Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2) Theories

Kentaro Hori, David Tong

TL;DR

The paper advances the understanding of non-Abelian dynamics in two-dimensional ${ m N}=(2,2)$ theories by computing the SU(k) Witten index and establishing a combinatorial criterion for IR non-singularity, revealing a rich structure of Coulomb branches and dualities. It demonstrates how gauged linear sigma models flow to Calabi–Yau target spaces embedded in Grassmannians, and uncovers multiple interior singularities in the Kahler moduli space that mirror discriminants of proposed mirrors. A key highlight is the Glop transition, proving Rodland’s conjecture that two distinct Calabi–Yau threefolds lie on the same moduli space yet are not birationally equivalent, connected via a Grassmannian flop and accompanied by derived-equivalent brane categories. The work also develops a robust framework of IR dualities, including a duality between SU(k) theories with N=k+1 flavors and free SCFT/LG models, and extends these ideas to a broader class of $U(k)$ linear sigma models with ramifications for mirror symmetry and D-brane categories.

Abstract

We study various aspects of N=(2,2) supersymmetric non-Abelian gauge theories in two dimensions, with applications to string vacua. We compute the Witten index of SU(k) SQCD with N>0 flavors with twisted masses; the result is presented as the solution to a simple combinatoric problem. We further claim that the infra-red fixed point of SU(k) gauge theory with N massless flavors is non-singular if (k,N) passes a related combinatoric criterion. These results are applied to the study of a class of U(k) linear sigma models which, in one phase, reduce to sigma models on Calabi-Yau manifolds in Grassmannians. We show that there are multiple singularities in the middle of the one-dimensional Kahler moduli space, in contrast to the Abelian models. This result precisely matches the complex structure singularities of the proposed mirrors. In one specific example, we study the physics in the other phase of the Kahler moduli space and find that it reduces to a sigma model for a second Calabi-Yau manifold which is not birationally equivalent to the first. This proves a mathematical conjecture of Rodland.

Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2) Theories

TL;DR

The paper advances the understanding of non-Abelian dynamics in two-dimensional theories by computing the SU(k) Witten index and establishing a combinatorial criterion for IR non-singularity, revealing a rich structure of Coulomb branches and dualities. It demonstrates how gauged linear sigma models flow to Calabi–Yau target spaces embedded in Grassmannians, and uncovers multiple interior singularities in the Kahler moduli space that mirror discriminants of proposed mirrors. A key highlight is the Glop transition, proving Rodland’s conjecture that two distinct Calabi–Yau threefolds lie on the same moduli space yet are not birationally equivalent, connected via a Grassmannian flop and accompanied by derived-equivalent brane categories. The work also develops a robust framework of IR dualities, including a duality between SU(k) theories with N=k+1 flavors and free SCFT/LG models, and extends these ideas to a broader class of linear sigma models with ramifications for mirror symmetry and D-brane categories.

Abstract

We study various aspects of N=(2,2) supersymmetric non-Abelian gauge theories in two dimensions, with applications to string vacua. We compute the Witten index of SU(k) SQCD with N>0 flavors with twisted masses; the result is presented as the solution to a simple combinatoric problem. We further claim that the infra-red fixed point of SU(k) gauge theory with N massless flavors is non-singular if (k,N) passes a related combinatoric criterion. These results are applied to the study of a class of U(k) linear sigma models which, in one phase, reduce to sigma models on Calabi-Yau manifolds in Grassmannians. We show that there are multiple singularities in the middle of the one-dimensional Kahler moduli space, in contrast to the Abelian models. This result precisely matches the complex structure singularities of the proposed mirrors. In one specific example, we study the physics in the other phase of the Kahler moduli space and find that it reduces to a sigma model for a second Calabi-Yau manifold which is not birationally equivalent to the first. This proves a mathematical conjecture of Rodland.

Paper Structure

This paper contains 24 sections, 101 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Splitting the Conifold Singularity
  3. The Models
  4. Abelian Theories
  5. Non-Abelian Theories
  6. Comparison with Mirror Symmetry Conjecture
  7. The Witten Index for Two Dimensional SQCD
  8. A First Attempt at Counting
  9. The Absence of Contributions from Strong Coupling Regimes
  10. Varying the Twisted Masses
  11. Complex Masses for $SU(2)$ Theories
  12. Infra-Red Dynamics of $SU(k)$ Gauge Theories
  13. IR Central Charge
  14. The Singularity Criterion
  15. $N=k+1$: Duality with Free SCFT and LG Models
  16. ...and 9 more sections

Figures (2)

  • Figure 1: The Kähler Moduli Space of the IR fixed points of the theory with superpotential (\ref{['starting']}). It has two large volume limits and three singularities
  • Figure 2: The Coulomb branch for $SU(2)$ SQCD with $N=3$. The scales of the left part ($|m|\ll |\tilde{m}_i|$) and the right part ($|m|\gg |\tilde{m}_i|$) are different.