Abstracts for Physical Math Seminar

Costello and Gwilliam developed a systematic approach to perturbative quantization of field theories that describes classical and quantum observables as factorization algebras on the space-time manifold. In the simple case of the linear sigma model whose fields are smooth maps from the real line to an inner product space $V$ (aka the free boson), the algebra of quantum observables is the Weyl algebra of differential operators on $V$. For the free fermion, where $V$ is replaced by its odd analog, the algebra of quantum observables is the Clifford algebra generated by $V$.

Freed and Moore's "Twisted Equivariant Matter" establishes a K-theory-based classification for free fermion symmetry-protected topological phases, building on the work of Kitaev. I present a generalization of their approach which does not assume the protecting symmetry group contains a $U(1)$. I will explain how the tenfold way arises using Wall's classification of $\mathbb{Z}_2$-graded division algebras. If time permits, I will explain how it relates to the tenfold way one can find in Freed-Moore, originally due to Ryu-Schnyder-Furusaki-Ludwig.

We discuss the language of functorial field theories, i.e. symmetric monoidal functors on geometric bordism categories, starting with the relation of 1-dimensional Euclidean theories to quantum mechanics. We explain how their concordance classes can lead to generalized cohomology and how this extends to the twisted, equivariant cases.