Line Operators in Supersymmetric Gauge Theories and the 2d-4d Relation

Abstract

Four-dimensional gauge theories with $\mathcal N=2$ supersymmetry admit half-BPS line operators. We review the exact localization methods for analyzing these operators. We also review the roles they play in the relation between four- and two-dimensional field theories, and explain how the two-dimensional CFT can be used to obtain the quantitative results for 4d line operators. This is a contribution to the special LMP volume on the 2d-4d relation, edited by J. Teschner.

A citation of the form [V:x] refers to article number x in this volume.

Appendix: Summary of Relevant Facts

1.1 Class $\mathcal S$ Theories of Type $A_1$

The low-energy theory in the world-volume of two M5-branes is a six-dimensional ${\mathcal N}=(0,2)$ supersymmetric theory with no known Lagrangian description. An $A_1$-theory of class $\mathcal S$ is believed to arise by compactifying the six-dimensional theory on the Riemann surface $C_{g,n}$ of genus g with n punctures, with each puncture carrying a codimension-two defect of the (0, 2) theory ([41, 61], [V:2]). The $A_1$-theories of class $\mathcal S$ provide basic examples of 2d-4d correspondence.

A weakly coupled description of such a theory may be encoded in a choice of decomposition of $C_{g,n}$ into $3g-3+n$ pairs of pants, and a trivalent graph $\Gamma$ drawn on $C_{g,n}$. Each pair of pants contains one vertex, and three edges come out through distinct boundary components (pants legs). An example is shown in Fig. 1a. We allow a pants leg to degenerate to a puncture. The graph $\Gamma$ has $3g-3+n$ internal edges and n external edges ending on the punctures. The field content in this description of the $\mathcal{N}=2$ theory can be read off from $\Gamma$ by associating to each internal edge an $\textit{SU}(2)$ gauge group and to each vertex eight half-hypermultiplets in the trifundamental representation of the $\textit{SU}(2)^3$ group associated to the three attached edges. When the edge is external the $\textit{SU}(2)$ symmetry corresponds to a flavor symmetry. A change of pants decomposition and $\Gamma$ corresponds to a S-duality transformation.

1.2 Liouville Theory

Liouville field theory is formally defined by the path integral over a single real field $\phi$ weighted by $e^{-S}$, where

$$\begin{aligned} S=\frac{1}{4\pi } \int _C \left( \partial ^\mu \phi \partial _\mu \phi +4\pi \mu e^{2b\phi }+QR \phi \right) . \end{aligned}$$

(57)

Here R is the scalar curvature, and $Q=b+1/b$ parametrizes the central charge $c=1+6Q^2$. The “cosmological constant” $\mu$ can be absorbed into a shift of $\phi$, and affects the theory in a very mild way. Liouville theory is a non-rational CFT, meaning that it contains infinitely many representations of the Virasoro algebra. The spectrum of representations is continuous, and the conformal dimension $\Delta$ is parametrized by the Liouville momentum $\alpha \in Q/2+i\mathbb R_{\ge 0}$ as $\Delta =\alpha (Q-\alpha )$. We denote the corresponding primary field by $V_\alpha$.

Fusion move coefficients $F_{s_1 s_2}=F_{s_1 s_2} \big [{\begin{matrix} \alpha _3 &{} -b/2\\ \alpha _4 &{} \alpha _1 \end{matrix}}\big ]$ in (39) are explicitly known:

$$\begin{aligned} \begin{aligned} F_{++}&=\frac{\Gamma (b(2\alpha _1-b))\Gamma (b(b-2\alpha _3)+1)}{\Gamma (b(\alpha _1-\alpha _3-\alpha _4+b/2)+1)\Gamma (b(\alpha _1-\alpha _3+\alpha _4-b/2))}, \\&\quad \,\text {etc.} \end{aligned} \end{aligned}$$

(58)