Manifestly unitary higher Hilbert spaces

Abstract: A key aspect of quantum theory its insistence that states evolve via unitary transformations. In order to understand the symmetries of higher dimensional quantum field theory, we need to develop higher dimensional analogues of unitarity. The language and theory of higher categories has greatly clarified the way we express these higher symmetries, but unfortunately this language imposes a certain dogma seems to be in conflict with various attempts at describing unitarity. In the nLab for example, there is a great debate over whether or not unitary structures on a (higher) category are `evil’; at term which is both dogmatic and technically precise.

Various attempts have been made to force these structures to `play nice’ with one another, to varying degrees of success. In this talk I will present our most recent contribution to these efforts: defining the notion of a 3-Hilbert space. Our work aims to encode a kind of evaluation on spheres of every dimension that plays nicely with duality structures that are imposed by the cobordism hypothesis. I will show how this compatibility is stronger than simply having daggers at all levels, thus differentiating our construction from previous attempts at higher unitarity. If time permits, we will discuss a roadmap for unitarity in any dimension via a unitary version of condensation completion.

mathematical physicsalgebraic topologycategory theory

Audience: researchers in the topic

Topology and Geometry Seminar (Texas, Kansas)