Quantum Cobordisms and Formal Group Laws

Abstract

The present paper is closely based on the lecture given by the second author at The Unity of Mathematics symposium and is based on our joint work in progress on Gromov-Witten invariants with values in complex cobordisms. We will mostly consider here only the simplest example, elucidating one of the key aspects of the theory. We refer the reader to [9] for a more comprehensive survey of the subject and to [5] for all further details. Consider Open image in new window , n ≥ 3, the Deligne-Mumford compactification of the moduli space of configurations of n distinct ordered points on the Riemann sphere C P 1. Obviously, Open image in new window , while Open image in new window is known to be isomorphic to C P 2 blown up at four points. In general, Open image in new window is a compact complex manifold of dimension n − 3, and it makes sense to ask what is the complex cobordism class of this manifold. The Thom complex cobordism ring, after tensoring with Q, is known to be isomorphic to U* = Q[C P 1,C P 2, ...], the polynomial algebra with generators C P k of degree −2k. Thus our question is to express Open image in new window , modulo the relation of complex cobordism, as a polynomial in complex projective spaces.