Domenico Fiorenza HomePage

This is the personal web of Domenico Fiorenza within the nLab.

Here is where I’ll discuss on-going projects and collect notes from talks. Currenly, my principal interest is in elliptic cohomology from an extended conformal field theory perspective. As a training I’m now working on 0-dimensional TQFT/singular cohomology and 1-dimensional TQFT/K-theory, and on the relations between them. I’ll write things here first, and then (as they are acceptably clean) on the Lab.

Below are more specific topics I’m thinking to at the moment. For each topic you’ll find a very short introduction and links to the page where it is developed. mother of the bride dresses

Is every connection flat?

Let $G$ be a Lie group, and $\nabla$ a $G$-connection on a principal $G$-bundle. The curvature $F_\nabla$ precisely measures how $\nabla$ fails to be flat. Therefore one can think to remedy the non-flatness of $\nabla$ by adding to it the datum of $F_\nabla$, i.e., by considering a 2-connection locally given by $(\nabla,F_\nabla)$. With this new connection the obstruction to flatness has been moved one step away. Repeating this procedure produces a flat $\infty$-connection in the sense of Schreiber. Details at flat ∞-connections.

The Dijkgraaf-Witten model

Mostly scrath for the moment, at Dijkgraaf-Witten model

Path integrals without integrals

An attempt to describe the abstract nonsense of path-integrals, extremizing the functorial point of view. Still very confused. Hopefully in March I’ll be able to work more seriously on this together with Urs. Details at functorial path integrals.

The standard model: a dictionary for the mathematician

With Giuseppe Malavolta, we are trying to write down a rather effective dictionary between the Physics parlance and the current mathematical description of the same concepts and constructions. There is no nPOV in it (well, maybe some shadow of it here and there…), but we thougt the nLab was the right place where to develop this work in progress after having bumped in this blog entry. Details at the standard model.

Chern character is dimension?

Let $Vect$ be the category of finite dimensional vector spaces over some fixed field $\mathbb{K}$. We can look at $Vect$ as to an $(\infty,1)$-category with trivial higher morphisms. It is a pointed infinity-category with finite limits, so it admits a stabiliziation $Sp(Vect)$, and we can consider the Waldhausen construcion on the stable infinity-category $Sp(Vect)$. This should be the K-theory spectrum.

Taking dimension is not a functor $Vect\to \mathbb{N}$ but it becomes a functor if one restricts morphisms in $Vect$ to be isomorphisms. Moreover, $dim$ is compatible with (direct) sums and (tensor) products. This suggests that there should be a stable version of dimension which is an infinity-functor $Sp(Vect)\to Sp(\mathbb{N})$ (this latter spectrum is the Eilenberg-Mac Lane spectrum). At the cohomology level, stable dimension should induce the Chern character from K-theory to singular cohomology (the fact that the leading term of $Ch$ is $dim$ seems to support this point of view). I’ll try to expand this point of view at Chern character.