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Abstract
Given a vector bundle F on a
smooth Deligne–Mumford stack X
and an invertible multiplicative characteristic class
c , we define orbifold
Gromov–Witten invariants of X
twisted by F
and c .
We prove a “quantum Riemann–Roch theorem” which expresses the generating
function of the twisted invariants in terms of the generating function of the untwisted
invariants. A quantum Lefschetz hyperplane theorem is derived from this by
specializing to genus zero. As an application, we determine the relationship between
genus–0 orbifold
Gromov–Witten invariants of X
and that of a complete intersection, under additional assumptions. This provides a
way to verify mirror symmetry predictions for some complete intersection
orbifolds.
Keywords
orbifold Gromov–Witten invariant, Deligne–Mumford stack,
Givental's formalism, Grothendieck–Riemann–Roch formula,
mirror symmetry
Mathematical Subject Classification 2000
Primary: 14N35
Secondary: 53D45, 14C40
Publication
Received: 16 July 2006
Revised: 20 May 2009
Accepted: 22 June 2009
Preview posted: 10 October 2009
Published: 2 January 2010
Proposed: Jim Bryan
Seconded: Richard Thomas, Frances Kirwan