Musings

According to Atiyah, a $d$-dimensional TQFT is a tensor functor, $Z$, from the category, $Cob(d)$, whose objects are closed $(d-1)$-manifolds¹, and whose morphisms are bordisms $$Hom (M,N) = \{B: \partial B \simeq M\amalg \overline{N}\}/\text{diffeomorphisms}$$ to $\Vect$, the category of complex vector spaces. $Cob(d)$ is a symmetric monoidal category, with $\otimes$ given by disjoint union of manifolds, and $Z$ preserves tensor products $$\begin{gathered} Z(M\amalg N) \simeq Z(M)\otimes Z(N)\\ Z(\emptyset) \simeq \mathbb{C} \end{gathered}$$

The vague idea of extended TQFT is to replace $Cob(d)$ with some sort of $n$-category (where $n=d$), consisting of manifolds of all dimension $m\leq d$, and replace $Vect$ with a similarly fancied-up $n$-category, $\mathcal{C}$.

Over the course of several lectures, Jacob proposes to tell us exactly what these all are, but the vague version is as follows:

For $m\leq d$, a $d$-framing of a manifold, $M$, of dimension $m$, is a trivialization of $T_M\oplus \mathbb{R}^{d-m}$.

$Cob(d)^{\text{framed}}_{\text{ext}}$ is a symmetric monoidal $d$-category (with tensor product given by disjoint union of manifolds).

• Objects are $0$-manifolds with a $d$-framing.
• Morphisms are $d$-framed bordisms between $d$-framed $0$-manifolds.
• 2-Morphisms are $d$-framed bordisms between $d$-framed $1$-manifolds.
• ⋮
• $d$-morphisms are $d$-framed $d$-manifolds (with corners) $/\text{diff}$

The statements which he proposes to prove are that

Given a $d$-category, $\mathcal{C}$, with tensor product, an ETQFT “with values in $\mathcal{C}$” is a tensor functor

1. $Z:\, Cob{d}^{\text{framed}}_{\text{ext}}(d) \to \mathcal{C}$
2. The “fully dualizable” objects, $X\in \mathcal{C}$ are given by $X = Z(\bullet)$.

In some sense, the whole ETQFT is determined by knowing what $Z$ of a point is. Here, “fully dualizable objects” is some condition analogous to demanding that the vectors spaces in an ordinary TQFT, associated to a $(d-1)$ manifold, are finite-dimensional.

There is, of course, a 110 page paper providing “an outline” of the idea. Probably, it will be more intelligible than my summary.