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

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.