I will discuss joint work with Inbar Klang in which 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. We use the notion of a "compact Calabi-Yau" ring spectrum to describe a certain duality phenomenon in string topology. This is a duality between the manifold string topology of Chas and Sullivan and the Lie group string topology of Chataur-Menichi. We also show how the gauge group of the principal bundle acts on this Calabi-Yau structure.

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 and their topological Hochschild homologies to study Lagrangian immersions of the sphere into its cotangent bundle.