Heisenberg groups in algebraic topology

We study the Madsen-Tillmann spectrum $\text{C}{P}_{-1}^{\mathrm{\infty }}$ as a quotient of the Mahowald pro-object $\text{C}{P}_{-\mathrm{\infty }}^{\mathrm{\infty }}$, which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten.