Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N} = 4$ Gauge Theories

Nathan Haouzi

Abstract

We analyze symmetries corresponding to separated topological sectors of 3d ${\cal} N=4$ gauge theories with Higgs vacua, compactified on a circle. The symmetries are encoded in Schwinger-Dyson identities satisfied by correlation functions of a certain gauge-invariant operator, the "vortex character." Such a character observable is realized as the vortex partition function of the 3d gauge theory, in the presence of a 1/2-BPS line defect. The character enjoys a double refinement, interpreted as a deformation of the usual characters of finite-dimensional representations of quantum affine algebras. We derive and interpret the Schwinger-Dyson identities for the 3d theory from various physical perspectives: in the 3d gauge theory itself, in a 1d gauged quantum mechanics, in 2d $q$-Toda theory, and in 6d little string theory. We establish the dictionary between all approaches. Lastly, we comment on the transformation properties of the vortex character under the action of three-dimensional Seiberg duality.

Non-Perturbative Schwinger-Dyson Equations for 3d ${\cal N} = 4$ Gauge Theories

Abstract

We analyze symmetries corresponding to separated topological sectors of 3d gauge theories with Higgs vacua, compactified on a circle. The symmetries are encoded in Schwinger-Dyson identities satisfied by correlation functions of a certain gauge-invariant operator, the "vortex character." Such a character observable is realized as the vortex partition function of the 3d gauge theory, in the presence of a 1/2-BPS line defect. The character enjoys a double refinement, interpreted as a deformation of the usual characters of finite-dimensional representations of quantum affine algebras. We derive and interpret the Schwinger-Dyson identities for the 3d theory from various physical perspectives: in the 3d gauge theory itself, in a 1d gauged quantum mechanics, in 2d -Toda theory, and in 6d little string theory. We establish the dictionary between all approaches. Lastly, we comment on the transformation properties of the vortex character under the action of three-dimensional Seiberg duality.

Paper Structure

This paper contains 22 sections, 170 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Schwinger-Dyson Equations: the Vortex Quantum Mechanics Perspective
  3. 3d $\mathcal{N}=4$ Gauge Theory, Vortices and Loop Defect
  4. The Vortex Quantum Mechanics
  5. The Index of the Quantum Mechanics
  6. The Index is a Vortex $qq$-character
  7. Physics of the Schwinger-Dyson Equations
  8. Schwinger-Dyson Equations: the 3-dimensional Perspective
  9. A Half-Index Presentation
  10. Relation between the 3d and 1d Expressions for the $qq$-character
  11. Schwinger-Dyson Equations: the ${\mathcal{W}}_{q,t}({\mathfrak{g}})$-Algebra Perspective
  12. The Deformed ${\mathcal{W}}_{q,t}({\mathfrak{g}})$-Algebra
  13. The Vortex $qq$-character is a Deformed ${\mathcal{W}}_{q,t}({\mathfrak{g}})$-Algebra Correlator
  14. Schwinger-Dyson Equations: the Little String Perspective
  15. Little String Basics
  16. ...and 7 more sections

Figures (12)

  • Figure 1: Example of the $G^{3d}$ theory $T_\rho[SU(N^{(n+1)})]$, and its vortex quantum mechanics $T^{1d}_{pure}$. Note we use a 3d $\mathcal{N}=4$ notation for the quiver on top, and a 1d $\mathcal{N}=4$ notation for the quiver on the bottom.
  • Figure 2: On the top, a loop defect is placed in the 3d theory $T_\rho[SU(N^{(n+1)})]$, with associated defect group $U(L^{(2)})$. We denoted the loop by a green cross. On the bottom, the vortex quantum mechanics $T^{1d}$ is displayed; black and green links denote 1d $\mathcal{N}=4$ chiral multiplets obtained by reduction of 2d $\mathcal{N}=(2,2)$ supersymmetry.
  • Figure 3: The black crosses denote poles in the set $\mathcal{M}^{pure}_k$, from the pure index, while the red dot denotes a pole in the set $\mathcal{M}_k\setminus \mathcal{M}^{pure}_k$. Such a pole is due to the factor $Z^{(a)}_{defect}$ in the integrand. On the left, we show a possible contour for the computation of the index at $k=1$. Note that by the JK prescription, we must in particular enclose the new pole in red. Remarkably, it is equivalent to trade this contour for the one on the right, which now only encloses the poles in the set $\mathcal{M}^{pure}_1$, but with a modified integrand; in the latter contour, the integrand will now contain insertions of additional $Y$-operators, with a vortex charge shift of one unit to account for the missing pole.
  • Figure 4: A vanishing 2-cycle of an $A_n$ singularity, labeled by $S_a$ (the black 2-sphere), and the dual non-compact 2-cycle $S_a^*$ (the black cigar).
  • Figure 5: The brane configuration in type IIB: there are $N$ D3$_{gauge}$ branes wrapping compact 2-cycles $S_{a}$ and $\mathbb{C}_q$ (yellow), $N_{f}$ D3$_{flavor}$ branes wrapping non-compact 2-cycles $S_{a}^{*}$'s and $\mathbb{C}_q$ (red). There are also $L$ D1$_{defect}$ branes wrapping the non-compact 2-cycles $S_{a}^{*}$'s and $\mathbb{C}_q$ (green). All branes are points on the cylinder $\mathcal{C}$. Later, we will also consider the quantum mechanics of $k$ D1$_{vortex}$ branes (not pictured) wrapping the compact 2-cycles $S_{a}$'s.
  • ...and 7 more figures