Supersymmetric Wilson Loops, Instantons, and Deformed ${\cal W}$-Algebras

Abstract

Let $\mathfrak{g}$ be a simple Lie algebra. We study 1/2-BPS Wilson loops of supersymmetric 5d $\mathfrak{g}$-type quiver gauge theories on a circle, in a non-trivial instanton background. The Wilson loops are codimension 4 defects of the gauge theory, and their interaction with self-dual instantons is captured by a modified 1d ADHM quantum mechanics. We compute the partition function as its Witten index. This index is a "$qq$-character" of a finite-dimensional irreducible representation of the quantum affine algebra $U_q(\hat{\mathfrak{g}})$. Using gauge/vortex duality, we can understand the 5d physics in 3d gauge theory terms. Namely, we reinterpret the 5d theory with vortex flux from the point of view of the vortices themselves. This vortex perspective has an advantage: it has yet another dual description in terms of deformed $\mathfrak{g}$-type Toda Theory on a cylinder, in free field formalism. We show that the gauge theory partition function is equal to a chiral correlator of the deformed Toda Theory, with stress tensor and higher spin operator insertions. We derive all the above results from type IIB string theory, compactified on a resolved $ADE$ singularity times a cylinder with punctures, with various branes wrapping the blown-up 2-cycles.

Figures (16)

• Figure 1: A vanishing 2-cycle of an $A_m$ singularity, labeled by $S_a$ (the black 2-sphere), and the dual non-compact 2-cycle $S_a^*$ (the red cigar).
• Figure 2: Brane configuration: there are $n$ D5 branes wrapping compact 2-cycles $S_{a}$'s (blue), $N_{f}$ D5 branes wrapping non-compact 2-cycles $S_{a}^{*}$'s (red). All D5 branes are points on the cyclinder $\mathcal{C}$ and extend in $\mathbb{C}^2$. There are also $L$ D1 branes wrapping the non-compact 2-cycles $S_{a}^{*}$'s (green), sitting at the origin of $\mathbb{C}^2$. All branes are points on the cylinder. Later, we will consider the quantum mechanics of $k$ D1$_{inst}$ branes (not pictured) wrapping the compact 2-cycles $S_{a}$'s.
• Figure 3: A non simply-laced Lie algebra $\mathfrak{g}$ is constructed as a subalgebra of a simply-laced Lie algebra $\mathfrak{g}_0$ that is invariant under the $A$-action on $\mathfrak{g}_0$. Note that $D_4$ is the only simply-laced Lie algebra that admits a $\mathbb{Z}_3$ outer automorphism action, resulting in the non simply-laced $G_2$.
• Figure 4: T-duality tells us that the D5 and D1 branes at points on the cylinder $\mathcal{C}$ in type IIB are the same as D6 and D2 branes wrapping the T-dual cylinder $\mathcal{C}$ in type IIA.
• Figure 5: The black crosses denote poles labeled by Young diagrams, while the red dot denotes a new pole due to a D1 brane insertion, resulting in the factor $Z^{(a)}_{D1}$ in the integrand. On the left, we show a possible contour for the computation of the 5d partition function, say for a $SU(3)$ gauge theory at instanton number $k$. By the JK residue prescription, we must in particular enclose the new pole in red. It turns out it is equivalent to trade this contour for the one on the right, at instanton number $k-1$, which only encloses the usual poles labeled by Young diagrams; this comes at the expense of inserting in the integrand new $Y$-operators and fundamental matter, with an instanton shift of one unit to account for the missing pole.

Theorems & Definitions (7)

• Example 2.1
• Example 2.2
• Example 2.3
• Example 3.1
• Example 4.1
• Example 4.2
• Example 4.3