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Surface Operators in N=2 4d Gauge Theories

Davide Gaiotto

TL;DR

This work develops a universal framework linking half-BPS surface operators in four-dimensional ${ m N}=2$ gauge theories to the Seiberg-Witten geometry of the bulk, via the twisted-chiral data of the 2d defect and its ${tt}^*$ structure. It shows that the parameter space of 2d twisted couplings fibers over a curve that reproduces the SW curve, with the surface operator's vacua providing SW-like periods through a holomorphic one-form $oldsymbol{ m λ}$. The authors extend the analysis to Hitchin-system-like structures in the 3d/tt* setting and propose a 2d-4d wall-crossing framework governing bound states of 2d and 4d BPS particles. Through a rich collection of examples including ${ m CP}^n$ sigma models, various ${ m N_f}$ SU(2) theories, and product theories, the paper demonstrates how surface operators encode bulk dynamics and suggests a broad, potentially universal structure underlying 4d ${ m N}=2$ theories and their defects.

Abstract

N=2 four dimensional gauge theories admit interesting half BPS surface operators preserving a (2,2) two dimensional SUSY algebra. Typical examples are (2,2) 2d sigma models with a flavor symmetry which is coupled to the 4d gauge fields. Interesting features of such 2d sigma models, such as (twisted) chiral rings, and the tt* geometry, can be carried over to the surface operators, and are affected in surprising ways by the coupling to 4d degrees of freedom. We will describe in detail a relation between the parameter space of twisted couplings of the surface operator and the Seiberg-Witten geometry of the bulk theory. We will discuss a similar result about the tt* geometry of the surface operator. We will predict the existence and general features of a wall-crossing formula for BPS particles bound to the surface operator.

Surface Operators in N=2 4d Gauge Theories

TL;DR

This work develops a universal framework linking half-BPS surface operators in four-dimensional gauge theories to the Seiberg-Witten geometry of the bulk, via the twisted-chiral data of the 2d defect and its structure. It shows that the parameter space of 2d twisted couplings fibers over a curve that reproduces the SW curve, with the surface operator's vacua providing SW-like periods through a holomorphic one-form . The authors extend the analysis to Hitchin-system-like structures in the 3d/tt* setting and propose a 2d-4d wall-crossing framework governing bound states of 2d and 4d BPS particles. Through a rich collection of examples including sigma models, various SU(2) theories, and product theories, the paper demonstrates how surface operators encode bulk dynamics and suggests a broad, potentially universal structure underlying 4d theories and their defects.

Abstract

N=2 four dimensional gauge theories admit interesting half BPS surface operators preserving a (2,2) two dimensional SUSY algebra. Typical examples are (2,2) 2d sigma models with a flavor symmetry which is coupled to the 4d gauge fields. Interesting features of such 2d sigma models, such as (twisted) chiral rings, and the tt* geometry, can be carried over to the surface operators, and are affected in surprising ways by the coupling to 4d degrees of freedom. We will describe in detail a relation between the parameter space of twisted couplings of the surface operator and the Seiberg-Witten geometry of the bulk theory. We will discuss a similar result about the tt* geometry of the surface operator. We will predict the existence and general features of a wall-crossing formula for BPS particles bound to the surface operator.

Paper Structure

This paper contains 19 sections, 33 equations, 4 figures.

Table of Contents

  1. Introduction and outline
  2. SUSY review
  3. 4d gauge theory in 2d language
  4. Surface operators in the Abelian IR theory
  5. Seiberg-Witten geometry from surface operators
  6. Hitchin systems from surface operators
  7. Wall-crossing and surface operators
  8. Examples
  9. The $\mathbb{C} \mathbb{P}^1$ sigma model
  10. The minimal surface operator in $SU(2)$ Seiberg-Witten theory
  11. Non-minimal surface operators in $SU(2)$ Seiberg-Witten theory: Coupling to an $\mathbb{C} \mathbb{P}^2$ sigma model.
  12. Non-minimal surface operators in $SU(2)$ Seiberg-Witten theory: Coupling to an $\mathbb{C} \mathbb{P}^n$ sigma model.
  13. Non-minimal surface operators in $SU(2)$ Seiberg-Witten theory: Triplet coupling to an $\mathbb{C} \mathbb{P}^2$ sigma model.
  14. Surface operators and flavors: $N_f=1$ $SU(2)$ Seiberg-Witten theory
  15. Surface operators and flavors: $N_f=2$ $SU(2)$ Seiberg-Witten theory
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Different brane realizations of a simple quiver gauge theory: $SU(3) \times SU(2)$ with a bifundmental and two $SU(2)$ fundamentals. Vertical lines represent NS5 branes, horizontal D4 branes, circles are D6 branes. (a) Simplest realization. Flavors from semiinfinite D4 branes. (b) Two D6 can also produce the flavors (c) D4 segments are created when moving the D6 branes. (d) An extra D6 has been added to the right, brought to the left.
  • Figure 2: A brane realization of the $\mathbb{C} \mathbb{P}^1$ sigma model. The dashed line represents a D2 brane ending on the system. If the brane ends on the NS5 brane, the linear sigma model can be recovered. Turning on an FI term moves the D2 along the D4 branes.
  • Figure 3: A brane realization of the minimal surface operator in the pure $SU(2)$ theory. If the D2 brane ends on the NS5 brane, the description as coupling to a $\mathbb{C} \mathbb{P}^1$ sigma model is recovered. If the D2 ends on the D4 the realization as a defect in the gauge theory is recovered.
  • Figure 4: A brane realization of a non-minimal surface operator in the pure $SU(2)$ theory.