Twisted Calabi–Yau ring spectra, string topology, and gauge symmetry

In this paper, we import the theory of "Calabi-Yau" algebras and categories from symplectic topology and topological field theories to the setting of spectra in stable homotopy theory. Twistings in this theory will be particularly important. There will be two types of Calabi-Yau structures in the setting of ring spectra: one that applies to compact algebras and one that applies to smooth algebras. The main application of twisted compact Calabi-Yau ring spectra that we will study is to describe, prove, and explain a certain duality phenomenon in string topology. This is a duality between the manifold string topology of Chas-Sullivan and the Lie group string topology of Chataur-Menichi. This will extend and generalize work of Gruher. Then, generalizing work of the first author and Jones, we show how the gauge group of the principal bundle acts on this compact Calabi-Yau structure, and compute some explicit examples. We then extend the notion of the Calabi-Yau structure to smooth ring spectra, and prove that Thom ring spectra of (virtual) bundles over the loop space, $\Omega M$, have this structure. In the case when $M$ is a sphere we will use these twisted smooth Calabi-Yau ring spectra to study Lagrangian immersions of the sphere into its cotangent bundle. We recast the work of Abouzaid-Kragh to show that the topological Hochschild homology of the Thom ring spectrum induced by the $h$-principle classifying map of the Lagrangian immersion, detects whether that immersion can be Lagrangian isotopic to an embedding. We then compute some examples. Finally, we interpret these Calabi-Yau structures directly in terms of topological Hochschild homology and cohomology.