THE INTERSECTION FORM IN H(M0n) AND THE EXPLICIT KÜNNETH FORMULA IN QUANTUM COHOMOLOGY

We prove a general formula for the intersection form of two arbitrary monomials in boundary divisors. Furthermore we present a tree basis of the cohomology of M0n. With the help of the intersection form we determine the Gram matrix for this basis and give a formula for its inverse. This enables us to calculate the tensor product of the higher order multiplications arising in quantum cohomology and formal Frobenius manifolds. In the context of quantum cohomology this establishes the explicit Künneth formula. 0. Introduction Let M0n be the moduli space of genus 0 curves with n marked points. Its cohomology ring was determined by Keel [Ke], who gave a presentation in terms of boundary divisors, their intersections and relations. A boundary divisor is specified by a 2-partition S1 ∐ S2 of n := {1, . . . , n}. The additive structure of this ring was studied and presented in [KM] and [KMK]. Although much about the structure of this ring is known there are still several open questions. The complete study of the intersection theory of this space however is of importance for the theory of quantum cohomology. In particular it is necessary in order to understand the Künneth formula for quantum cohomology, which is given by the tensor product of Cohomological Field Theories, cf. [KM] and [KMK]. In §2 of this paper we prove a formula for the intersection form for any two polynomials in the boundary divisors of complementary degree. More precisely, after the introduction of the notion of trees with multiplicities and good multiplicity orientations we can formulate the following Theorem. Let mon(σ1, m1) and mon(σ2, m2) be two monomials of complementary degree in H(M0n). If there is no good multiplicity orientation of (τ,m) := τ(σ1 ∪ σ2, m1 +m2) then 〈mon(σ1, m1)mon(σ1, m1)〉 = 0. If there does exist one then: 〈mon(σ1, m1)mon(σ1, m1)〉 = ∏ v∈Vτ (−1) (|v| − 3)! ∏ f∈F (v)(mult(f))! 2 ∏