Quantum Cohomology and Virasoro Algebra

TL;DR

This paper explores how the Virasoro algebra governs quantum cohomology of Fano manifolds, providing explicit constructions for projective spaces and extending to other Fano varieties, linking algebraic structures to geometric invariants.

Contribution

It introduces a framework where the Virasoro algebra controls quantum cohomology and partition functions of Fano manifolds, including explicit operators for projective spaces and a broader class of Fano varieties.

Findings

Virasoro operators reproduce known genus-0,1 instanton numbers.

Constructed Virasoro operators for complex projective spaces.

Extended the construction to a wider class of Fano varieties.

Abstract

We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds $M$ ($c_1(M)>0$) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of Kontsevich-Manin, Getzler etc. on the genus-0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to $\chi(M)$, the Euler characteristic of the manifold $M$.