| In this dissertation, we study twisted Gromov-Witten invariants of orbifolds. Our main goal is to generalize the Quantum Riemann-Rock theorem of Coates-Givental [11] to orbifold target spaces.; We work over the field of complex numbers. Let chi be an algebraic orbifold (smooth Deligne-Mumford stack), F a vector orbi-bundle on chi and c(·) an invertible multiplicative characteristic class of complex vector bundles. Following Coates-Givental, we define the (c, F)-twisted orbifold Gromov-Witten invariants of chi by replacing the virtual fundamental class of the moduli space Mg,n (chi, d) of n-pointed genus- g degree-d orbifold stable maps to chi with the cap-product [ Mg,n (chi, d)]vir∩ c(Fg,n,d). Here Fg,n,d is a virtual bundle (a class in the K-theory) on the moduli space Mg,n (chi, d) constructed from the bundle F on chi (by applying K-theoretic pushforward along the fibers of the universal family of stable maps). Our main result, the orbifold Quantum Riemann-Roch theorem, determines twisted orbifold Gromov-Witten invariants with gravitational descendants in terms of the untwisted ones. This result extends the Quantum Riemann-Roch theorem of Coates-Givental. The theorem is stated in terms of Givental's quantization formalism applied to orbifold Gromov-Witten theory. The proof is based on an application, similar to the work of Coates-Givental, of the Grothendieck-Riemann-Roch formula to universal families of stable maps. However, our results features Bernoulli polynomials in place of Bernoulli numbers.; In the special case of genus 0 orbifold Gromov-Witten invariants twisted by the Euler class, we derive a general form of Quantum Lefschetz hyperplane section theorem for orbifolds, which expresses---in a fashion related to the mirror conjecture---certain untwisted genus 0 orbifold Gromov-Witten invariants of complete intersections in terms of those of the ambient space.; As another application of the orbifold Quantum Riemann-Roch theorem, we extend Quantum Serre duality to orbifolds, giving a relationship between (c, F)-twisted and ( c∨,F∨ )-twisted Gromov-Witten invariants. |
