## Curves in Calabi-Yau 3-folds and topological quantum field theory

Citations: | 3 - 1 self |

### BibTeX

@TECHREPORT{Bryan_curvesin,

author = {Jim Bryan and Rahul P},

title = {Curves in Calabi-Yau 3-folds and topological quantum field theory},

institution = {},

year = {}

}

### OpenURL

### Abstract

We continue our study of the local Gromov-Witten invariants of curves in Calabi-Yau 3-folds. We define relative invariants for the local theory which give rise to a 1+1-dimensional TQFT taking values in the ring Q[[t]]. The associated Frobenius algebra over 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 case either the target genus or the degree equals 1. We prove there exist degree 2 rigid curves of any genus. Hence, our degree 2 theory agrees with the double cover contributions to the Gromov-Witten invariants of the ambient 3-folds. 1 Notation, definitions and results A central problem in Gromov-Witten theory is to determine the structure of the Gromov-Witten invariants. Of special interest is the case where the

### Citations

121 | Hodge integrals and Gromov-Witten theory
- Faber, Pandharipande
(Show Context)
Citation Context ... the constant series p(d). In particular, the genus two and higher multiple cover contributions of a super-rigid elliptic curve are all zero. by By the localization calculation of Faber-Pandharipande =-=[8]-=-, Zd(0) is given Zd(0) = ∑ α⊢d t 2d z(α) ℓ(α) ∏ i=1 ( 2 sin( αit 2 ) ) −2 where the sum is over all partitions α of d. Here, ℓ(α) is the length of the α, and z(α) is a combinatorial factor (see Defini... |

117 | Topological gauge theories and group cohomology
- Dijkgraaf, Witten
- 1990
(Show Context)
Citation Context ...e t = 0 specialization of Zd(−) is a well-known TQFT obtained from the gauge theory of the symmetric group Sd, see Lemma 4.3. The TQFT determined by Sd was studied by Dijkgraaf-Witten and Freed-Quinn =-=[5, 10]-=-. Our TQFT may be viewed as a 1-parameter deformation of the DijkgraafWitten/Freed-Quinn theory. Corresponding to any 1+1-dimensional TQFT is a Frobenius algebra. In our case, the dimension of the cor... |

108 | Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds
- Li, Ruan
- 2001
(Show Context)
Citation Context ... and thus we have constructed the desired isomorphism of Frobenius algebras: A ∼ = Rλ1 ⊕ · · · ⊕ Rλn. 3 Relative local invariants and gluing Motivated by the symplectic theory of A.-M. Li and Y. Ruan =-=[16]-=-, J. Li has developed an algebraic theory of relative stable maps to a pair (X, B). This theory compactifies the moduli space of maps to X with prescribed ramification along a non-singular divisor B ⊂... |

81 | Relative Gromov-Witten Invariants
- Ionel, Parker
(Show Context)
Citation Context ...ual fundamental cycle of the usual stable map moduli space of X in terms of virtual cycles for relative stable maps of (X1, B) and (X2, B). The theory of relative stable maps has also been pursued in =-=[12, 13]-=-, [6]. In our case, the target is a non-singular curve X of genus g, and the divisor B is a collection of points b1, . . .,br ∈ X. Definition 3.1. Let (X, b1, . . .br) be a fixed non-singular genus g ... |

77 |
Frobenius Algebras and 2D Topological Quantum Field Theories
- Kock
- 2003
(Show Context)
Citation Context ...us algebras. The result goes back to Dijkgraaf’s thesis, and has been proven in various contexts by Sawin [24], Abrams [1], and Quinn [23]. The form of the correspondence that we quote is due to Kock =-=[15]-=-: Theorem 2.1. The category of 1+1-dimensional TQFTs taking values in R is equivalent to the category of commutative Frobenius algebras over R. A commutative Frobenius algebra over R is a commutative ... |

61 |
The symplectic sum formula for Gromov– Witten invariants
- Ionel, Parker
- 2004
(Show Context)
Citation Context ...ual fundamental cycle of the usual stable map moduli space of X in terms of virtual cycles for relative stable maps of (X1, B) and (X2, B). The theory of relative stable maps has also been pursued in =-=[12, 13]-=-, [6]. In our case, the target is a non-singular curve X of genus g, and the divisor B is a collection of points b1, . . .,br ∈ X. Definition 3.1. Let (X, b1, . . .br) be a fixed non-singular genus g ... |

59 | Two-dimensional topological quantum field theories and Frobenius algebras
- Abrams
- 1996
(Show Context)
Citation Context ...∼ = W ′ preserving diffeomorphism, then Z(W) = Z(W ′). by a boundary2 SEMISIMPLE TQFTS OVER COMPLETE LOCAL RINGS 6 (iv) The trivial oriented cobordism corresponds to the identity homomorphism, Z(Y × =-=[0, 1]-=-) = IdZ(Y ). (v) The concatenation of cobordisms corresponds to the composition of the corresponding R-module homomorphisms. (vi) The disjoint union of n-manifolds corresponds to the tensor product ∐ ... |

54 |
Chern-Simons theory with finite gauge group
- Freed, Quinn
- 1993
(Show Context)
Citation Context ...e t = 0 specialization of Zd(−) is a well-known TQFT obtained from the gauge theory of the symmetric group Sd, see Lemma 4.3. The TQFT determined by Sd was studied by Dijkgraaf-Witten and Freed-Quinn =-=[5, 10]-=-. Our TQFT may be viewed as a 1-parameter deformation of the DijkgraafWitten/Freed-Quinn theory. Corresponding to any 1+1-dimensional TQFT is a Frobenius algebra. In our case, the dimension of the cor... |

53 |
Mirror symmetry and elliptic curves, The moduli space of curves (Texel Island
- Dijkgraaf
- 1994
(Show Context)
Citation Context ...rface. The Hurwitz numbers can be viewed as counting principal Sd-bundles over punctured surfaces. The associated TQFT has been studied in detail and is well-known to be semisimple [5], [10] (but see =-=[4]-=- for a short explanation). The proofs of both Lemma 4.3 and Theorem 1.5 are complete. The Frobenius algebra H = ⊕αQeα obtained from Z0 d (−) is isomorphic to the center of Q[Sd], the group ring of the... |

49 | A degeneration formula of GW-invariants
- Li
- 2002
(Show Context)
Citation Context ... − 2 = (2g − 2)d + b. ⎞ (g)t b q d⎠ The generating function for the degree d, local, disconnected invariants is ∞∑ Zd(g) = Z b=0 b d (g)tb. In Section 3, we use J. Li’s theory of relative stable maps =-=[17, 18]-=- to construct relative versions of the local invariants. These relative invariants obey a gluing law which allows us to construct a Topological Quantum Field Theory (TQFT):1 NOTATION, DEFINITIONS AND... |

45 |
Introduction to symplectic field theory. Geom. Funct
- Eliashberg, Givental, et al.
- 1999
(Show Context)
Citation Context ...ental cycle of the usual stable map moduli space of X in terms of virtual cycles for relative stable maps of (X1, B) and (X2, B). The theory of relative stable maps has also been pursued in [12, 13], =-=[6]-=-. In our case, the target is a non-singular curve X of genus g, and the divisor B is a collection of points b1, . . .,br ∈ X. Definition 3.1. Let (X, b1, . . .br) be a fixed non-singular genus g curve... |

37 |
Lectures on Axiomatic Topological Quantum Field Theory, Geometry and Quantum Field theory
- Quinn
- 1995
(Show Context)
Citation Context ...nsion 1+1 are in bijective correspondence with commutative Frobenius algebras. The result goes back to Dijkgraaf’s thesis, and has been proven in various contexts by Sawin [24], Abrams [1], and Quinn =-=[23]-=-. The form of the correspondence that we quote is due to Kock [15]: Theorem 2.1. The category of 1+1-dimensional TQFTs taking values in R is equivalent to the category of commutative Frobenius algebra... |

36 | Hodge integrals and degenerate contributions
- Pandharipande
- 1999
(Show Context)
Citation Context ...ondition on the normal bundle NX/Y . Assuming NX/Y is generic, we may ask for which pairs (d, h) does (d, h)rigidity hold? The 1-rigidity of a generic normal bundle is straightforward and was used in =-=[22]-=-. We prove the following positive result: Theorem 1.3. If X ⊂ Y is a genus g curve in a 3-fold Y and NX/Y is a generic stable bundle of degree 2g − 2, then X is 2-rigid.1 NOTATION, DEFINITIONS AND RE... |

20 | Open string instantons and relative stable morphisms
- Li, Song
(Show Context)
Citation Context ... regard I(X, α) as a (virtual) obstruction bundle and cb(I(X, α)) as the associated Euler class. Remark 3.2. In case g = 0 and r = 1, the above Euler class was previously defined by J. Li and Y. Song =-=[19]-=-. Li and Song were modeling “open string” Gromov-Witten theory using relative stable maps. They argued that the integral Z b d (0)α should compute the multiple cover formula for maps to a disk in a Ca... |

10 |
and Rahul Pandharipande. BPS states of curves in Calabi-Yau 3folds
- Bryan
(Show Context)
Citation Context ...s the universal curve, f : U → X is the universal map, and [ ] vir denotes the virtual fundamental class. The (d, h)-rigidity of X guarantees that R 1 π∗f ∗ N is a bundle (the obstruction bundle). In =-=[3]-=-, we defined an integral depending only on d, h, and g that agrees with Equation (1) whenever X ⊂ Y is a (d, h)-rigid curve of genus g. The integral is given by N h−g ∫ d (g) = [Mh(X,d[X])] vir c(I(X)... |

9 | Direct sum decompositions and indecomposable TQFTs
- Sawin
- 1995
(Show Context)
Citation Context ...nvariant of W. TQFTs of dimension 1+1 are in bijective correspondence with commutative Frobenius algebras. The result goes back to Dijkgraaf’s thesis, and has been proven in various contexts by Sawin =-=[24]-=-, Abrams [1], and Quinn [23]. The form of the correspondence that we quote is due to Kock [15]: Theorem 2.1. The category of 1+1-dimensional TQFTs taking values in R is equivalent to the category of c... |

5 | A degeneration of stable morphisms and relative stable morphisms (2000) arXiv: math.AG/0009097
- Li
(Show Context)
Citation Context ... − 2 = (2g − 2)d + b. ⎞ (g)t b q d⎠ The generating function for the degree d, local, disconnected invariants is ∞∑ Zd(g) = Z b=0 b d (g)tb. In Section 3, we use J. Li’s theory of relative stable maps =-=[17, 18]-=- to construct relative versions of the local invariants. These relative invariants obey a gluing law which allows us to construct a Topological Quantum Field Theory (TQFT):1 NOTATION, DEFINITIONS AND... |

4 |
Chiu-Chu Melissa Liu. Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
- Katz
(Show Context)
Citation Context ...gued that the integral Z b d (0)α should compute the multiple cover formula for maps to a disk in a Calabi-Yau 3-fold where the boundary of the disk lies on a Lagrangian 3-manifold (see also Katz-Liu =-=[14]-=-). They arrived at the integrand by considering the obstruction theory of maps to the disk with the Lagrangian boundary conditions. It is plausible that such an open-string theory interpretation of th... |

3 | and Rahul Pandharipande - Bryan |

3 |
Stable maps and branch divisors. Preprint: math.AG/9905104
- Fantechi, Pandharipande
(Show Context)
Citation Context ...is omitted as we consider all domain genera: the moduli space is a countable union of connected components with varying expected dimensions. The branch points of a stable map to X are well-defined by =-=[9]-=-. The number of branch points b of map f equals the expected dimension of the moduli space at the moduli point [f]. We define the (possibly disconnected) local Gromov-Witten invariants to be Z b ∫ d(g... |

2 |
Relative maps and tautological classes. arXiv:math.AG/0304485
- Faber, Pandharipande
(Show Context)
Citation Context ...C ∗ -action on P canonically lifts to a C ∗ -action on the moduli space of maps Mg(P, (a)) relative to ∞. Discussions of the virtual localization formula in the relative context can be found in [19], =-=[7]-=-, [11]. An equivariant lifting of C ∗ to a line bundle L over P is uniquely determined by the weights [l0, l∞] of the fiber representations at the fixed points L0, L∞. The canonical lifting of C ∗ to ... |