Finite gauge state spaces and bordism spans

Fix an integer $d\ge 1$ and a finite group $G$. A triangulated $d$-bordism $M:{\mathrm{\Sigma }}_0\to {\mathrm{\Sigma }}_1$ is a compact oriented piecewise-linear $d$-manifold with product collars on its boundary, where ${\mathrm{\Sigma }}_0$ and ${\mathrm{\Sigma }}_1$ are closed oriented $(d-1)$-manifolds with finite triangulations and the incoming boundary is identified with $-{\mathrm{\Sigma }}_0$ while the outgoing boundary is identified with ${\mathrm{\Sigma }}_1$. The boundary triangulations agree with the induced triangulation of $M$.

For a closed $(d-1)$-manifold $\mathrm{\Sigma }$, define the state space

$V_G(\mathrm{\Sigma }):=\mathrm{Fun}({\pi }_0({\mathcal{C}}_G(\mathrm{\Sigma })),\mathbb{C}),⟨f,g{⟩}_{\mathrm{\Sigma }}:={\sum }_{[A]\in {\pi }_0({\mathcal{C}}_G(\mathrm{\Sigma }))}\frac{\overline{f([A])}g([A])}{\mathrm{\mid }{\mathrm{Aut}}_{{\mathcal{C}}_G(\mathrm{\Sigma })}(A)\mathrm{\mid }}\mathrm{.}$

The orientation reversal $-\mathrm{\Sigma }$ has the same underlying triangulated cell complex and the same chosen directions on its edges, with only the manifold orientation reversed. Since the flat-connection groupoid uses the cell complex, directed edge labels, flatness equations, and vertex gauge action, not the manifold orientation, the identity on these data gives ${\mathcal{C}}_G(-\mathrm{\Sigma })={\mathcal{C}}_G(\mathrm{\Sigma })$ and hence the same function set for their state spaces. For a complex vector space $W$, write $\bar{W}$ for its conjugate vector space, with the same addition and scalar multiplication $\lambda {\cdot }_{\bar{W}}w:=\bar{\lambda }w$. The map $J_{\mathrm{\Sigma }}:V_G(-\mathrm{\Sigma })\to \overline{V_G(\mathrm{\Sigma })}$ given by $J_{\mathrm{\Sigma }}(f)=\bar{f}$ pointwise is complex-linear. Reversing the manifold orientation does not replace any fixed directed edge label by its inverse; inversion occurs only when a chosen edge direction itself is reversed.

The specified boundary triangulation identifications induce restriction functors. On the incoming end, $r_0$ is restriction to ${\mathrm{\partial }}_{\mathrm{i}\mathrm{n}}M$, transport through its identification with $-{\mathrm{\Sigma }}_0$, and then the identity on the common underlying cell data ${\mathcal{C}}_G(-{\mathrm{\Sigma }}_0)={\mathcal{C}}_G({\mathrm{\Sigma }}_0)$. The outgoing functor $r_1$ is defined by restriction and its specified identification with ${\mathrm{\Sigma }}_1$. In either transport, a chosen edge mapped to the opposite chosen direction uses the inverse-edge rule from the flat-connection Definition. These functors give a span of finite groupoids

${\mathcal{C}}_G({\mathrm{\Sigma }}_0)\stackrel{r_0}{\leftarrow }{\mathcal{C}}_G(M)\stackrel{r_1}{\to }{\mathcal{C}}_G({\mathrm{\Sigma }}_1)\mathrm{.}$

The associated linear map is

$Z_M:=(r_1{)}_{!}{r}_0^{\ast }:V_G({\mathrm{\Sigma }}_0)\to V_G({\mathrm{\Sigma }}_1),$

where pullback and pushforward are the finite homotopy-fiber operations just defined. For empty boundary, the amplitude is $Z_G(M)=\mathrm{\mid }{\mathcal{C}}_G(M)\mathrm{\mid }$.

Finite flat connection groupoid on a cell complex

Finite groupoid cardinality and homotopy-fiber integration