Gauge Theories with Time Dependent Couplings and their Cosmological Duals

TL;DR

This paper investigates the behavior of N=4 SYM theory with a time-dependent coupling that vanishes at t=0, revealing singularities and potential resolutions, and explores the implications for its cosmological gravity duals.

Contribution

It analyzes the singular behavior of gauge theories with vanishing couplings at t=0 and discusses conditions under which a smooth continuation or singularity occurs, linking to cosmological duals.

Findings

Energy diverges at t=0 for generic states

Wave function phase oscillates wildly if p>1

Singularity is genuine and not easily resolvable

Abstract

We consider the N=4 SYM theory in flat 3+1 dimensional spacetime with a time dependent coupling constant which vanishes at $t=0$, like $g_{YM}^2=t^p$. In an analogous quantum mechanics toy model we find that the response is singular. The energy diverges at $t=0$, for a generic state. In addition, if $p>1$ the phase of the wave function has a wildly oscillating behavior, which does not allow it to be continued past $t=0$. A similar effect would make the gauge theory singular as well, though nontrivial effects of renormalization could tame this singularity and allow a smooth continuation beyond $t=0$. The gravity dual in some cases is known to be a time dependent cosmology which exhibits a space-like singularity at $t=0$. Our results, if applicable in the gauge theory for the case of the vanishing coupling, imply that the singularity is a genuine sickness and does not admit a meaningful continuation. When the coupling remains non-zero and becomes small at $t=0$, the curvature in the bulk becomes of order the string scale. The gauge theory now admits a time evolution beyond this point. In this case, a finite amount of energy is produced which possibly thermalizes and leads to a black hole in the bulk.