Smooth Fock bundles, and spinor bundles on loop space (Q6593663)

From the Introduction: ``A fermionic Fock space \(F_L\) is the Hilbert completion of the exterior alebra of a Lagrangian subspace \(L\) of a Hilbdert space \(V\) (compolex, but equipped with a real structure), and it carries an irreducible represengtation of a Clifford \(C^\ast\)-algebra \(Cl(V)\). A typical situation that appears in fermionic gauge theories is that the Hilbert space \(V\) is the fibre \(\mathcal{H}_x\) of a continuous bundle \(\mathcal{H}\) of Hilbert spaces over a parameter space \(\mathcal{M}\) , but Lagrangians \(L_x \subset \mathcal{H}_x\) cannot be chosen to depend continuously on the parameter \(x\in \mathcal{M}\). Consequently, the Fock spaces \(F_{L_x}\) do not automatically form a continuous bundle of Hilbert spaces over \(\mathcal{M}\). [[\dots]] \NIn this article, we set up a general theory of constructing bundles of Fock spaces, which reproduces, completes, and generalizes the treatments of \textit{S. Stolz} and \textit{P. Teichner} [Lond. Math. Soc. Lect. Note Ser. 308, 247--343 (2004; Zbl 1107.55004)] and \textit{S. Ambler} [A bundle gerbe construction of a spinor bundle from the smooth free loop of a vector bundle. University of Notre Dame (PhD Thesis) (2012)]. However, we go one step beyond the construction of \textit{continuous} Fock bundles, and construct \textit{smooth} ones. Our main motivation for this is that at some point one may want to study certain \textit{differential} operators acting on sections of such bundles.''\N\NThe main results of the article are two theorems. \N\NTheorem 1.1 contains the construction of a smooth Fock bundle \(F^\infty_L(\widetilde{P})\) which is a rigged Cl\(^\infty(\mathcal{P})\)-module bundle, where \(\mathcal{P}\) is a Fréchet principal \(\mathcal{G}\)-bundle over a Fréchet manifolf \(\mathcal{M}\) and \(\widetilde{\mathcal{P}}\) is a lift of the structure group of \(\mathcal{P}\). \N\NTheorem 1.2 proves the relation between \(F^\infty_L(\widetilde{P})\) and the twisted Fock bundle \(F^\infty_L(\widetilde{P},\widetilde{\mathcal{G}})\) obtained by replacing \(G\) by its central expansion \(\widetilde{\mathcal{G}}\). \N\NCorollary 1.1 establishes the connection with the results of \textit{S. Stolz} and \textit{P. Teichner} [``The spinor bundle on loop space'', Preprint, \url{https://people.mpim-bonn.mpg.de/teichner/Math/ewExternalFiles/MPI.pdf}] and \textit{S. Ambler} [``A bundle gerbe construction of a spinor bundle from the smooth free loop of a vector bundle'', Preprint, \url{arXiv:1207.4418}].