Duality, Partial Supersymmetry, and Arithmetic Number Theory

TL;DR

This paper explores dualities in quantum theories connected to arithmetic functions, introducing partial supersymmetry and deriving number theoretic identities, including results related to the Riemann zeta function.

Contribution

It develops the concept of partial supersymmetry in quantum theories and links duality and partition functions to number theory and arithmetic functions.

Findings

Identified dual quantum theories with identical partition functions.

Constructed fermionic and parafermionic thermal partition functions.

Derived number theoretic identities related to the Riemann zeta function.

Abstract

We find examples of duality among quantum theories that are related to arithmetic functions by identifying distinct Hamiltonians that have identical partition functions at suitably related coupling constants or temperatures. We are led to this after first developing the notion of partial supersymmetry-in which some, but not all, of the operators of a theory have superpartners-and using it to construct fermionic and parafermionic thermal partition functions, and to derive some number theoretic identities. In the process, we also find a bosonic analogue of the Witten index, and use this, too, to obtain some number theoretic results related to the Riemann zeta function.