A Higher Spin-Statistics Theorem for Invertible Quantum Field Theories

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.