D=0 TQFT in nLab

Contents

Contents

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 Cob 0Cob_0 of 0-dimensional cobordisms is the symmetric monoidal category Cob 0Cob_0 having the −1-1-dimensional manifold ∅\emptyset as the only object and isomorphism classes of compact 00-dimensional manifolds as morphisms. Clearly Cob 0Cob_0 is equivalent to Bℕ\mathbf{B}\mathbb{N}

0-Dimensional TQFT

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

Z:Cob 0→ℤMod. Z\colon Cob_0\to \mathbb {Z} Mod \,.

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

Over a manifold

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

In dimension 00, morphisms in Cob 0(X)Cob_0(X) are the topological monoid ⋃ n≥1Sym n(X)\bigcup_{n\geq 1} Sym^n(X). In particular, continuous tensor functors from Cob 0(X)Cob_0(X) to ℤ\mathbb{Z}-modules are naturally identified with degree 0 integral cohomology H 0(X;ℤ)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 −1-1-dimensional vacuum is assigned the symmetric monoidal 0-category ℤ\mathbb{Z}, and consequently, the infinity-version of the space of all 00-dimensional TQFTs is the Eilenberg-Mac Lane spectrum. It follows that the space of extended 00-dimensional TQFTs with target XX (taking values in ℤ\mathbb{Z}-modules) is the graded integral cohomology ring H *(X;ℤ)H^*(X;\mathbb{Z}).

Super version

From the differential geometry point of view, a relation between de Rham cohomology of a smooth manifold XX and 00-dimensional functorial field theories arises if one moves from topological field theory to (0|1)(0|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