Riemann-Roch theorems in Gromov-Witten theory

Gromov-Witten invariants of a compact almost-Kahler manifold X are intersection numbers in moduli spaces of stable maps to X. These spaces, introduced by Kontsevich, are compactifications of spaces of pseudo-holomorphic maps from marked Riemann surfaces to X. Gromov-Witten invariants encode information about the enumerative geometry of X---roughly speaking, they count the number of curves in X which pass through various cycles and satisfy certain conditions on their complex structure. These invariants have important applications in both symplectic topology and enumerative algebraic geometry.; In this dissertation we use various Riemann-Roch theorems, together with Givental's formalism of quantized quadratic Hamiltonians, to develop tools for computing Gromov-Witten invariants and their generalizations. As a consequence, we obtain a new proof of the Mirror Conjecture of Candelas, de la Ossa, Green and Parkes, concerning the genus-0 Gromov-Witten invariants of quintic hypersurfaces in C P4.; Following Kontsevich, we introduce a notion of Gromov-Witten invariant twisted by a holomorphic vector bundle E over X and an invertible multiplicative characteristic class c. Special cases of this construction are closely related to Gromov-Witten invariants of hypersurfaces and to local Gromov-Witten invariants (these measure the contribution to the Gromov-Witten invariants of a space Y coming from curves in a neighbourhood of a submanifold X, where the normal bundle to X in Y is E). We express all twisted Gromov-Witten invariants, of all genera, in terms of untwisted Gromov-Witten invariants. This result (Theorem 1) is a consequence of the Grothendieck-Riemann-Roch formula applied to the universal family of stable maps.; As an application, we obtain the Quantum Lefschetz Hyperplane Principle (Theorem 2 and Corollary 5). This determines genus-0 Gromov-Witten invariants of a large class of complete intersections in terms of genus-0 Gromov-Witten invariants of the ambient space. It is more general than earlier versions, due to Givental, Kim, Lian-Liu-Yau, Bertram, Lee and Gathmann, as it applies to complete intersections of arbitrary Fano index and does not require "restriction to the small parameter space". In particular, this gives a new proof of the Mirror Conjecture of Candelas et al. We also establish "non-linear Serre duality" in a very general form.; Tangent-twisted Gromov-Witten invariants are intersection numbers involving characteristic classes of virtual tangent bundles to moduli spaces of stable maps. They give a rich supply of symplectic invariants of X. We determine all tangent-twisted Gromov-Witten invariants, of all genera, in terms of untwisted Gromov-Witten invariants. A key step is to interpret tangent-twisted Gromov-Witten invariants in terms of Gromov-Witten invariants with values in complex cobordism. We extend Givental's quantization formalism to the cobordism-valued setting, and combine this with various Riemann-Roch calculations to give a formula (Theorem 3) expressing cobordism-valued Gromov-Witten invariants in terms of usual (cohomological, untwisted) Gromov-Witten invariants. This determines all Gromov-Witten invariants with values in any complex-oriented cohomology theory in terms of cohomological Gromov-Witten invariants. Theorem 3 reduces "quantum extraordinary cohomology" to quantum cohomology, and in this sense can be regarded as a "quantum" version of the Hirzebruch-Riemann-Roch theorem.