Twisted Calabi-Yau ring spectra, string topology, and gauge symmetry (Q1999648)

In this paper, the authors introduce a homotopy theoretic version of compact Calabi-Yau algebras and use it for example to study the string topology of the classifying space of a compact Lie group. More specifically, they introduce the notion of a twisted compact Calabi-Yau ring spectrum of dimension \(n\). It consists of an associative ring spectrum \(R\), a ``twisting'' \(R\)-bimodule \(Q\), and a map \(t: \mathrm{THH}(R;Q) \to \Sigma^{-n}\mathbb S\) satisfying certain conditions. Primary examples for this structure arise from the Spanier-Whitehead duals of closed \(n\)-manifolds and the manifold string topology spectrum associated with a certain principal bundle over a compact Lie group. In the second example, the authors show that a certain equivalence of string topology spectra induced by this structure respects the gauge group action studied by \textit{R. L. Cohen} and \textit{J. D. S. Jones} [Bol. Soc. Mat. Mex., III. Ser. 23, No. 1, 233--255 (2017; Zbl 1378.55006)]. The authors also show that under suitable orientability conditions, the structure of a twisted compact Calabi-Yau ring spectrum gives rise to a Frobenius algebra structure in homology. Moreover, they discuss a ``smooth'' version of Calabi-Yau ring spectra and show that Thom spectra of loop maps provide examples for this structure, leading to an equivalence of topological Hochschild homology spectra that is related to work of \textit{A. J. Blumberg} et al. [Geom. Topol. 14, No. 2, 1165--1242 (2010; Zbl 1219.19006)] and \textit{M. Abouzaid} and \textit{T. Kragh} [J. Topol. 9, No. 1, 232--244 (2016; Zbl 1405.53120)].