One-Loop Free Energy of Tensionless Type IIB String in AdS$_5\times$S$^5$

TL;DR

This paper computes the one-loop free energy of tensionless type IIB string theory in AdS$_5\times$S$^5$ at zero 't Hooft coupling, revealing a finite result that matches the boundary super-Yang-Mills theory's expected behavior.

Contribution

It introduces a method to calculate the one-loop free energy of tensionless strings in AdS$_5\times$S$^5$ using character integral representation, and demonstrates the matching with boundary theory results.

Findings

The free energy for each Regge trajectory is proportional to a single Maxwell multiplet.

The summed free energy over all trajectories yields a finite result.

The bulk free energy matches the -1 term in the boundary theory's N^2-1 factor.

Abstract

Considering the zero 't Hooft coupling limit of ${\cal N}=4$ super-Yang-Mills theory, the exact spectrum of all single-trace operators can be accessed in terms of the underlying $so(2,4)$ character. This makes it possible in turn to compute the one-loop free energy of the tensionless type IIB string theory in AdS$_5\times$S$^5$ background, with help of the recently developed method of character integral representation of zeta function (CIRZ). We calculate first the one-loop free energy of the string states in the $(p-1)$-th Regge trajectory and find the result to be $p$ times the free energy of a single ${\cal N}=4$ Maxwell multiplet. The full one-loop free energy is hence proportional to the divergent series $\sum_{p=2}^\infty p\,$. The divergence arises as a result of interrupting the regularization procedure in an intermediate stage. With a reorganization of states, we extract the finite part of free energy after summing over the Regge trajectories. This way gives us a finite result which is minus of the free energy of the ${\cal N}=4$ multiplet. Hence, this bulk one-loop result matches the -1 term in the $N^2-1$ factor of the boundary result.