Quantum Riemann–Roch, Lefschetz and Serre

By Tom Coates, Alexander Givental. 2001

291 citations recorded by the source corpus.

Given a holomorphic vector bundle E over a compact Kahler manifold X, one defines twisted Gromov-Witten invariants of X to be intersection numbers in moduli spaces of stable maps f :Σ → X with the cap product of the virtual fundamental class and a chosen multiplicative invertible characteristic class of the virtual vector bundle H 0 (Σ ,f ∗ E) � H 1 (Σ ,f ∗ E). Using the formalism of quantized quadratic Hamiltonians (25), we express the descendant potential for the twisted theory in terms of that for X. 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 invari- ants. When E is concave and the C × -equivariant inverse Euler class is chosen as the characteristic class, the twisted invariants of X give Gromov-Witten invariants of the total space of E. "Nonlinear Serre duality" (21), (23) expresses Gromov-Witten invariants of E in terms of those of the super-manifold ΠE: it relates Gromov-Witten invariants of X twisted by the inverse Euler class and E to Gromov-Witten invariants of X twisted by the Euler class and E ∗ . We derive from Theorem 1 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 of E, the genus-zero Gromov-Witten invariants of ΠE coin- cide with those of Y. We establish a "quantum Lefschetz hyperplane section principle" (Theorem 2) expressing genus-zero Gromov-Witten invariants of a complete intersection Y in terms of those of X. This extends earlier results (4), (9), (18), (29), (33) and yields most of the known mirror formulas for toric complete intersections.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c67b264396c6e280f508521f02f42e9f58a5e8e6

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Citations in the corpus

QUANTUM KIRWAN MORPHISM AND GROMOV-WITTEN INVARIANTS OF QUOTIENTS I (2012, methodology): “We also describe twisted versions of GromovWitten invariants arising from vector bundles on the target, see for example Coates-Givental [6].”
Quantum Cohomology and Periods (2011, methodology): “We remark that one can calculate Le and Υ̃v from the functions IvV using the Birkhoff factorization in the theory of loop groups, as observed by Coates-Givental [17] and Guest [28].”
An integral structure in quantum cohomology and mirror symmetry for toric orbifolds (2009, background): “ Jacobi ring J(Wq) of the mirror. The conclusion follows from Proposition 3.10 (ii). Remark 4.9. When X is not weak Fano, the mirror theorem Conjecture 4.3 should be replaced with the Coates-Givental [24] style statement that the I-function is on the Givental’s Lagrangian cone (30). The D-module iso”
The Abelian/Nonabelian correspondence and Frobenius manifolds (2006, methodology): “Now apply the quantum Lefschetz theorem of Coates and Givental [ CG ] and use a similar argument to the one in the proof of Theorem 5.3.3 to conclude that i ∗”
Symplectic geometry of Frobenius structures (2003, methodology): “subject, as usual, to the dilaton shift q(z) = −z + P ¯ tkzk. Theorem 2 (see Appendix 2 in [ 7 ]).”
Invariance of tautological equations II: Gromov-Witten theory (2006, background): “(iv) Coates and Givental [ 4 ] (see also [11]) give a beautiful geometric reformation of the genus zero axiomatic theory in terms of Lagrangian cones in H. When viewed in the Lagrangian cone formulation, Theorem 1 becomes transparent and a proof is almost immediate.”
The Crepant Transformation Conjecture for Toric Complete Intersections (2014, methodology): “4 relies on the Mirror Theorem for toric complete intersections [31], and on non-linear Serre duality [33, 43, 44, 75], which relates the quantum cohomology of Y± to the quantum cohomology of the total space of the dual bundles E∨ ±.”
Gromov–Witten Invariants of Toric Fibrations (2009, methodology): “They can be stated in terms of the cone LB as follows [3, 7].”
The structure of 2D semi-simple field theories (2007, methodology): “The various inhomogeneous theories matching a given semi-simple Frobenius manifold can be related geometrically by means of a Hodge bundle twist, as in [ CG1 ].”