The spin-statistics theorem asserts that in a unitary quantum field theory, the spin of a particle—characterized by its transformation under the central element of the spin group, which corresponds to a 360-degree rotation—determines whether it obeys bosonic or fermionic statistics. This relationship can be formalized mathematically as equivariance for a geometric and algebraic action of the 2-group $B\mathbf{Z}_2$. In my talk, I will present a refinement of these actions, extending from $B\mathbf{Z}_2$ to appropriate actions of the stable orthogonal group $O$, and demonstrate that every unitary invertible quantum field theory intertwines these geometric and algebraic $O$-actions.

The proof draws on tools from stable homotopy theory, which offer a powerful framework for describing invertible quantum field theories. I will also provide a gentle introduction to functorial quantum field theories, which frame quantum field theory as the representation theory of bordism categories, and explain the connections between invertible quantum field theories and stable homotopy theory.