Ralph M. Kaufmann

Higher Weil-Petersson volumes of moduli spaces of stable

Moduli spaces of compact stablen-pointed curves carry a hierarchy of cohomology classes of top dimension which generalize the Weil-Petersson volume forms and constitute a version of Mumford classes. We give various new formulas for the integrals of these forms and their generating functions.

n

The operation of tensor product of Cohomological Field Theories (or algebras over genus zero moduli operad) introduced in an earlier paper by the authors is described in full detail, and the proof of a theorem on additive relations between strata classes is given. This operation is a version of the Kuenneth formula for quantum cohomology. In addition, rank one CohFTs are studied, and a generalization of Zografs formula for Weil-Petersson volumes is suggested.

-pointed curves

We construct two new G-equivariant rings:

Quantum cohomology of a product

\mathcal{K}(X,G)

Stringy K-theory and the Chern character

, called the stringy K-theory of the G-variety X, and

Orbifolding Frobenius Algebras

\mathcal{H}(X,G)

Arc operads and arc algebras

, called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne–Mumford stack

Closed/open string diagrammatics

\mathcal{X}

Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads

, we also construct a new ring

Pointed admissible G -covers and G -equivariant cohomological field theories

\mathsf{K}_{\mathrm{orb}}(\mathcal{X})