Static Charges in the Low-Energy Theory of the S-Duality Twist

TL;DR

The paper probes the low-energy Tr-S theory, formed by a 2+1D N=6 supersymmetric theory obtained from an S-duality twist and an R-symmetry twist of 4D N=4 SYM, by inserting static quark/antiquark sources and computing the Witten index on a torus. Using a chain of string dualities, the authors map the system to a type-IIA D2-F1 setup and describe the ground states via a quantum-mechanical model on T^{2m}, encoded by a binding matrix and a permutation. They find that for abelian cases the results reproduce U(1) Chern-Simons theory at level k, but for nonabelian U(n) theories the index does not generally match Chern-Simons predictions, signaling a richer structure beyond a pure CS description. The work also provides a framework for extracting Wilson-loop eigenvalues from the Witten index and discusses SL(2,Z) mapping-class symmetry implications, offering insights into the topological character and potential knot-invariant interpretations of Tr-S. Overall, the results indicate that Tr-S is topological for small n (relative to the twist order) but does not simply decompose into CS theories in general, motivating further exploration of its relation to duality-based constructions like T(U(n)) and potential knot-theoretic invariants.

Abstract

We continue the study of the low-energy limit of N=4 super Yang-Mills theory compactified on a circle with S-duality and R-symmetry twists that preserve N=6 supersymmetry in 2+1D. We introduce external static supersymmetric quark and anti-quark sources into the theory and calculate the Witten Index of the resulting Hilbert space of ground states on a torus. Using these results we compute the action of simple Wilson loops on the Hilbert space of ground states without sources. In some cases we find disagreement between our results for the Wilson loop eigenvalues and previous conjectures about a connection with Chern-Simons theory.

Figures (12)

• Figure 1: External quark and anti-quark sources are realized as endpoints of fundamental strings. At low-energy, the strings are described by free 1+1D fields $\scX_\iStIIB^\mu(x_9,t),\fpsi_\iStIIB(x_9,t)$ and the low-energy modes of the compact interacting system of D$3$-branes and charges are described by periodic variables $\pvar^a(t),\qvar^\alpha(t).$
• Figure 2: The type-IIA configuration following the U-duality transformation of Table \ref{['tab:Dualities']}. The $m$ pairs of type-IIB open strings become $m$ continuous D$2$-branes. The $n$ D$3$-branes become $n$ fundamental strings, which in the presence of the D$2$-branes can break up into open strings. At least one pair of open strings must be attached to each D$2$-brane. In this example $m=4$ and $n=2.$
• Figure 3: (a) A fundamental string (F1) bound to the D$2$-brane. The D$2$-brane wraps the compact direction $x_{10}$ (the vertical direction) and extends indefinitely in direction $x_9$ (not shown in the picture). The fundamental string is at $x_9=0$ and extends in direction $x_3$ (perpendicular to the plane of the drawing). Because of the S-duality twist, which in the type-IIA picture translates to a rotation, the fundamental string's endpoint $z"$ can be different from its starting point $z'.$ (b) Configurations of the D$2$-brane with the two endpoints $z',z"$ of the string marked as oppositely charged points. As the D$2$-brane changes its $x_1$-position, the positions of the charges change accordingly.
• Figure 4: A variable $\pvar^\iDb$ ($\iDb=1,\dots,m$) is associated with each D2-brane, ordered in the direction of increasing $x_3$. Let every string break at every brane into a total of $m+1$ open segments. The constant $x_{10}$ coordinate of each of these segments is denoted by $\yvar^{\iSt\iDbz}$ with $\iSt=1,\dots,n$ and $\iDbz=0,\ldots,m$ in the direction of increasing $x_3$. However, the strings actually break only at positions marked with a • and the main contribution to the action comes from these break points. The strings are coincident, but we separated them in the picture for clarity.
• Figure 5: (a) A junction of two open fundamental strings, one starting and one ending on a D$2$-brane. (b) The D$2$-brane wraps direction $10\equiv\tenx$ and the configuration can be deformed so that it lifts to a smooth holomorphic curve in M-theory.