Schwarzschild radius from Monte Carlo calculation of the Wilson loop in supersymmetric matrix quantum mechanics

Abstract

In the string/gauge duality it is important to understand how the space-time geometry is encoded in gauge theory observables. We address this issue in the case of the D0-brane system at finite temperature T. Based on the duality, the temporal Wilson loop operator W in gauge theory is expected to contain the information of the Schwarzschild radius R_{Sch} of the dual black hole geometry as log <W> = R_{Sch} / (2 pi alpha' T). This translates to the power-law behavior log <W> = 1.89 (T/lambda^{1/3})^{-3/5}, where lambda is the 't Hooft coupling constant. We calculate the Wilson loop on the gauge theory side in the strongly coupled regime by performing Monte Carlo simulation of supersymmetric matrix quantum mechanics with 16 supercharges. The results reproduce the expected power-law behavior up to a constant shift, which is explainable as alpha' corrections on the gravity side.