Gromov-Witten invariants for abelian and nonabelian quotients

Authors:
Aaron Bertram, Ionut Ciocan-Fontanine and Bumsig Kim

Journal:
J. Algebraic Geom. **17** (2008), 275-294

DOI:
https://doi.org/10.1090/S1056-3911-07-00456-0

Published electronically:
October 1, 2007

MathSciNet review:
2369087

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Abstract | References | Additional Information

Abstract: Conjectural formulas are given expressing the genus zero Gromov-Witten invariants of the quotient of a complex projective manifold by a reductive group (the nonabelian quotient) in terms of the invariants of the quotient by a maximal torus (the abelian quotient). The ``-function'' version of the formulas is proved when is a (generalized) flag manifold.

**[BF]**K. Behrend and B. Fantechi,*The intrinsic normal cone*, Invent. Math.**128**(1997), no. 1, 45-88. MR**1437495 (98e:14022)****[Ber]**A. Bertram,*Another way to enumerate rational curves with torus actions*, Invent. Math.**142**(2000), no. 3, 487-512. MR**1804158 (2001m:14077)****[BCK]**A. Bertram, I. Ciocan-Fontanine and B. Kim,*Two proofs of a conjecture of Hori and Vafa*, Duke Math. J.**126**(2005), 101-136. MR**2110629 (2006e:14077)****[Bri]**M. Brion,*The push-forward and Todd class of flag bundles*, in ``Parameter spaces (Warsaw, 1994)'', 45-50, Banach Center Publ., 36, Polish Acad. Sci., Warsaw, 1996. MR**1481478 (98h:14059)****[CG]**T. Coates and A. Givental,*Quantum Riemann-Roch, Lefschetz and Serre*, Ann. of Math. (2)**165**(2007), 15-53. MR**2276766 (Review)****[ES]**G. Ellingsrud and S. A. Strømme,*On the Chow ring of a geometric quotient*, Ann. of Math.**130**(1989), 159-187. MR**1005610 (90h:14019)****[Giv1]***A mirror theorem for toric complete intersections*, in*Topological field theory, primitive forms and related topics (Kyoto, 1996)*, 141-175, Progr. Math. vol. 160, Birkhäuser, Boston, MA, 1998. MR**1653024 (2000a:14063)****[Giv2]**A. Givental,*Gromov-Witten invariants and quantization of quadratic Hamiltonians*, Mosc. Math. J.**1**(2001), no. 4, 551-568, 645. MR**1901075 (2003j:53138)****[HV]**K. Hori and C. Vafa,*Mirror symmetry*, preprint (2000), hep-th/0002222.**[Kim1]**B. Kim,*Quantum hyperplane section theorem for homogeneous spaces*, Acta Math.**4**(1999), 71-99. MR**1719555 (2001i:14076)****[Kim2]**B. Kim,*Quantum hyperplane section principle for concavex decomposable vector bundles*, J. Korean Math. Soc.**37**(2000), no. 3, 455-461. MR**1760373 (2002b:14070)****[LT]**J. Li and G. Tian,*Virtual moduli cycles and Gromov-Witten invariants*, Jour. Amer. Math. Soc.**11**(1998), 119-174. MR**1467172 (99d:14011)****[Lee]**Y. P. Lee,*Quantum Lefschetz hyperplane theorem*, Invent. Math.**145**(2001), 121-149. MR**1839288 (2002i:14049)****[LLY]**C. H. Liu, K. Liu and S. T. Yau,*-fixed-points for hyper-Quot-schemes and an exact mirror formula for flag manifolds from the extended mirror principle diagram*, preprint(2004), math.AG/0401367.**[Mar]**S. Martin,*Symplectic quotients by a nonabelian group and by its maximal torus*, preprint (2000), math.SG/0001002.

Additional Information

**Aaron Bertram**

Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Email:
bertram@math.utah.edu

**Ionut Ciocan-Fontanine**

Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
ciocan@math.umn.edu

**Bumsig Kim**

Affiliation:
School of Mathematics, Korea Institute for Advanced Study, 207-43 Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722, Korea

Email:
bumsig@kias.re.kr

DOI:
https://doi.org/10.1090/S1056-3911-07-00456-0

Received by editor(s):
February 1, 2006

Received by editor(s) in revised form:
April 6, 2006

Published electronically:
October 1, 2007

Additional Notes:
The first two authors were partially supported by NSF grants DMS-0200895 and DMS-0303614, respectively. The third author was supported by KOSEF R01-2004-000-10870-0.