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

TL;DR

This paper explores deep connections between string theory, supersymmetric gauge theories, and the Riemann hypothesis, suggesting that bounds derived from the hypothesis influence the counting of BPS states and lead to cancellations in superstring models.

Contribution

It establishes novel links between the Riemann hypothesis and supersymmetric state counting in $ ext{SYM}$, and proposes implications for string theory via AdS/CFT correspondence.

Findings

Riemann hypothesis bounds affect BPS state counts in $ ext{SYM}$

Connections between divisor sum inequalities and supersymmetric indices

Potential for physics to provide insights into the 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.