Integrable Wilson loops

Abstract

The generalized quark-antiquark potential of $\mathcal{N}=4$ supersymmetric Yang-Mills theory on ${{\mathbb{S}}^3}\times \mathbb{R}$ calculates the potential between a pair of heavy charged particles separated by an arbitrary angle on ${{\mathbb{S}}^3}$ and also an angle in flavor space. It can be calculated by a Wilson loop following a prescribed path and couplings, or after a conformal transformation, by a cusped Wilson loop in flat space, hence also generalizing the usual concept of the cusp anomalous dimension. In $\mathbb{A}\mathrm{d}{{\mathrm{S}}_5}\times {{\mathbb{S}}^5}$ this is calculated by an infinite open string. I present here an open spin-chain model which calculates the spectrum of excitations of such open strings. In the dual gauge theory these are cusped Wilson loops with extra operator insertions at the cusp. The boundaries of the spin-chain introduce a non-trivial reflection phase and break the bulk symmetry down to a single copy of $\mathfrak{p}\mathfrak{s}\mathfrak{u}\left( {\left. 2 \right|2} \right)$. The dependence on the two angles is captured by the two embeddings of this algebra into $\mathfrak{p}\mathfrak{s}\mathfrak{u}{{\left( {\left. 2 \right|2} \right)}^2}$, i.e., by a global rotation. The exact answer to this problem is conjectured to be given by solutions to a set of twisted boundary thermodynamic Bethe ansatz integral equations. In particular the generalized quark-antiquark potential or cusp anomalous dimension is recovered by calculating the ground state energy of the minimal length spin-chain, with no sites. It gets contributions only from virtual particles reflecting off the boundaries. I reproduce from this calculation some known weak coupling perturtbative results.