## Batalin–Vilkovisky algebras and two-dimensional topological field theories (1994)

Venue: | 265–285. AND ALGEBRAS 231 |

Citations: | 126 - 4 self |

### BibTeX

@INPROCEEDINGS{Getzler94batalin–vilkoviskyalgebras,

author = {E. Getzler},

title = {Batalin–Vilkovisky algebras and two-dimensional topological field theories},

booktitle = {265–285. AND ALGEBRAS 231},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract: By a Batalin-Vilkovisky algebra, we mean a graded commutative algebra A, together with an operator A: A.-+ A. such that A +1 2 = 0, and \_A,d \ — Aa is a graded derivation of A for all a e A. In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads. Batalin-Vilkovisky algebras are a new type of algebraic structure on graded vector spaces, which first arose in the work of Batalin and Vilkovisky on gauge fixing in quantum field theory: a Batalin-Vilkovisky algebra is a differential graded commutative algebra together with an operator A: A.-+A such that A m+ί 2 = 0, and Δ{abc) = A(ab)c + (- V)^aA{bc) + (- l) (|α |-ίm

### Citations

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(Show Context)
Citation Context ... particularly explicit type of functor, which may be studied by means of their "Taylor coefficients," the S-module a. The notion of an operad evolved through the work of Boardman and Vogt [4] and May =-=[17]-=-. Their definition is equivalent to the following one. Definition 4.2. An operad in a symmetric monoidal category %> is an S-module a together with the structure of a triple on the analytic functor Γ(... |

248 |
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Citation Context ... and Zuckerman employ the term Gerstenhaber algebra for what we call a braid algebra. Gerstenhaber showed that the Hochschild cohomology H • (A, A) of a graded algebra is what we call a braid algebra =-=[9]-=-. This cohomology is defined using the Hochschild cochains, which are the multilinear maps from A to itself: C • ∞∑ (A, A) = Hom(A k=0 (k) , A). Gerstenhaber defines an operation c1 ◦ c2 on C • (A, A)... |

244 |
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(Show Context)
Citation Context ...s of this article through his joint work with John Jones [10], where we show that the braid operad (defined in Sect. 4) satisfies a certain self-duality property. Independently, Ginzburg and Kapranov =-=[11]-=- have introduced the notion of a Koszul operad, and shown that the associative, commutative and Lie operads are Koszul. (This is an analogue of the notion of a Koszul algebra, that is, an algebra with... |

225 |
links and the mapping class group
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(Show Context)
Citation Context ...ogy class ω ί<7 is dual to the image under the Hurewicz map of the element A tj e F fc given by the formula Aij = Vj-lVj-l - -Gi+l<TiGi+l Gj-2 σ j-l For a picture of this braid, see page 21 of Birman =-=[3]-=-.282 E. Getzler Our proof of Theorem 4.4 in [10] proceeds by showing that b(/c) is quasiisomorphic to a chain complex with summands indexed by trees with k leaves. These trees correspond to strata of... |

198 |
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Citation Context ...nctors are a particularly explicit type of functor, which may be studied by means of their "Taylor coefficients," the S-module a. The notion of an operad evolved through the work of Boardman and Vogt =-=[4]-=- and May [17]. Their definition is equivalent to the following one. Definition 4.2. An operad in a symmetric monoidal category %> is an S-module a together with the structure of a triple on the analyt... |

197 |
Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation, Nucl. Phys
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(Show Context)
Citation Context ...c) = A(ab)c + (- V)^aA{bc) + (- l) (|α| - ίm bA(ac) - (Aa)bc - (- l) (Some references for Batalin-Vilkovisky algebras, with diverse applications to physics, are Schwarz [20], Witten [22] and Zwiebach =-=[24]-=-.) In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. Lian and Zuckerman [16] have ... |

183 |
R.Macpherson, A compactification of configuration space
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Citation Context ...b(/c) is quasiisomorphic to a chain complex with summands indexed by trees with k leaves. These trees correspond to strata of the Fulton-MacPherson compactification of the configuration space Fk ((D) =-=[8]-=- (or rather, its analogue in the category of differentiable manifolds with corners). The theorem follows from the collapse of the spectral sequence for the homology of this stratified space at its £ 2... |

157 |
The geometry of tensor calculus
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Citation Context ... Coxeter matrix (dij)\ is isomorphic to the hyperoctahedral group Skl(Z/2) ^ (Z/2) k xSk . The corresponding generalized braid group is isomorphic to B j2^ίχlB . As is exfc fc plained in Chapter 4 of =-=[14]-=-, this group may be realized as the group of braids on k ribbons: the generator σ 1 ^ i: ^ k — 1, braids the i i9 th ribbon beneath the i + 1 th ribbon, while σ takes the fc k th ribbon through a half... |

154 |
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Citation Context ...ct to the total differential d + β. Witten's formulas [2,5] = 0, and τ [β GΔ relating the degree zero component of the functional integral S and the coefficients of its degree one component G μx (see =-=[23]-=- ) are interpreted as the lowest two terms of the formula {d + Q)(S - G μvdg^ + •) = 0 for the full functional integral ω = S — Gμvdg μv + . To complete the definition of a topological conformal field... |

107 |
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Citation Context ...ons of a braid algebra are satisfied. In the special case that A is the algebra of differentiable functions C∞ (M) on a manifold M, the Hochschild cohomology was shown by Hochschild-Kostant-Rosenberg =-=[12]-=- to be naturally isomorphic to the space of multivectors Γ(M, ∧• TM). With this identification, the cup product may be identified with the wedge product on Γ(M, ∧• TM), while the Gerstenhaber bracket ... |

102 | Geometry of Batalin–Vilkovisky Quantization
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(Show Context)
Citation Context ...such that A m+ί 2 = 0, and Δ{abc) = A(ab)c + (- V)^aA{bc) + (- l) (|α| - ίm bA(ac) - (Aa)bc - (- l) (Some references for Batalin-Vilkovisky algebras, with diverse applications to physics, are Schwarz =-=[20]-=-, Witten [22] and Zwiebach [24].) In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions... |

98 |
New perspectives on the BRST-algebraic structure of string theory
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- 1993
(Show Context)
Citation Context ...ebach [24].) In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. Lian and Zuckerman =-=[16]-=- have constructed a Batalin-Vilkovisky structure on the cohomology of a topological chiral field theory, and calculated it explicitly in the case of D = 2 string theory. Our approach to the study of t... |

88 |
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Citation Context ...ng a map from D to itself to the image of its centre, is a fibration with contractible fibres, hence a homotopy equivalence (May [17], page 34). It may be shown by induction on k (Fadell and Neuwirth =-=[7]-=- ) that Fk (D) ~ K(Ψk , 1). This gives the §-module b(/c) = HΛ (^(k)) the structure of a linear operad. Arnold [1] has calculated the homology of the operad ^(k), or rather, of the configuration space... |

80 |
The cohomology structure of an associative
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(Show Context)
Citation Context ...an and Zuckerman employ the term Gerstenhaber algebra for what we call a braid algebra. Gerstenhaber showed that the Hochschild cohomology H*(A, A) of a graded algebra is what we call a braid algebra =-=[9]-=-. This cohomology is defined using the Hochschild cochains, which are the multilinear maps from A to itself: C Λ {A,A)= f Hom(A {k \A). fc = 0 Gerstenhaber defines an operation c x ° c 2 on C'(A, A) b... |

78 |
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(Show Context)
Citation Context ...ll adopt a definition of a topological conformal field theory inspired by the operator formalism of bosonic string theory, in which the functional integrals are differential forms on Segal's category =-=[21]-=-. Let Jίg%n denote the moduli space of connected Riemann surfaces of genus g, together with a biholomorphic map from the disjoint union JJ" = λ D of n discs, such that the images of the discs are disj... |

61 |
The cohomology ring of the colored braid group
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(Show Context)
Citation Context ...uivalence (May [17], page 34). It may be shown by induction on k (Fadell and Neuwirth [7] ) that Fk (D) ~ K(Ψk , 1). This gives the §-module b(/c) = HΛ (^(k)) the structure of a linear operad. Arnold =-=[1]-=- has calculated the homology of the operad ^(k), or rather, of the configuration spaces Fk(€). Proposition 4.6. For 1 :g i Φj ^ /c, let co^ e H * (F k ((E) 9 Έ) be the inverse image of the generator o... |

47 |
The homology of Cn+1-spaces, n ≥ 0, in ‘The Homology of Iterated Loop Spaces
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Citation Context ...ket is known as the Browder operation. Let us illustrate this in the case where M is a suspension Σ 2 X, where X is connected and carries a circle action preserving the base-point. By Cohen’s results =-=[5]-=-, the homology of Ω 2 Σ 2 X is the free braid algebra on ˜ H•(X), the reduced homology of X. The action of ∆ on H•(Ω 2 Σ 2 X) is determined by its action on ˜ H•(X), together with the formula ∆(xy) = ... |

39 |
On some algebraic structures arising in string theory”, UC Davis preprint hepth/9212072
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Citation Context ...finition of a trace contained in the earlier version, and Larry Breen for a number of helpful remarks. Since the original preprint, some related papers have appeared: Horava [13] and Penkava-Schwartz =-=[18]-=-. 1. Batalin-Vilkovisky Algebras By a chain complex, we mean a vector space over (C graded by integers with differential Q lowering degree by 1. Eventually, we will specialize to the case where V is t... |

36 |
Koszul resolutions
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- 1970
(Show Context)
Citation Context ... commutative and Lie operads are Koszul. (This is an analogue of the notion of a Koszul algebra, that is, an algebra with quadratic relations whose Koszul268 E. Getzler complex, as defined by Priddy =-=[19]-=-, is acyclic: examples of Koszul algebras are the universal enveloping algebra of a Lie algebra and the Steenrod algebra.) Implicit in Theorem 4.5 is the statement that the Batalin-Vilkovisky operad i... |

28 |
Infinitesimal structure of moduli spaces of G– bundles
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- 1992
(Show Context)
Citation Context ...with differential Q, the functional integral is a differential form on Mg%n with values in $ {n \ and the functional integral is closed with respect to the total differential d + β. Witten's formulas =-=[2,5]-=- = 0, and τ [β GΔ relating the degree zero component of the functional integral S and the coefficients of its degree one component G μx (see [23] ) are interpreted as the lowest two terms of the formu... |

22 |
A note on the antibracket formalism
- Witten
- 1990
(Show Context)
Citation Context ...+ί 2 = 0, and Δ{abc) = A(ab)c + (- V)^aA{bc) + (- l) (|α| - ίm bA(ac) - (Aa)bc - (- l) (Some references for Batalin-Vilkovisky algebras, with diverse applications to physics, are Schwarz [20], Witten =-=[22]-=- and Zwiebach [24].) In this article, we show that there is a natural structure of a Batalin-Vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. Lian and Zu... |

11 |
and Mapping Class Groups (Ann
- Birman, Links
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(Show Context)
Citation Context ...is dual to the image under the Hurewicz map of the element Aij ∈ Pk given by the formula Aij = σj−1σj−2 . . .σi+1σ 2 i σ−1 i+1 . . .σ−1 j−2σ−1 j−1 . For a picture of this braid, see page 21 of Birman =-=[3]-=-. Our proof of Theorem 4.4 in [10] proceeds by showing that b(k) is quasi-isomorphic to a chain complex with summands indexed by trees with k leaves. These trees correspond to strata of the Fulton-Mac... |

9 |
Notes on conformal field theory. (unpublished
- Segal
(Show Context)
Citation Context ...ll adopt a definition of a topological conformal field theory inspired by the operator formalism of bosonic string theory, in which the functional integrals are differential forms on Segal’s category =-=[21]-=-. Let ̂ Mg,n denote the moduli space of connected Riemann surfaces of genus g, together with a biholomorphic map from the n∐ disjoint union D of n discs, such that the images of the discs are disjoint... |

1 |
The homology of n + 1 -spaces, n ^ 0. In: The homology of iterated loop spaces
- Cohen
- 1976
(Show Context)
Citation Context ...with differential Q, the functional integral is a differential form on Mg%n with values in $ {n \ and the functional integral is closed with respect to the total differential d + β. Witten's formulas =-=[2,5]-=- = 0, and τ [β GΔ relating the degree zero component of the functional integral S and the coefficients of its degree one component G μx (see [23] ) are interpreted as the lowest two terms of the formu... |

1 |
J.D.S.: Operads and homotopy algebras
- Getzler, Jones
- 1993
(Show Context)
Citation Context ...provided by (bosonic) string theories, which tend to have infinite-dimensional cohomology and non-vanishing A. The author was led to the results of this article through his joint work with John Jones =-=[10]-=-, where we show that the braid operad (defined in Sect. 4) satisfies a certain self-duality property. Independently, Ginzburg and Kapranov [11] have introduced the notion of a Koszul operad, and shown... |

1 |
A.: Differential forms on regular afrine algebras
- Hochschild, Kostant, et al.
- 1962
(Show Context)
Citation Context ...ons of a braid algebra are satisfied. In the special case that A is the algebra of differentiate functions C CO {M) on a manifold M, the Hochschild cohomology was shown by Hochschild-KostantRosenberg =-=[12]-=- to be naturally isomorphic to the space of multivectors Γ(M, f\ TM). With this identification, the cup product may be identified with the wedge product on Γ(M, f\ TM), while the Gerstenhaber bracket ... |

1 |
Spacetime diffeomorphisms and topological W^-symmetry in two dimensional topological string theory
- Hofava
- 1993
(Show Context)
Citation Context ... and for correcting the definition of a trace contained in the earlier version, and Larry Breen for a number of helpful remarks. Since the original preprint, some related papers have appeared: Horava =-=[13]-=- and Penkava-Schwartz [18]. 1. Batalin-Vilkovisky Algebras By a chain complex, we mean a vector space over (C graded by integers with differential Q lowering degree by 1. Eventually, we will specializ... |

1 |
N.P.: Chiral rings in N — 2 superconformal field theories
- Lerche, Vafa, et al.
- 1989
(Show Context)
Citation Context ...rsymmetric conformal field theory, the Batalin-Vilkovisky algebra which we obtain is naturally isomorphic to the chiral primary ring of the original supersymmetric field theory (see LercheVafa-Warner =-=[15]-=- and Dijkgraaf-Verlinde-Verlinde [6] for thorough discussions of this ring). This is a rather degenerate example of a Batalin-Vilkovisky algebra, since the operator A vanishes; in this, it is analogou... |

1 |
n-algebras
- Getzler, Jones
(Show Context)
Citation Context ...provided by (bosonic) string theories, which tend to have infinite-dimensional cohomology and non-vanishing ∆. The author was led to the results of this article through his joint work with John Jones =-=[10]-=-, where we show that the braid operad (defined in Section 4) satisfies a4 E. GETZLER certain self-duality property. Independently, Ginzburg and Kapranov [11] have introduced the notion of a Koszul op... |

1 |
Spacetime diffeomorphisms and topological W∞-symmetry in two dimensional topological string theory
- Hoˇrava
- 1993
(Show Context)
Citation Context ...and for correcting the definition of a trace contained in the earlier version, and Larry Breen for a number of helpful remarks. Since the original preprint, some related papers have appeared: Hoˇrava =-=[13]-=- and Penkava-Schwartz [18]. 1. Batalin-Vilkovisky algebras By a chain complex, we mean a vector space over C graded by integers with differential Q lowering degree by 1. Eventually, we will specialize... |