×

Twisted orbifold Gromov-Witten invariants. (English) Zbl 1303.14065

This paper defines and studies twisted orbifold Gromov-Witten invariants of a smooth proper Deligne-Mumford stack, extending earlier work of H.-H. Tseng [Geom. Topol. 14, No. 1, 1–81 (2010; Zbl 1178.14058)]. The paper also contains some results which are used in a joint paper with Givental on quantum \(K\)-theory.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Citations:

Zbl 1178.14058
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] D. Abramovich, “Lectures on Gromov-Witten invariants of orbifolds” in Enumerative Invariants in Algebraic Geometry and String Theory , Lecture Notes in Math. 1947 , Springer, Berlin, 2008, 1-48. · Zbl 1151.14005 · doi:10.1007/978-3-540-79814-9_1
[2] D. Abramovich, T. Graber, M. Olsson, and H.-H. Tseng, On the global quotient structure of the space of twisted stable maps to a quotient stack , J. Algebraic Geom. 16 (2007), 731-751. · Zbl 1126.14002 · doi:10.1090/S1056-3911-07-00443-2
[3] D. Abramovich, T. Graber, and A. Vistoli, “Algebraic orbifold quantum products” in Orbifolds in Mathematics and Physics (Madison, Wis., 2001) , Contemp. Math. 310 , Amer. Math. Soc., Providence, 2002, 1-24. · Zbl 1067.14055
[4] D. Abramovich, T. Graber, and A. Vistoli, Gromov-Witten theory of Deligne-Mumford stacks , Amer. J. Math. 130 (2008), 1337-1398. · Zbl 1193.14070 · doi:10.1353/ajm.0.0017
[5] D. Abramovich and A. Vistoli, Compactifying the space of stable maps , J. Amer. Math. Soc. 15 (2002), 27-75. · Zbl 0991.14007 · doi:10.1090/S0894-0347-01-00380-0
[6] K. Behrend and B. Fantechi, The intrinsic normal cone , Invent. Math. 128 (1997), 45-88. · Zbl 0909.14006 · doi:10.1007/s002220050136
[7] W. Chen and Y. Ruan, “Orbifold Gromov-Witten theory” in Orbifolds in Mathematics and Physics (Madison, Wis., 2001) , Contemp. Math. 310 , Amer. Math. Soc., Providence, 2002, 25-85. · Zbl 1091.53058
[8] T. H. Coates, Riemann-Roch theorems in Gromov-Witten theory , Ph.D. dissertation, University of California, Berkeley, Berkeley, Calif., 2003.
[9] T. Coates and A. Givental, Quantum Riemann-Roch, Lefschetz and Serre , Ann. of Math. (2) 165 (2007), 15-53. · Zbl 1189.14063 · doi:10.4007/annals.2007.165.15
[10] A. B. Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians , Mosc. Math. J. 1 (2001), 551-568, 645. · Zbl 1008.53072
[11] A. B Givental, “Symplectic geometry of Frobenius structures” in Frobenius Manifolds , Aspects Math. E36 , Friedr. Vieweg, Wiesbaden, 2004, 91-112. · Zbl 1075.53091
[12] A. Givental and V. Tonita, The Hirzebruch-Riemann-Roch theorem in true genus \(0\) quantum \(K\)-theory , preprint, [math.AG]. · Zbl 1335.19002
[13] T. J. Jarvis and T. Kimura, “Orbifold quantum cohomology of the classifying space of a finite group” in Orbifolds in Mathematics and Physics (Madison Wis., 2001) , Contemp. Math. 310 , Amer. Math. Soc., Providence, 2002, 123-134. · Zbl 1065.14069
[14] A. Kabanov and T. Kimura, A change of coordinates on the large phase space of quantum cohomology , Comm. Math. Phys. 217 (2001), 107-126. · Zbl 1042.53061 · doi:10.1007/s002200000359
[15] Y.-P. Lee, A formula for Euler characteristics of tautological line bundles on the Deligne-Mumford moduli spaces , Int. Math. Res. Not. IMRN 1997 , 393-400. · Zbl 0949.14016 · doi:10.1155/S1073792897000263
[16] D. McDuff and D. Salamon, \(J\)-Holomorphic Curves and Quantum Cohomology , Univ. Lecture Ser. 6 , Amer. Math. Soc., Providence, 1994. · Zbl 0809.53002
[17] C. Teleman, The structure of 2D semi-simple field theories , Invent. Math. 188 (2012), 525-588. · Zbl 1248.53074
[18] B. Toën, Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford , \(K\)-Theory 18 (1999), 33-76. · Zbl 0946.14004 · doi:10.1023/A:1007791200714
[19] H.-H. Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre , Geom. Topol. 14 (2010), 1-81. · Zbl 1178.14058 · doi:10.2140/gt.2010.14.1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.