The continuum gluing conjecture for spin foam nets

Let $M$ and $N$ be $d$-dimensional manifolds such that

$$\partial M=\Sigma_1^*\sqcup\Sigma,\qquad \partial N=\Sigma^*\sqcup\Sigma_2.$$

Let $\psi_1\otimes\psi_2\in\mathcal{D}_{\Sigma_1^*\sqcup\Sigma_2}$ be decomposable, and let $\{\varphi_\alpha\}_{\alpha\in\mathscr I}$ be an orthonormal basis of $\mathcal{D}_{\Sigma}$. For every closed $(d-1)$-dimensional manifold $\Sigma$, let $P_\Sigma$ be the rigging map on $\mathcal{D}_\Sigma$.

Gluing conjecture. The continuum amplitudes satisfy

$$Z(M\cup_\Sigma N)[\psi_1\otimes\psi_2]=\sum_{\alpha,\beta\in\mathscr I}Z(M)[\psi_1\otimes\varphi_\alpha]P_\Sigma(\varphi_\alpha)[\varphi_\beta]Z(N)[R_\Sigma(\varphi_\beta)\otimes\psi_2].$$

Moreover, the rigging map satisfies the convolution property

$$P_\Sigma(\phi)[\psi]=\sum_{\alpha\in\mathscr I}P_\Sigma(\phi)[\varphi_\alpha]P_\Sigma(\varphi_\alpha)[\psi]$$

for all $\phi,\psi\in\mathcal{D}_\Sigma$.

This proposal is the continuum analogue of convolution for path-integral propagators, with gluing implemented by summing over intermediate boundary geometric degrees of freedom. Its justification requires interchanging a limit with a sum, which may impose additional conditions on spin foam models and their treatment of divergences; the source does not establish whether the proposal is proved or remains open.