Definition. The determinant action of a fixed connection [fdip-0070]

✍️sourceAGENTDRAFTED

Use a connection background with its fixed invariant fiber metric and mass \(m>0\). Let \(\mathbb K=\mathbb R\) with an orthogonal representation, or \(\mathbb K=\mathbb C\) with a unitary representation. The matter space is \(\mathcal H=F^V\), and \(N=\dim _{\mathbb K}\mathcal H\) includes the fiber dimension. Write \(d\operatorname {vol}\) for the induced real volume on this space, of real dimension \(N\) or \(2N\), respectively. Set \[\begin {array}{c|cc} \mathbb K&\kappa &d\nu \\\hline \mathbb R&1/2&(2\pi )^{-N/2}d\operatorname {vol}\\ \mathbb C&1&\pi ^{-N}d\operatorname {vol} \end {array}\] and define the matter partition function and determinant action by \[\begin {aligned} Z_{\mathrm {mat}}(U) &=\int _{\mathcal H}e^{-\kappa \langle \phi ,Q_U\phi \rangle }\,d\nu (\phi ),\\ W(U)&=-\log Z_{\mathrm {mat}}(U). \end {aligned}\] The connection is fixed in this integral; only \(\phi \) is integrated.

Orthogonal or unitary diagonalization reduces the integral to positive eigenvalues \(\lambda _1,\ldots ,\lambda _N\). Each real coordinate contributes \(\lambda _j^{-1/2}\) by the real Gaussian integral; each complex coordinate contributes \(\lambda _j^{-1}\) by the complex integral. Therefore \[Z_{\mathrm {mat}}(U)=(\det _{\mathbb K}Q_U)^{-\kappa },\qquad W(U)=\kappa \log \det _{\mathbb K}Q_U.\] The determinant is positive even in the complex case, since \(Q_U\) is positive Hermitian. These reference measures make \(Z_{\mathrm {mat}}=1\) when \(Q_U=I\). The normalized matter law has density \(Z_{\mathrm {mat}}(U)^{-1}e^{-\kappa \langle \phi ,Q_U\phi \rangle }\) with respect to \(\nu \), and covariance \(Q_U^{-1}\) over the chosen field. In the complex convention this means \(\mathbb E[\phi \phi ^*]=Q_U^{-1}\).

A gauge transformation acts unitarily on \(\mathcal H\), and gauge covariance gives \(Q_{U^g}=T_gQ_UT_g^{-1}\). Thus \(W(U^g)=W(U)\). Unitary coordinate changes also preserve the declared real volume. The metric, mass, weights and graph remain supplied data; this determinant does not choose them or yet define a probability law on connections.