One-loop free energy of tensionless type IIB string in AdS$_5\times$S$^5$
Résumé
Considering the zero ’t Hooft coupling limit of $ \mathcal{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$_{5}$×S$^{5}$ background, with help of the recently developed method of character integral representation of zeta function (CIRZ). We calcu-late 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 $ \mathcal{N}=4 $ Maxwell multiplet. The full one-loop free energy is hence proportional to the divergent series $ {\displaystyle {\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 $ \mathcal{N}=4 $ multiplet. Hence, this bulk one-loop result matches the −1 term in the N$^{2}$ − 1 factor of the boundary result.