Gauge Theories with Time Dependent Couplings and their Cosmological Duals

TL;DR

This work investigates cosmological singularities within the gauge/gravity duality by studying N=4 SYM with a time-dependent dilaton that vanishes at the singularity. Through a quantum-mechanical toy model and a detailed field-theory analysis, it finds that energy input diverges as $t\to 0$ and, for $p>1$, the wavefunction phase becomes wildly oscillatory, hindering continuation beyond the singularity. In the gravity dual, spacelike singularities arise when the dilaton vanishes, while if the coupling remains nonzero and becomes small, the bulk curvature nears the string scale and time evolution can proceed, potentially yielding a black hole after thermalization. The results suggest that genuine sick singularities occur for vanishing coupling, whereas non-vanishing, small couplings could lead to finite energy production and black hole formation, with renormalization offering possible taming mechanisms; universality in near-singularity behavior is highlighted via BKL-type analyses. These insights clarify the boundary theory’s fate at bulk singularities and inform how cosmological solutions may evolve into or avoid black-hole geometries within AdS/CFT.

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.

Figures (1)

• Figure 1: Here p$_3$ and p$_2$ are plotted as functions of p$_1$ for $\alpha=0$ case. Notice how the regions $-1/3 \leq p_1 \leq 0$, $0 \leq p_1 \leq 2/3$, $2/3 \leq p_1 \leq 1$ have the same values for $p_i$'s