String theory, $\mathcal{N}=4$ SYM and Riemann hypothesis

Abstract

We discuss new relations among string theory, four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) and the Riemann hypothesis. It is known that the Riemann hypothesis is equivalent to an inequality for the sum of divisors function $\sigma (n)$. Based on previous results in literature, we focus on the fact that $\sigma (n)$ appears in a problem of counting supersymmetric states in the $\mathcal{N}=4$ SYM with $SU(3)$ gauge group: the Schur limit of the superconformal index plays a role of a generating function of $\sigma (n)$. Then assuming the Riemann hypothesis gives bounds on information on the $1/8$-BPS states in the $\mathcal{N}=4$ SYM. The AdS/CFT correspondence further connects the Riemann hypothesis to the type IIB superstring theory on $AdS_5 \times S^5$. In particular, the Riemann hypothesis implies a miraculous cancellation among Kaluza-Klein modes of the supergravity multiplet and D3-branes wrapping supersymmetric cycles in the string theory. We also discuss possibilities to gain new insights on the Riemann hypothesis from the physics side.