Precision lattice test of the gauge/gravity duality at large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>

Abstract

We perform a systematic, large-scale lattice simulation of D0-brane quantum mechanics. The large-$N$ and continuum limits of the gauge theory are taken for the first time at various temperatures $0.4\ensuremath{\le}T\ensuremath{\le}1.0$. As a way to test the gauge/gravity duality conjecture we compute the internal energy of the black hole as a function of the temperature directly from the gauge theory. We obtain a leading behavior that is compatible with the supergravity result $E/{N}^{2}=7.41{T}^{14/5}$: the coefficient is estimated to be $7.4\ifmmode\pm\else\textpm\fi{}0.5$ when the exponent is fixed and stringy corrections are included. This is the first confirmation of the supergravity prediction for the internal energy of a black hole at finite temperature coming directly from the dual gauge theory. We also constrain stringy corrections to the internal energy.