Duke Mathematical Journal

Curves in Calabi-Yau threefolds and topological quantum field theory

Jim Bryan and Rahul Pandharipande

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau threefolds.

We define relative invariants for local theory which give rise to a (1+1)-dimensional topological quantum field theory (TQFT) taking values in the ring $\mathbb{Q}[[t]]$. The associated Frobenius algebra over $\mathbb{Q}[[t]]$ is semisimple. Consequently, we obtain a structure result for the local invariants. As an easy consequence of our structure formula, we recover the closed formulas for the local invariants in the case where either the target genus or the degree equals 1.

Article information

Duke Math. J. Volume 126, Number 2 (2005), 369-396.

First available in Project Euclid: 21 January 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]


Bryan, Jim; Pandharipande, Rahul. Curves in Calabi-Yau threefolds and topological quantum field theory. Duke Math. J. 126 (2005), no. 2, 369--396. doi:10.1215/S0012-7094-04-12626-0. https://projecteuclid.org/euclid.dmj/1106332724

Export citation


  • L. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5 (1996), 569--587.
  • J. Bryan and R. Pandharipande, BPS states of curves in Calabi-Yau $3$-folds, Geom. Topol. 5 (2001), 287--318.
  • --------, Local Gromov-Witten theory of curves, preprint.
  • --------, ``Rigidity of curves in Calabi-Yau 3-folds'' to appear in The Interaction of Finite Type and Gromov-Witten Invariants (Banff, Canada, 2003), preprint.
  • R. Dijkgraaf, ``Mirror symmetry and elliptic curves'' in The Moduli Space of Curves (Texel Island, Netherlands, 1994), Progr. Math. 129, Birkhäuser, Boston, 1995, 149--163.
  • R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), 393--429.
  • Y. Eliashberg, A. Givental, and H. Hofer, ``Introduction to symplectic field theory'' in GAFA 2000: Visions in Mathematics: Towards 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, Birkhäuser, Basel, 560--673.
  • C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math. 139 (2000), 173--199.
  • --------, Relative maps and tautological classes, preprint.
  • B. Fantechi and R. Pandharipande, Stable maps and branch divisors, Compositio Math. 130 (2002), 345--364.
  • D. S. Freed and F. Quinn, Chern-Simons theory with finite gauge group, Comm. Math. Phys. 156 (1993), 435--472.
  • T. Graber and R. Vakil, Relative virtual localization and vanishing of tautological classes on moduli spaces of curves, preprint.
  • E.-N. Ionel and T. H. Parker, Relative Gromov-Witten invariants, Ann. of Math. (2) 157 (2003), 45--96.
  • --------, The symplectic sum formula for Gromov-Witten invariants, preprint.
  • S. Katz and C.-C. M. Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc, Adv. Theor. Math. Phys. 5 (2001), 1--49.
  • J. Kock, Frobenius algebras and $2$D Topological Quantum Field Theories, London Math. Soc. Stud. Texts 59, Cambridge Univ. Press, Cambridge, 2004.
  • A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants of Calabi-Yau $3$-folds, Invent. Math. 145 (2001), 151--218.
  • J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differential Geom. 57 (2001), 509--578.
  • --. --. --. --., A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002), 199--293.
  • J. Li and Y. S. Song, Open string instantons and relative stable morphisms, Adv. Theor. Math. Phys. 5 (2001), 67--91.
  • G. Moore, Kavli Institute for Theoretical Physics lectures on branes, K-theory, and RR-charges, preprint, 2001, http://online.kitp.ucsb.edu/online/mp01/moore1/
  • A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, and completed cycles, preprint.
  • R. Pandharipande, Hodge integrals and degenerate contributions, Comm. Math. Phys. 208 (1999), 489--506.
  • F. Quinn, ``Lectures on axiomatic topological quantum field theory'' in Geometry and Quantum Field Theory (Park City, Utah, 1991), IAS/Park City Math. Ser. 1, Amer. Math. Soc., Providence, 1995, 323--453.
  • S. Sawin, Direct sum decompositions and indecomposable TQFTs, J. Math. Phys. 36 (1995), 6673--6680.