A Higher Spin-Statistics Theorem for Invertible Quantum Field Theories

TL;DR

This paper generalizes the spin-statistics theorem to invertible quantum field theories by introducing higher spin and higher statistics actions of the orthogonal group, demonstrating their intertwining in unitary cases.

Contribution

It formulates a higher spin-statistics theorem for invertible quantum field theories, extending classical concepts through new group actions on spacetime and algebraic structures.

Findings

Proves that unitary invertible QFTs satisfy the higher spin-statistics relation.

Defines higher spin and higher statistics actions of the orthogonal group.

Shows that invertible QFTs intertwine these actions.

Abstract

We prove that every unitary invertible quantum field theory satisfies a generalization of the famous spin-statistics theorem. To formulate this extension, we define a `higher spin' action of the stable orthogonal group $O$ on appropriate spacetime manifolds, which extends both the reflection involution and spin flip. On the algebraic side, we define a `higher statistics' action of $O$ on the universal target for invertible field theories, $I\mathbb{Z}$, which extends both complex conjugation and fermion parity $(-1)^F$. We prove that every unitary invertible quantum field theory intertwines these actions.