Connes fusion of spinors on loop space (Q6615207)

The authors solve a problem formulated by Stolz and Teichner's preprint [The spinor bundle on loop space (2025)], that is, they construct the Connes fusion product on the spinor bundle on the loop space of a string manifold. More precisely, let \(M\) be a string manifold, \(PM\) the space of smooth paths in \(M\), \(LM\) the loop space of \(M\) and \(F(LM)\) the bundle of an infinite-dimensional Hilbert space that is constructed by the authors' previous work [\textit{P. Kristel} and \textit{K. Waldorf}, J. Differ. Geom. 128, No. 1, 193--255 (2024; Zbl 1547.53031)]. They firstly construct a hyperfinite type \(\mathrm{III}_{1}\) von Neumann algebra bundle \(\mathcal{N}\) over \(PM\) and prove (Theorem 5.2.5) that the bundle \(F(LM)\) is a \(p_{1}^{\ast}\mathcal{N}\text{-}p_{2}^{\ast}\mathcal{N}\)-bimodule bundle, where \(p_{1},p_{2} \colon LM \to PM\) are the maps that divide a loop into its two halves. The theorem imples that the fiber \(L(FM)_{\beta_{1} \cup \beta_{2}}\) is an \(\mathcal{N}_{\beta_{1}}\text{-}\mathcal{N}_{\beta_{2}}\)-bimodule for any pair of paths \(\beta_{1},\beta_{2} \in PM\) with a common initial point and a common endpoint.\N\NThe main result (Theorem 5.3.1) of this paper is the existence and unique characterization of the unitary interwiners\N\[\N\chi_{\beta_{1}, \beta_{2}, \beta_{3}} \colon F(LM)_{\beta_{2} \cup \beta_{3}} \boxtimes F(LM)_{\beta_{1} \cup \beta_{2}} \to F(LM)_{\beta_{1} \cup \beta_{3}}\N\]\Nof \(\mathcal{N}_{\beta_{3}}\text{-}\mathcal{N}_{\beta_{1}}\)-bimodules, where \(\boxtimes\) is the Connes fusion of bimodules over \(\mathcal{N}_{\beta_{2}}\), for any trple of paths \(\beta_{1}, \beta_{2}, \beta_{3} \in PM\) with a common initial point and a common endpoint.