Quantum Riemann – Roch

Given a holomorphic vector bundle E : EX → X over a compact Kähler manifold, one introduces twisted GW-invariants of X replacing virtual fundamental cycles of moduli spaces of stable maps f : Σ → X by their cap-product with a chosen multiplicative characteristic class of H(Σ, fE)− H(Σ, fE). Using the formalism [18] of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for X. The result (Theorem 1) is a consequence of Mumford’s Riemann – Roch – Grothendieck formula [32, 14] applied to the universal stable map. When E is concave, and the inverse C-equivariant Euler class is chosen, the twisted theory yields GW-invariants of EX. The “non-linear Serre duality principle” [20, 21] expresses GW-invariants of EX via those of the supermanifold ΠEX, where the Euler class and E replace the inverse Euler class and E. We derive from Theorem 1 the nonlinear Serre duality in a very general form (Corollary 2). When the bundle E is convex, and a submanifold Y ⊂ X is defined by a global section, the genus 0 GW-invariants of ΠEX coincide with those of Y . We prove a “quantum Lefschetz hyperplane section principle” (Theorem 2) expressing genus 0 GW-invariants of a complete intersection Y via those of X. This extends earlier results [5, 26, 8, 29, 16] and yields most of the known mirror formulas for toric complete intersections. Introduction. The mirror formula of Candelas et al [9] for the virtual numbers nd of degree d = 1, 2, 3, ... holomorphic spheres on a quintic 3-fold Y ⊂ X = CP 4 can be stated [17] as the coincidence of the 2-dimensional cones over the following two curves in H(Y ;Q) = Q[P ]/(P ): JY (τ) := e Pτ + P 2 5 ∑