## Topological conformal field theories and Calabi-Yau categories (2004)

Citations: | 87 - 6 self |

### BibTeX

@MISC{Costello04topologicalconformal,

author = {Kevin Costello},

title = {Topological conformal field theories and Calabi-Yau categories},

year = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-

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associativity of H-Spaces I
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The definition of conformal field theory
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Citation Context ...ms of C∗(M) are defined by Define Mor C∗(M)(a,b) = C∗(MorM(a,b)) C def = C∗(M) C , like M, is a symmetric monoidal category. The following definition is due independently to Getzler [Get94] and Segal =-=[Seg99]-=-. 12 KEVIN COSTELLO Definition. A topological conformal field theory is a tensor functor F from the differential graded category C to the category of chain complexes. What this means is the following... |

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Citation Context ...ferential of degree −1), and the composition maps are bilinear and compatible with the differential. Call these dgsm categories, for short. A good reference for the general theory of dg categories is =-=[Kel94]-=-.�� �� �� � � � �� �� �� 18 KEVIN COSTELLO The dgsm categories controlling topological conformal field theory are strictly monoidal. On objects, (α ∐ β) ∐ γ = α ∐ (β ∐ γ), and similarly the diagram H... |

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Citation Context ...uld give a functor from C∗(M) → Comp K; pulling back via the functor C∗(M ′ ) → C∗(M) will give the required TCFT. The model M ′ for M we need was first constructed by Kimura, Stasheff and Voronov in =-=[8]-=-. It can be constructed by performing a real blow up of the Deligne-Mumford spaces along their boundary. More precisely, we can take for M ′ the moduli space of curves Σ ∈ M, together with at each mar... |

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Citation Context ...form the Lagrangians Li by a cycle bi ∈ C∗(Li), which satisfies a type of Maurer-Cartan equation. Parts of this conjectural theory have previously been constructed by P. Siedel [19, 18] and C.-C. Liu =-=[12]-=-. Seidel constructs the “topological field theory” version with fixed complex structure on the source Riemann surface. This corresponds to working with H0 of moduli spaces. The part dealing with only ... |

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Citation Context ...aces of Riemann surfaces on the Hochschild chains. Results of this form were conjectured by Kontsevich as far back as 1994 [8], and have also been conjectured by Getzler, Segal, Kapustin and Rozansky =-=[4]-=-.... A different approach to the proof of this result has been given by Kontsevich [9] using the ribbon graph decomposition of moduli spaces of curves, in a lecture at the Hodge centenary conference i... |

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Citation Context ...he diagram H∗(M(I,J),det d ) ⊗ HH∗(Fuk(X)) ⊗I �� H∗(M(I,J),det d ) ⊗ HF∗(X) ⊗I HH∗(Fuk(X)) ⊗J � HF∗(X) ⊗J commutes. The map from Hochschild to Floer homology is the same as that proposed by Seidel in =-=[20]-=-. The homology of a TCFT has the structure of cocommutative coalgebra, coming from the pair-of-pants coproduct. This coproduct structure on HH∗(Fuk(X)) is dual to the standard cup product on Hochschil... |

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Citation Context ...s setting: we should deform the Lagrangians Li by a cycle bi ∈ C∗(Li), which satisfies a type of Maurer-Cartan equation. Parts of this conjectural theory have previously been constructed by P. Siedel =-=[19, 18]-=- and C.-C. Liu [12]. Seidel constructs the “topological field theory” version with fixed complex structure on the source Riemann surface. This corresponds to working with H0 of moduli spaces. The part... |

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Mirror symmetry and noncommutative geometry of A∞-categories
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Citation Context ... and non-degenerate. If αi : Ai → Ai+1mod n are morphisms, then 〈mn−1(α0 ⊗ ... ⊗ αn−2),αn−1〉 is required to be cyclically symmetric. One can think of this as a “non-commutative Calabi-Yau space”, see =-=[21]-=-. The notion of extended CY A∞ category is a reasonably small generalisation of this definition, and will be explained later. If X is a compact symplectic manifold of dimension 2d, then its Fukaya cat... |

7 |
Talk at the Hodge Centennial conference
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Citation Context ...ed by Kontsevich as far back as 1994 [8], and have also been conjectured by Getzler, Segal, Kapustin and Rozansky [4].... A different approach to the proof of this result has been given by Kontsevich =-=[9]-=- using the ribbon graph decomposition of moduli spaces of curves, in a lecture at the Hodge centenary conference in 2003. 1.3. Higher genus B model. A Calabi-Yau category is the categorical generalisa... |

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Citation Context ...i-isomorphism. The notion of extended CY A∞ category will be explained later. The proof of part (1) uses the operadic version of the ribbon graph decomposition of moduli space proved by the author in =-=[1]-=-. This categorical equivalence implies the existence of the ribbon graph decomposition. The homotopy universal closed TCFT Li∗Φ has the property that for every openclosed TCFT Ψ, with a map Φ → i ∗ Ψ ... |

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Citation Context ...(in a precise sense) categories of functors, so that up to homotopy there is no ambiguity. 1.2. Open and open-closed TCFTs. Open-closed conformal field theory was first axiomatised by Moore and Segal =-=[15, 16]-=-. A Riemann surface with open-closed boundary is a Riemann surface Σ, some of whose boundary components are parameterised,TCFTS AND CALABI-YAU CATEGORIES 3 Figure 1. A Riemann surface with open-close... |

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A-infinity categories and non-commutative geometry
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Citation Context ...with one incoming and one outgoing boundary. This is a variant of an unpublished result of Kontsevich [9]. I have recently learned that a proof of this is being written by Kontsevich and Soibelman in =-=[10]-=-. I will sketch here a different approach to the proof of this result. This approach is more homotopical and categorical in flavour; but it shares with Kontsevich’s the key idea that the compactified ... |

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1 |
Topological field theory. 1999, http://www.cgtp.duke.edu/ITP99/segal/. Notes of lectures at Stanford university
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1 | The A∞ operad and the moduli space of curves - Cohen, theory, et al. - 2004 |