Topological quantum field theory

A quantum field theory assigning vector spaces to closed spatial manifolds and linear maps to interpolating spacetime cobordisms, compatibly with composition and disjoint union. Its TQFT partition function on $\Sigma \times S^1$ is the trace of the identity of the state space on $\Sigma$, hence its dimension.

The cobordism hypothesis classifies fully extended framed topological quantum field theories by the object assigned to a point. Precisely, for a suitable symmetric monoidal $(\infty ,n)$-category $\mathcal{C}$, evaluating a framed theory at a positively framed point identifies its space of theories with the space of fully dualizable objects of $\mathcal{C}$. The word “fully” records duals and adjoints at the lower-dimensional levels needed to cut and glue manifolds.^[ref]

In one dimension, a finite-dimensional vector space $V$ has a dual $V^{\ast }$. Evaluation and coevaluation describe interval pieces; closing them into a circle gives the trace of the identity, $Z(S^1)=\dim V$. This is the simplest glimpse of how assigning data to a point controls closed-manifold invariants.

The framing hypothesis matters. Passing to oriented or other tangentially structured theories requires corresponding extra compatibility data, often expressed as a homotopy-fixed-point condition. Nor does the statement classify arbitrary unextended field theories by their point values: those theories do not include all the lower-dimensional data in the first place.

The scalar assigned to a closed spacetime manifold by a topological quantum field theory. Gluing the two ends of a cylinder over a spatial manifold $\Sigma$ gives $Z(\Sigma \times S^1)=\mathrm{tr}(1_{V(\Sigma )})=\dim V(\Sigma )$. This is the ground-state degeneracy when the state space describes the topological ground sector of a gapped system.

A topological quantum field theory TQFT partition function for a closed oriented three-manifold presented by Dehn surgery on a framed link. With normalized Kirby color on each surgery component and the convention $Z(S^3)=1/\mathcal{D}$, it is ${\mathcal{D}}^{-1}{e}^{-2\pi ic\sigma /8}$ times the colored link evaluation. Here $\sigma$ is the signature of the linking matrix and $c$ is the chiral central charge.