Seminários de até

I will begin by reviewing geometric and deformation quantization of a symplectic vector space. The goal will be to explain an analogy between these objects and Rozansky–Witten theory (along with a certain four-dimensional TQFT). This analogy will factor through an analogy concerning three-dimensional TQFTs generated by pointed fusion categories. Throughout, there will be an emphasis on equivariance and anomalies.

The “alpha” version of factorization homology pairs framed n-manifolds with $E_n$-algebras. This construction generalizes the classical homology of a manifold, yields novel results concerning configuration spaces of points in a manifold, and supplies a sort of state-sum model for sigma-models (i.e., mapping spaces) to (n-1)-connected targets. This “alpha” version of factorization homology novelly extends Poincaré duality, shedding light on deformation theory and dualities among field theories. Being defined using homotopical mathematical foundations, “alpha” factorization homology is manifestly functorial and continuous in all arguments, notably in moduli of manifolds and embeddings between them, and it satisfies a local-to-global expression that is inherently homotopical in nature.

Now, $E_n$-algebras can be characterized as $(\infty,n)$-categories equipped with an (n-1)-connected functor from a point. The (full) “beta” version of factorization homology pairs framed n-manifolds with pointed $(\infty,n)$-categories with adjoints. Applying 0th homology, or $\pi_0$, recovers a version of the string net construction on surfaces, as well as skein modules of 3-manifolds. In some sense, the inherently homotopical nature of (full) “beta” factorization homology affords otherwise unforeseen continuity in all arguments, and local-to-global expressions.

In this talk, I will outline a definition of “beta” factorization homology, focusing on low-dimensions and on suitably reduced $(\infty,n)$-categories (specifically, braided monoidal categories). I will outline some examples, and demonstrate some features of factorization homology. Some of this material is established in the literature, some a work in progress, and some conjectural — the status of each assertion will be made clear. I will be especially interested in targeting this talk to those present, and so will welcome comments and questions.

A what?!

We will give a gentle introduction to gerbes and other assorted "higher structures" from topology and mathematical physics. Gerbes are a generalization of line bundles. For a line bundle, the space of sections forms a vector space, and a little extra geometric structure can make it into a Hilbert space, beloved by quantum physicists everywhere. After introducing gerbes, we will ponder the analogous construction: how do we define a "Hilbert space of sections" for a gerbe?

This expository talk is based on the work of Bunk and Szabo.