Virasoro Constraints For Quantum Cohomology

In [EHX2], Eguchi, Hori and Xiong, proposed a conjecture that the partition function of topological sigma model coupled to gravity is annihilated by infinitely many differential operators which form half branch of the Virasoro algebra. A similar conjecture was also proposed by S. Katz [Ka] (See also [EJX]). Assuming this conjecture is true, they were able to reproduce certain instanton numbers of some projective spaces known before (cf. the above cited references and [EX] for details). This conjecture is also referred to as the Virasoro conjecture by some authors. The main purpose of this paper is to give a proof of this conjecture for the genus zero part. The theory of topological sigma model coupled to gravity has been extensively studied recently by both mathematicians and physicists. This theory is built on the intersection theory of moduli spaces of stable maps from Riemann surfaces to a fixed manifold V ,which is a smooth projective variety (or more generally, a symplectic manifold). To each cohomology class of V (denoted by O) and a non-negative integer n, there is associated a quantum field theory operator, denoted by τn(O). When n = 0, the corresponding operator is simply denoted by O and is called a primary field. For n > 0, τn(O) is called the n-th (gravitational) descendent of O. The so called k-point genus-g correlators in topological field theory can be defined via the Gromov-Witten invariants as follows: