Quantum Riemann–Roch, Lefschetz and Serre

This is the authors' abstract. We don't add key points for this paper.

Given a holomorphic vector bundle E over a compact Khler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f : X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (, f * E) H 1 (, f * E). Using the formalism of quantized quadratic Hamiltonians This result (Theorem 1) is a consequence of Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family over the moduli space of stable maps. It determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants.