## The local Gromov-Witten theory of curves (2008)

Citations: | 14 - 4 self |

### BibTeX

@MISC{Bryan08thelocal,

author = {J. Bryan},

title = {The local Gromov-Witten theory of curves},

year = {2008}

}

### OpenURL

### Abstract

We study the equivariant Gromov-Witten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the Gromov-Witten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix.

### Citations

177 | Pandharipande.Localization of virtual classes
- Graber, R
- 1999
(Show Context)
Citation Context ...th [2], we will not adopt the latter convention. 7The T-fixed part of the perfect obstruction theory for M • h (Y, β) induces a perfect obstruction theory for M • h (Y, β)T and hence a virtual class =-=[9]-=-. The equivariant virtual normal bundle of the embedding, M • h(Y, β) T ⊂ M • h(Y, β), is Norm vir with equivariant Euler class e(Norm vir ). The integral in (2) denotes equivariant push-forward. Let ... |

172 |
Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon
- Macdonald
- 1995
(Show Context)
Citation Context ...the content c(□) to be i − j, and the hook length h(□) to be λi + λ ′ j − i − j + 1. The total content and the total hooklength cλ = ∑ c(□) □∈λ ∑ h(□) □∈λ satisfy the following identities (page 11 of =-=[13]-=-): ∑ h(□) = n(λ) + n(λ ′ ) + d, cλ = n(λ ′ ) − n(λ), (3) □∈λ where l(λ) ∑ n(λ) = (i − 1)λi. i=1 3.2 Relative invariants To formulate our gluing laws for the residue theory of rank 2 bundles on X, we r... |

122 | Topological gauge theories and group cohomology
- Dijkgraaf, Witten
- 1990
(Show Context)
Citation Context ... P1 with prescribed ramification α, β, and γ at 0, 1, and ∞. Modulo factors of s1 and s2, the quotient F/mF is the Frobenius algebra associated to the TQFT studied by Dijkgraaf-Witten and Freed-Quinn =-=[5, 8]-=-. The latter Frobenius algebra is isomorphic to Q[Sd] Sd, the center of the group algebra of the symmetric group, and well-known to be semisimple. We derive below an explicit idempotent basis for F/mF... |

122 | Hodge integrals and Gromov – Witten theory
- Faber, Pandharipande
(Show Context)
Citation Context ...irreducible curve of genus g over C, and let N → X 3be a rank 2 vector bundle with det N ∼ = KX. Then, N is a non-compact Calabi-Yau 1 threefold, and the Gromov-Witten theory, defined and studied in =-=[2, 3, 4, 6, 20]-=-, is called the local Calabi-Yau theory of X. We study here the local theory of curves without imposing the Calabi-Yau condition det N ∼ = KX on the bundle N. The study of non Calabi-Yau local theorie... |

111 | A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations
- Thomas
(Show Context)
Citation Context ...ave proper support. 41Let Y be equipped with an action by an algebraic torus T. The moduli space In(Y, β) carries a T-equivariant perfect obstruction theory obtained from (traceless) Ext0(I, I), see =-=[29]-=-. Though Y is quasi-projective, Ext0(I, I) is well-behaved since the associated quotient scheme Z ⊂ Y is proper. Alternatively, for any T-equivariant compactification, the obstruction theory on Y ⊂ Y ... |

83 |
Frobenius algebras and 2D topological quantum field theories
- Kock
- 2004
(Show Context)
Citation Context ...heory of rank 2 bundles on curves is most concisely formulated as a functor of tensor categories, Zd(−) : 2Cob L1,L2 → Rmod. Our discussion follows Sections 2 and 4 of [2] and draws from Chapter 1 of =-=[10]-=-. Modifications of the categories have to be made to accommodate the more complicated objects studied here. 144.2 2Cob and 2Cob L1,L2 We first define the category 2Cob of 2-cobordisms. The objects of... |

83 | GromovWitten theory and Donaldson-Thomas theory
- Maulik, Nekrasov, et al.
- 2006
(Show Context)
Citation Context ...n the exponent is defined via localization on Y , ∫ Y ∫ c3(TY ⊗ KY ) = Y T c3(TY ⊗ KY ) e(N Y T /Y ) ∈ Q(s1, . . .,sr). The subvariety Y T is compact as a consequence of Assumption 2. By Theorem 1 of =-=[15]-=-, Conjecture 1 holds for toric Y . Conjecture 2. The reduced series Z ′ DT (Y )β is a rational function of the equivariant parameters si and q. The GW/DT correspondence for absolute residue invariants... |

81 | Relative Gromov–Witten invariants
- Ionel, Parker
- 2003
(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 =-=[7, 12, 13]-=-. In our case, the target is a non-singular curve X of genus g, and the divisor B is a collection of points x1, . . .,xr ∈ X. 11Definition 3.1. Let (X, x1, . . .xr) be a fixed non-singular genus g cu... |

72 |
Two dimensional Yang-Mills, black holes and topological strings,” hepth/0406058
- Vafa
(Show Context)
Citation Context ...anagic, Ooguri, Saulina, and Vafa have recently found that the local Gromov-Witten theory of curves is closely related to q-deformed 2D Yang-Mills theory and bound states of BPS black holes [1] (c.f. =-=[23]-=-). The anti-diagonal action is exactly opposite to the original motivations of the project. It would be very interesting to find connections between the anti-diagonal case and the original questions o... |

66 | Black holes, q-deformed 2d YangMills, and non-perturbative topological strings,” Nucl
- Aganagic, Ooguri, et al.
- 2005
(Show Context)
Citation Context ...ormula, Aganagic, Ooguri, Saulina, and Vafa have recently found that the local Gromov-Witten theory of curves is closely related to q-deformed 2D Yang-Mills theory and bound states of BPS black holes =-=[1, 30]-=-. The anti-diagonal action is exactly opposite to the original motivations of the project. It would be very interesting to find connections between the anti-diagonal case and the original questions of... |

64 |
Symmetric Functions and Hall Polynomials, The Clarendon
- Macdonald
- 1979
(Show Context)
Citation Context ... the content c(□) to be i − j, and the hooklength h(□) to be λi + λ ′ j − i − j + 1. The total content and the total hooklength cλ = ∑ c(□) □∈λ ∑ h(□) □∈λ satisfy the following identities (page 11 of =-=[19]-=-): ∑ h(□) = n(λ) + n(λ ′ ) + d, cλ = n(λ ′ ) − n(λ), (4) □∈λ where l(λ) ∑ n(λ) = (i − 1)λi. i=1 3.2 Relative invariants To formulate our gluing laws for the residue theory of rank 2 bundles on X, we r... |

56 | Stable morphisms to singular schemes and relative stable morphisms
- Li
(Show Context)
Citation Context .... 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 over a non-singular divisor B ⊂ X, =-=[16, 17]-=-. Li constructs a moduli space of relative stable maps together with a virtual fundamental cycle and proves a gluing formula. Consider a degeneration of X to X1 ∪B X2, the union of X1 and X2 along a s... |

54 |
Simmons theory with finite gauge group
- Freed, Quinn
- 1993
(Show Context)
Citation Context ... P1 with prescribed ramification α, β, and γ at 0, 1, and ∞. Modulo factors of s1 and s2, the quotient F/mF is the Frobenius algebra associated to the TQFT studied by Dijkgraaf-Witten and Freed-Quinn =-=[5, 8]-=-. The latter Frobenius algebra is isomorphic to Q[Sd] Sd, the center of the group algebra of the symmetric group, and well-known to be semisimple. We derive below an explicit idempotent basis for F/mF... |

52 | R Pandharipande, Gromov–Witten theory, Hurwitz numbers, and matrix models I arXiv:math.AG/0101147
- Okounkov
(Show Context)
Citation Context ...ion of the gluing formulas in [5]. The only difference is the modified metric term z(λ)(t1t2) l(λ) . The first factor, z(λ), is obtained from the degeneration formula for the virtual class [17] as in =-=[23]-=-. The second factor, (t1t2) l(λ) , arises from normalization sequences associated to the fractured domains. Let f : C → X be an element of M • h (X, µ1 , . . ., µ s , ν 1 , . . .,ν t ). Consider a red... |

50 |
Introduction to symplectic field theory, Geom. Funct. Anal
- Eliashberg, Hofer, et al.
(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 =-=[7, 12, 13]-=-. In our case, the target is a non-singular curve X of genus g, and the divisor B is a collection of points x1, . . .,xr ∈ X. 11Definition 3.1. Let (X, x1, . . .xr) be a fixed non-singular genus g cu... |

49 | A degeneration formula of GW-invariants
- Li
- 2002
(Show Context)
Citation Context ...ows the derivation of the gluing formulas in [2]. The only difference is the modified metric term z(λ)(s1s2) l(λ) . The first factor, z(λ), is obtained from degeneration formula for the virtual class =-=[11]-=- as in [18]. The second factor, (s1s2) l(λ) , arises from normalization sequences associated to the fractured domains. Let f : C → X be an element of M • h (X, µ1 , . . ., µ s , ν 1 , . . .,ν t ). Con... |

38 | Hodge integrals and degenerate contributions
- Pandharipande
- 1999
(Show Context)
Citation Context ...irreducible curve of genus g over C, and let N → X 3be a rank 2 vector bundle with det N ∼ = KX. Then, N is a non-compact Calabi-Yau 1 threefold, and the Gromov-Witten theory, defined and studied in =-=[2, 3, 4, 6, 20]-=-, is called the local Calabi-Yau theory of X. We study here the local theory of curves without imposing the Calabi-Yau condition det N ∼ = KX on the bundle N. The study of non Calabi-Yau local theorie... |

34 | Three questions in Gromov-Witten theory
- Pandharipande
(Show Context)
Citation Context ...e study here the local theory of curves without imposing the Calabi-Yau condition det N ∼ = KX on the bundle N. The study of non Calabi-Yau local theories has several motivations. The calculations of =-=[6, 20, 22]-=- predict a uniform structure for all threefold theories closely related to the Calabi-Yau case. The introduction of non CalabiYau bundles N yields a more flexible mathematical framework in which new m... |

30 |
Logarithmic series and Hodge integrals in the tautological ring, preprint math.AG/0002112v3
- Faber, Pandharipande
(Show Context)
Citation Context ...2) = i s1 ( + s2 s1s2 = i s1 + s2 s1s2 = i s1 + s2 s1s2 Mh,2 → Mh. ∫ 6u ( u 4 λ1 + ∞∑ (2h + 2)! (2h − 1)! u2h−1 ∫ M1,1 h=2 ∞∑ u h=2 2h+2 ∫ )′′′ λhλh−1 H 96 + ( u 2 H(u)) )′′′ where H(u) is defined in =-=[7]-=- on page 222. By Corollary 2 of [7], and thus (u 2 H(u)) ′′ ( ( )) u = − log cos , 2 ̂Z2(0 | 0, 0)(2),(2),(2) = i s1 + s2 tan 2 s1s2 ( ) u . 2 λhλh−1 H ) 30We conclude The function cot ( u 2 We defin... |

29 | The crepant resolution conjecture
- Bryan, Graber
- 2009
(Show Context)
Citation Context ...(ii) The local theory of the trivial rank 2 bundle over P 1 is equivalent to the quantum cohomologies of the Hilbert scheme Hilb n (C 2 ) and the orbifold (C 2 ) n /Sn. Our results here together with =-=[2, 24]-=- prove the equivalences, see Section 10. We expect further connections will likely be found in the future. 1.3 Results Let N be a rank 2 bundle on a curve X of genus g. We assume N is decomposable as ... |

24 | The Toda equations and the Gromov-Witten theory of the Riemann sphere, Lett.Math.Phys. 53 (2000) 59-74; A.Okounkov, Toda equations for Hurwitz numbers
- Pandharipande
(Show Context)
Citation Context ...t yields ρ ( ∂M h(P 1 , (d), (d), (2)) ) ⊂ ǫ −1 (∂Mh,1). The restriction of the virtual class to Mh(P 1 , (d), (d), (2)) is well-known to equal the ordinary fundamental class of the moduli space, see =-=[21]-=-. Since ρ : Mh(P 1 , (d), (d), (2)) → Hd is a proper cover of degree 2h, we conclude ρ∗[Mh(P 1 , (d), (d), (2))] vir = 2h[Hd] + B (17) where B is a cycle supported on ǫ−1 (∂M h,1). Since λhλh−1 vanish... |

22 |
The quantum cohomology of the Hilbert scheme of points of the plane
- Okounkov, Pandharipande
(Show Context)
Citation Context ...(ii) The local theory of the trivial rank 2 bundle over P 1 is equivalent to the quantum cohomologies of the Hilbert scheme Hilb n (C 2 ) and the orbifold (C 2 ) n /Sn. Our results here together with =-=[2, 24]-=- prove the equivalences, see Section 10. We expect further connections will likely be found in the future. 1.3 Results Let N be a rank 2 bundle on a curve X of genus g. We assume N is decomposable as ... |

16 | Hodge integrals and invariants of the unknots
- Okounkov, Pandharipande
(Show Context)
Citation Context ...ed in the anti-diagonal case. We define the Q-dimension of ρ, an irreducible representation of the symmetric group, indicated dimQ ρ, as follows: dimQ ρ d! = ∏ □∈ρ ( i Q h(□) 2 − Q −h(□) ) −1 2 , see =-=[22]-=-. Under the substitution Q = eiu , the Q-dimension can be expressed as: ( dimQ ρ ∏ = 2 sin d! h(□)u ) −1 . 2 □∈ρ By the hook length formula for dimρ, the leading term in u of the above dim ρ expressio... |

15 |
BPS states of curves
- Bryan, Pandharipande
(Show Context)
Citation Context ...r, irreducible curve of genus g over C, and let N → X be a rank 2 vector bundle with det N ∼ = KX. Then N is a non-compact Calabi-Yau 1 threefold, and the Gromov-Witten theory, defined and studied in =-=[3, 4, 5, 8, 26]-=-, is called the local Calabi-Yau theory of X. We study here the local theory of curves without imposing the Calabi-Yau condition det N ∼ = KX on the bundle N. The study of non Calabi-Yau local theorie... |

14 | On the tautological ring of Mg
- LOOIJENGA
- 1995
(Show Context)
Citation Context ...Mh,2] ∩ Pd) ∈ A∗(Mh,2). By a result of Looijenga using the Fourier-Mukai transform, the locus of d-torsion points of any family of Abelian varieties is a multiple of the zero section in the Chow ring =-=[13]-=-. Hence, [Pd] = d2h − 1 2 2h − 1 [P2] 29and [Hd] = d2h − 1 2 2h − 1 [H2]. We conclude ch(d) = d2h − 1 22h − 1 ch(2). Consider the d = 2 case. In genus 1, the class [H2] ∈ A∗(M1,2) pushes forward to 3... |

12 |
R Pandharipande, Curves in Calabi-Yau threefolds and topological quantum field theory
- Bryan
(Show Context)
Citation Context ...nt version of the Gromov-Witten/Donaldson-Thomas correspondence is formulated and discussed in detail for the case of N. The theory generalizes the local Calabi-Yau theory of X defined and studied in =-=[2, 4]-=-. 1Contents 1 Introduction 3 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Acknowledgments . ... |

12 | Yang-Mills Theory and Topological Field Theory," hep-th/9409044 - Moore |

11 |
and Rahul Pandharipande. BPS states of curves in Calabi-Yau 3folds
- Bryan
(Show Context)
Citation Context ...nt version of the Gromov-Witten/Donaldson-Thomas correspondence is formulated and discussed in detail for the case of N. The theory generalizes the local Calabi-Yau theory of X defined and studied in =-=[2, 4]-=-. 1Contents 1 Introduction 3 1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Acknowledgments . ... |

8 | The local Donaldson-Thomas theory of curves, math.AG/0512573 - Okounkov, Pandharipande |

5 |
Hirosi Ooguri, Natalia Saulina, and Cumrun Vafa. Black holes, q-deformed 2d yang-mills, and non-perturbative topological strings
- Aganagic
- 2005
(Show Context)
Citation Context ...ormula, Aganagic, Ooguri, Saulina, and Vafa have recently found that the local Gromov-Witten theory of curves is closely related to q-deformed 2D Yang-Mills theory and bound states of BPS black holes =-=[1]-=- (c.f. [23]). The anti-diagonal action is exactly opposite to the original motivations of the project. It would be very interesting to find connections between the anti-diagonal case and the original ... |

3 |
On the rigidity of stable maps to CalabiYau threefolds, in The interaction of finite-type and Gromov-Witten invariants (BIRS 2003
- Bryan, Pandharipande
(Show Context)
Citation Context ...q 8 +316q 7 +9438q 6 +63792q 5 +117353q 4 +63792q 3 +9438q 2 +316q+3). We note Z3(3 |2, 2) does not have integer coefficients. Because of the existence of infinitessimal deformations of triple covers =-=[3]-=-, the precise local to global formalism is not yet clear. The fractions in Z3(3 |2, 2) therefore do not contradict the conjectured global integral structure of Calabi-Yau threefolds. Empirically, for ... |

3 | Curves in Calabi-Yau 3-folds and Topological Quantum Field Theory
- Bryan, Pandharipande
(Show Context)
Citation Context ...r, irreducible curve of genus g over C, and let N → X be a rank 2 vector bundle with det N ∼ = KX. Then N is a non-compact Calabi-Yau 1 threefold, and the Gromov-Witten theory, defined and studied in =-=[3, 4, 5, 8, 26]-=-, is called the local Calabi-Yau theory of X. We study here the local theory of curves without imposing the Calabi-Yau condition det N ∼ = KX on the bundle N. The study of non Calabi-Yau local theorie... |

1 |
Symplectic surgery and Gromov-Witten
- Li, Ruan
(Show Context)
Citation Context ...variants To formulate our gluing laws for the residue theory of rank 2 bundles on X, we require relative versions of the residue invariants. Motivated by the symplectic theory of A.-M. Li and Y. Ruan =-=[15]-=-, 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 over a non-singular divisor B ⊂ ... |