nLab
0-dimensional TQFT

Idea

A 0-dimensional TQFT is a TQFT regarded in the sense of FQFT as a representation of the category of 0-dimensional cobordisms.

This degenerate case turns out to exhibit a nontrivial amount of interesting information, in particular if regarded in the context of super QFT.

Definition

0-Dimensional Cobordisms

The category ${\mathrm{Cob}}_0$ of 0-dimensional cobordisms is the symmetric monoidal category ${\mathrm{Cob}}_0$ having the $-1$-dimensional manifold $\varnothing$ as the only object and isomorphism classes of compact $0$-dimensional manifolds as morphisms. Clearly ${\mathrm{Cob}}_0$ is equivalent to $B\mathbb{N}$

0-Dimensional TQFT

A 0-dimensional TQFT (with values in $\mathbb{Z}$-modules) is a monoidal functor

$$Z:{\mathrm{Cob}}_0\to \mathbb{Z}\mathrm{Mod}.$$Z\colon Cob_0\to \mathbb {Z} Mod \,.

By definition of monoidal functor, one has $Z(\varnothing )=\mathbb{Z}$ and so $Z$ is completely (and freely) determined by the assignment $Z(\{\mathrm{pt}\}\in {\mathrm{End}}_{\mathbb{Z}}(\mathbb{Z})=\mathbb{Z}$. In other words, the space of 0-dimensional TQFTs is $\mathbb{Z}$.

Over a manifold

One can consider TQFTs with a target manifold $X$: all bordisms are required to have a map to $X$.

In dimension $0$, morphisms in ${\mathrm{Cob}}_0(X)$ are the topological monoid ${\bigcup }_{n\ge 1}{\mathrm{Sym}}^n(X)$. In particular, continuous tensor functors from ${\mathrm{Cob}}_0(X)$ to $\mathbb{Z}$-modules are naturally identified with degree 0 integral cohomology $H^0(X;\mathbb{Z})$.

Extended version

The picture becomes more interesting if one goes from topological field theory to extended topological quantum field theory. Indeed, from this point of view, to the $-1$-dimensional vacuum is assigned the symmetric monoidal 0-category $\mathbb{Z}$, and consequently, the infinity-version of the space of all $0$-dimensional TQFTs is the Eilenberg-Mac Lane spectrum. It follows that the space of extended $0$-dimensional TQFTs with target $X$ (taking values in $\mathbb{Z}$-modules) is the graded integral cohomology ring $H^{*}(X;\mathbb{Z})$.

Super version

From the differential geometry point of view, a relation between de Rham cohomology of a smooth manifold $X$ and $0$-dimensional functorial field theories arises if one moves from topological field theory to $(0\mid 1)$-supersymmetric field theory, see Axiomatic field theories and their motivation from topology.

It would be interesting to describe a direct connection between the extended and the susy theory; it should parallel the usual Cech-de Rham argument