't Hooft Operators in Gauge Theory from Toda CFT

TL;DR

This paper establishes an exact correspondence between loop operators in 2d A_{N-1} Toda CFT and supersymmetric loop observables in 4d ${\cal N}=2$ SU(${N}$) gauge theories on $S^4$, using Verlinde loop operators and the AGT framework. It derives explicit fusion and braiding matrices for degenerate representations and applies them to compute the exact expectation values of 't Hooft and dyonic loops in ${\cal N}=2^*$ and ${\cal N}=2$ conformal SQCD, including their S-duality relations to Wilson loops. The results express the loop vevs as Nekrasov partition data (classical, 1-loop, and instanton pieces) shifted by magnetic charges, with detailed formulas for diagonal and off-diagonal contributions and a discussion of monopole bubbling. The authors argue that higher-rank theories may require topological web operators in Toda CFT to capture loop operators not in the Wilson-duality orbit, signaling a rich structure and suggesting future exploration of topological webs and quiver generalizations.

Abstract

We construct loop operators in two dimensional Toda CFT and calculate with them the exact expectation value of certain supersymmetric 't Hooft and dyonic loop operators in four dimensional \Ncal=2 gauge theories with SU(N) gauge group. Explicit formulae for 't Hooft and dyonic operators in \Ncal=2^* and \Ncal=2 conformal SQCD with SU(N) gauge group are presented. We also briefly speculate on the Toda CFT realization of arbitrary loop operators in these gauge theories in terms of topological web operators in Toda CFT.

Figures (8)

• Figure 1: The punctured Riemann surfaces $C_{1,1}$ and $C_{0,4}$ associated to ${\cal N}=2^*$ and ${\cal N}=2$ conformal SQCD.
• Figure 2: Decomposition of the punctured Riemann surfaces $C_{1,1}$ and $C_{0,4}$ into pairs of pants, and the corresponding trivalent graphs $\Gamma_\sigma$. The momenta ${\rm \hat{m}}, {\rm \hat{m}}_2, {\rm \hat{m}}^*_3$ are semi-degenerate, and $\alpha, {\rm m}_1, {\rm m}^*_4$ are non-degenerate.
• Figure 3: Toda loop operators describing 't Hooft operators in ${\cal N}=2^*$ and ${\cal N}=2$ conformal SQCD. The green curves denote the support of the loop operator and the black ones define the pants decomposition.
• Figure 4: 't Hooft loop as the monodromy associated with moving $V_\mu(z)$ along the b-cycle of $C_{1,1}$. The Toda momenta labeling the edges are $\mu=-bh_1$, $\mu^* = b h_N$, ${\rm \hat{m}}= N\left({q/2}+i\hat{m}\right)\, \omega_{N-1}$, $\alpha'_l = \alpha - b h_l$, and $\alpha'_k = \alpha-b h_k$ for $l,k=1,\ldots,N$.
• Figure 5: 't Hooft loop as the monodromy associated with moving $V_\mu(z)$ along $p_t$ on $C_{0,4}$. The Toda momenta labeling the edges are described in the main text: $\alpha'_l = \alpha - b h_l$, and $\alpha"_{l,k} = \alpha-b h_l + b h_k$ for $1\leq l,k\leq N$. For the curve $p_u$, the monodromies are replaced by: