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Elliptic Genera of 2d $\mathcal{N}=(0,1)$ Gauge Theories

Jiakang Bao, Masahito Yamazaki, Dongao Zhou

Abstract

We derive an exact residue formula for the elliptic genera of 2d $\mathcal{N}=(0,1)$ gauge theories. We find a new residue prescription which recovers the Jeffery-Kirwan residue prescription for $\mathcal{N}=(0,2)$ theories. We apply the formula to the Gukov-Pei-Putrov model and analyze the phase structure of the theory.

Elliptic Genera of 2d $\mathcal{N}=(0,1)$ Gauge Theories

Abstract

We derive an exact residue formula for the elliptic genera of 2d gauge theories. We find a new residue prescription which recovers the Jeffery-Kirwan residue prescription for theories. We apply the formula to the Gukov-Pei-Putrov model and analyze the phase structure of the theory.

Paper Structure

This paper contains 41 sections, 1 theorem, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setups and Notations
  3. The (0,1) Supersymmetry
  4. The Lagrangian
  5. Reality and Weight Spaces
  6. Anomaly Cancellations
  7. Technical Assumptions
  8. Results for N=(0,1) Elliptic Genera
  9. Elliptic Genera
  10. One-Loop Determinants
  11. Residues
  12. Comments
  13. The overall sign
  14. Global gauge anomalies
  15. Modular Transformation Properties
  16. ...and 26 more sections

Key Result

Theorem A.1

Any irreducible real representation $\rho:\mathfrak{g}_0\rightarrow\mathfrak{gl}(V_0)$ of a semisimple real Lie algebra $\mathfrak{g}_0$ satisfies one of the followings: Conversely, any real representation $\rho$ satisfying either of the above two conditions is an irreducible representation.

Figures (3)

  • Figure 4.1: The contour for $D$. In this illustration, there is a pole in the upper half plane approaching the origin when $\epsilon\rightarrow0$. This pushes the contour to the lower half plane with an extra circle surrounding the origin.
  • Figure 5.1: The left picture indicates the flow from the upper right phase to the lower right supersymmetry breaking phase. In the right picture, theories related by trialities are schematically shown with the red arrows. Each pair of upper left and lower right corners connected by a blue arrow has elliptic genera that differ by a sign only, which could be interpreted as different choices of orientations.
  • Figure D.1: The tadpole diagram for the 1-loop correction of $\langle D \rangle$

Theorems & Definitions (1)

  • Theorem A.1