Properties of Chiral Wilson Loops

TL;DR

The paper develops a non-renormalization framework for a class of chiral Wilson loops in maximally supersymmetric Yang-Mills theories by embedding them in a ${ m N}=2$, $d=1$ chiral ring. Using a lower-dimensional superspace, it derives a chiral loop equation and shows that the anomaly vanishes for $D<7$, leading to shape-independence and, via factorization in the chiral ring, $ig\langle\frac{1}{N}{\rm tr}W\big\rangle = 1$. The results extend to $D=3,5,6$, while in $D=7$ a generalized Konishi anomaly yields loop-equation terms analogous to those in Chern-Simons theory, signaling a breakdown of the previous non-renormalization. The work provides a holomorphic, non-perturbative constraint on BPS Wilson loops and connects field-theoretic loop equations to geometric and topological structures, with potential implications for AdS/CFT and holomorphic matrix-model descriptions.

Abstract

We study a class of Wilson Loops in N =4, D=4 Yang-Mills theory belonging to the chiral ring of a N=2, d=1 subalgebra. We show that the expectation value of these loops is independent of their shape. Using properties of the chiral ring, we also show that the expectation value is identically 1. We find the same result for chiral loops in maximally supersymmetric Yang-Mills theory in three, five and six dimensions. In seven dimensions, a generalized Konishi anomaly gives an equation for chiral loops which closely resembles the loop equations of the three dimensional Chern-Simons theory.

Figures (2)

• Figure 1: (Almost) loop equation.
• Figure 2: Using zig-zag symmetry and shape independence to show that $\langle\frac{1}{N}{\rm}trW(C)\rangle= 1$. This assumes that shape independence still holds when cusps are introduced. An argument which avoids this assumption relies on properties of the chiral ring.