Using the formalism of quantized quadratic Hamiltonians [A. B. Givental, Mosc. Math. J. 1, No. 4, 551–568 (2001; Zbl 1008.53072)], the authors are able to prove quantum versions of three classical theorems in algebraic geometry; namely, the Riemann-Roch theorem, Serre duality, and the Lefschetz hyperplane section theorem. The key ingredient consists in introducing a notion of twisted Gromov-Witten invariants of a compact projective complex manifold ; the quantum version of the aforementioned theorems can then be seen as relations between the twisted and the nontwisted Gromov-Witten theory of .
More precisely, let be the moduli space of genus , -pointed stable maps to of degree , where is an element in , and let be a holomorphic vector bundle on . Since a point in is represented by a pair , where is a complex curve and a holomorphic map, one can use to pull back on and then consider the -theory Euler character of , i.e., the virtual vector space , as the fiber over of a virtual vector bundle over . This intuitive construction is made completely rigorous by considering -theory push-pull along the diagram
A rational invertible multiplicative characteristic class of a complex vector bundle is an expression of the form
where are the components of the Chern character, and the are arbitrary parameters. These data determine a cohomology class (actually, a formal family of cohomology classes parametrized by the ) in , and one can define the total -twisted descendant potential as
Here is the representative of in the semigroup ring of degrees of holomorphic curves in , are rational cohomology classes on , and is the first Chern class of the universal cotangent bundle over corresponding to the -th marked point of . For the zero element in , the twisted potential reduces to , the total descendant potential of .
At this point the formalism of quantized quadratic hamiltonians enters the picture. One considers the symplectic space of Laurent polynomials in with coefficients in the cohomology of , endowed with the symplectic form
The subspace is a Lagrangian subspace, and is identified with the canonical symplectic structure on . Finally, given an infinitesimal symplectic transformation of , one can consider the differential operator of order on functions on , which is associated by quantization to the quadratic Hamiltonian on . By the inclusion , the operator acts on asymptotic elements of the Fock space, i.e., on functions of the formal variable in . By the dilaton shift, i.e., setting , with , the operator acts on any function of , notably on the descendant potentials.
Having introduced this formalism, the authors are able to express the relation between twisted and untwisted Gromov-Witten invariants in an extremely elegant way: up to a scalar factor,
where is the linear symplectic transformation defined by the asymptotic expansion of
under the identification of the variable with the first Chern class of the universal line bundle . This is the quantum Riemann-Roch theorem; it explicitly determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants. The result is a consequence of Mumford’s Grothendieck-Riemann-Roch theorem applied to the universal family . If is the trivial line bundle, then , where is the Hodge bundle, and one recovers from quantum Riemann-Roch results of D. Mumford [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271–328 (1983; Zbl 0554.14008)] and C. Faber, R. Pandharipande [Invent. Math. 139, No.1, 173–199 (2000; Zbl 0960.14031)] on Hodge integrals.
If is the multiplicative characteristic class
then , and one the following quantum version of Serre duality:
Finally, if is a convex vector bundle and a submanifold is defined by a global section of , then the genus zero Gromov-Witten invariants of can be expresssed in terms of the invariants of twisted by the Euler class of . These are in turn related to the untwisted Gromov-Witten invariants of by the quantum Riemann-Roch theorem, so the authors end up with a quantum Lefschetz hyperplane section principle, expressing genus zero Gromov-Witten invariants of a complete intersection in terms of those of . This extends earlier results [V. V. Batyrev, I. Ciocan-Fontanine, B. Kim and D. van Straten, Acta Math. 184, No. 1, 1–39 (2000; Zbl 1022.14014); A. Bertram, Invent. Math. 142, No. 3, 487–512 (2000; Zbl 1031.14027); A. Gathmann, Math. Ann. 325, No. 2, 393–412 (2003; Zbl 1043.14016); B. Kim, Acta Math. 183, No. 1, 71–99 (1999; Zbl 1023.14028); Y.-P. Lee, Invent. Math. 145, No. 1, 121–149 (2001; Zbl 1082.14056)], and yields most of the known mirror formulas for toric complete intersections. The idea of deriving mirror formulas by applying the Grothendieck-Riemann-Roch theorem to universal stable maps is not new: according to the authors it can be traced back at least to Kontsevich’s investigations in the early 1990s, and to Faber’s and Pandharipande’s work on Hodge integrals.