted indices. These indices are computed for $N=4$ gauge theories on $\Sigma_g \times S^1$, where, importantly, the $A$- or $B$-twist is applied along the Riemann surface, respecting the symmetry between $SU(2)_H$ and $SU(2)_C$. Mirror symmetry predicts the equivalence of these indices across dual theories, verified through direct computations and symmetry arguments.

Implications for Future Research

The computation methods and theoretical insights presented have broad implications for the paper of supersymmetric theories in three dimensions. These techniques provide a robust tool for exploring various theoretical aspects, from quantum algebras to dualities, which could further influence research in related fields such as quantum gravity, string theory, and topological quantum field theory.

Notably, the paper opens avenues for detailed exploration into the connection with Hilbert series of moduli spaces and implications in holographic duality. The authors also suggest potential continued research into the complex relations between quantum field theory descriptions and quantum integrable models, hinting at interdisciplinary advancements.

This exploration of twisted indices in gauge theories offers a thorough and rigorous approach to understanding and applying supersymmetric localization, contributing to the broader understanding of quantized gauge theories and their dualities.