Lattice and string worldsheet in AdS/CFT: a numerical study

TL;DR

This paper explores a lattice discretization of the Green-Schwarz superstring in AdS/CFT, measuring the cusp anomalous dimension through numerical simulations, revealing qualitative agreement at large coupling and challenges at small coupling.

Contribution

It introduces a lattice approach for the superstring sigma-model in AdS/CFT and performs numerical measurements of the cusp anomalous dimension, highlighting issues like the sign problem at small coupling.

Findings

Qualitative agreement with expected behavior at large coupling

Obstruction of the continuum limit at small coupling due to divergence

Detection of a phase in the fermion determinant causing a sign problem

Abstract

We consider a possible discretization for the gauge-fixed Green-Schwarz (two-dimensional) sigma-model action for the Type IIB superstring and use it for measuring the cusp anomalous dimension of planar $\mathcal{N}=4$ SYM as derived from string theory. We perform lattice simulations employing a Rational Hybrid Monte Carlo (RHMC) algorithm and a Wilson-like fermion discretization. In this preliminary study, we compare our results with the expected behavior for very large values of $g=\frac{\sqrt{\lambda}}{4\pi}$, which is the perturbative regime of the sigma-model, and find a qualitative agreement at finite lattice spacing. For smaller $g$ the continuum limit is obstructed by a divergence. We also detect a phase in the fermion determinant, whose origin we explain, which for small $g$ leads to a sign problem not treatable via standard reweigthing. Results presented here are discussed thoroughly in~\cite{toappear}.