## Noncommutative homotopy algebras associated with open strings

Venue: | Rev. Math. Phys |

Citations: | 17 - 5 self |

### BibTeX

@ARTICLE{Kajiura_noncommutativehomotopy,

author = {Hiroshige Kajiura},

title = {Noncommutative homotopy algebras associated with open strings},

journal = {Rev. Math. Phys},

year = {},

pages = {1--99}

}

### OpenURL

### Abstract

We discuss general properties of A∞-algebras and their applications to the theory of open strings. The properties of cyclicity for A∞-algebras are examined in detail. We prove the decomposition theorem, which is a stronger version of the minimal model theorem, for A∞algebras and cyclic A∞-algebras and discuss various consequences of it. In particular it is applied to classical open string field theories and it is shown that all classical open string field theories on a fixed conformal background are cyclic A∞-isomorphic to each other. The same results hold for classical closed string field theories, whose algebraic structure is governed by cyclic L∞-algebras. Contents 1 Introduction and Summary 2 1.1 A∞-space and A∞-algebras.............................. 2 1.2 A∞-structure and classical open string field theory................. 6 1.3 Dual description; formal noncommutative supermanifold.............. 13

### Citations

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Citation Context ...has 2 + 3 + · · · + (n − 1) terms corresponding to the boundary components. Other interesting examples of topological operads and their connection to the compactification can be found for instance in =-=[30, 100]-=-. Let H be a Z-graded vector space and m := {mn : (H) ⊗n → H}n≥1 be a collection of multilinear maps. (H,m) is then an A∞-algebra iff m satisfies the following relations (see also Definition 2.4) m1mn... |

223 |
Rational homotopy theory
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Citation Context ...s purpose requires the off-shell extension of string theory as above. A typical off-shell physics is tachyon condensation[89]. Recently, string field theory is applied to such a direction successfully=-=[93, 77]-=- (see also [59]). Though we assumed the existence of σ which is required many consistency conditions as above, actually there exists many Lorentz-covariant string field theories (SFTs) 6 ; the covaria... |

195 |
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Citation Context ...d open strings, respectively. We use the term ‘classical’ (resp. quantum) for theory without loop (resp. with loops). There exists an abstract standard way for constructing these string field theories=-=[68, 117]-=-. We shall review it briefly in the case of classical open string field theories below. The essence is the same for the other ones. Let {ei} be open string states. They are given canonically for a fix... |

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Citation Context ...bsection 5.3. Though the minimal model theorem follows from the decomposition theorem, the proof relies on inductive arguments and the form of the minimal model is not explicit. On the other hand, in =-=[58]-=- 44for any A∞-algebra its minimal model can be given explicitly by using some Feynman diagrams (see also [73, 36, 34]). We demonstrate in subsection 5.4 that it arises naturally from the issue of fin... |

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Citation Context ... In order to discuss the algebraic properties of cyclic A∞-algebras on a formal noncommutative supermanifold, we need some notions of noncommutative symplectic supergeometry. Such objects appeared in =-=[53, 54]-=-, where constant symplectic structure is introduced. We shall extend it to nonconstant one in the way inspired from the physics of open strings and examine those various mathematical properties such a... |

141 |
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Citation Context ... In order to discuss the algebraic properties of cyclic A∞-algebras on a formal noncommutative supermanifold, we need some notions of noncommutative symplectic supergeometry. Such objects appeared in =-=[53, 54]-=-, where constant symplectic structure is introduced. We shall extend it to nonconstant one in the way inspired from the physics of open strings and examine those various mathematical properties such a... |

141 |
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Citation Context ...see section 4). In string theory, the nonconstant symplectic structure here is relevant to background independent string field theory (recently preferable to be called a boundary string field theory) =-=[115]-=- (see also [39, 45]). Consequently, our definition as above seems to be natural also mathematically. 1.6 Plan of this paper Section 2 is devoted mostly to fixing our conventions for A∞-algebras. The p... |

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Citation Context ...quations are the equation of motion of the action, and the procedure relates to the way of finding some classical solutions in closed string field theory [75, 63] or constructing the tachyon potential=-=[74]-=- (see [45]). Consider solving the Maurer-Cartan equation (MC-eq.), ∑ mk(Φ) = 0 . (5.12) k≥1 Hereafter we often use a shorthand notation mk(Φ) for mk(Φ, · · · ,Φ) as above. This is an extended MC-eq. o... |

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Citation Context ... i } will then be treated as fields in the sense of field theory. The pair of Z-graded vector space H and the algebra of formal power series of the coordinates C(φ) on H is called formal supermanifold=-=[3, 57]-=-. We call so, though this may be an infinitesimal neighborhood or gerb of a more general global supermanifold. Though usually the term ‘super’ indicates Z230graded, we use it for Z-graded object. The... |

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Citation Context ...ed commutative. A L∞-algebra is then obtained by defining degree one coderivation so that it is compatible with the graded commutativity, that is, by graded symmetrizing each multi-linear map mk (see =-=[67, 23]-=-, etc.). 2.3 Cyclic A∞-structure In this subsection A∞-structures with cyclic symmetry are defined. We consider a graded vector space H equipped with an odd constant symplectic inner product. An origi... |

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Citation Context ...s purpose requires the off-shell extension of string theory as above. A typical off-shell physics is tachyon condensation[89]. Recently, string field theory is applied to such a direction successfully=-=[93, 77]-=- (see also [59]). Though we assumed the existence of σ which is required many consistency conditions as above, actually there exists many Lorentz-covariant string field theories (SFTs) 6 ; the covaria... |

123 |
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Citation Context ...1). The odd Poisson bracket associated to eq.(6.1) is then written as ←− ∂ ( , ) = ∂φiωij −→ ∂ = ∂φj ←− ∂ ∂φ a −→ ∂ ∂φ ∗ a − ←− ∂ ∂φ ∗ a −→ ∂ . (6.2) ∂φa This is just the situation in the BV-formalism=-=[10, 11, 40, 32]-=-. In the context of the BV-formalism, {φ a } consists of usual fields of degree zero, ghost fields of degree one, and ghosts of ghosts with degree two, ... and also so-called antighosts whose degree i... |

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Operads in algebra, topology and physics
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Citation Context ...ristic zero. For more simplicity we set k = C. A∞- (and L∞-) algebras are defined in different ways. One way is the operads. An A∞algebra is obtained by an algebra over a non-symmetric dg operad (see =-=[71]-=-). Another one is the bar construction and then A∞-algebras are defined as coalgebras with some additional structures. The bar construction is useful to define A∞-algebras in a simple manner and we ta... |

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Citation Context ... for k ≥ 1. One can see that extending α to j1···jk φ-dependent one leads a Lie algebra which closes even off-shell. A natural extension of gauge transformation in differential graded Lie algebra case=-=[31]-=- to A∞-algebras leads the choice (a) (see [23]). Thus, (a) is also natural and in fact sufficient for our purpose. However, in this subsection we shall use (b) as the definition of the gauge transform... |

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Citation Context ...owers of φ. Repeating this process completes the proof. � Since the Poincaré’s lemma (Lemma 4.1) holds, one can also extend the above results to a noncommutative version of Darboux-Weinstein’s theorem=-=[112]-=-. In contrast, one can also prove Proposition 4.1 in the previous subsection by power expansion as in Theorem 4.1. 4.5 Cyclic A∞-algebras from the dual pictures Here we shall reconsider the meaning of... |

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Citation Context ...l below consider the moduli space of an A∞-algebra in H 0 , the degree zero part of H, though our formalism may be appropriate to extend it to the whole degree (cf. subsection 3.3, subsection 5.4 and =-=[7]-=-). In order to avoid the problem of convergence, one often considers in general the tensor product of maximal ideal of an Artin algebra with H (see [23]). For simplicity, we shall introduce a formal p... |

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Citation Context ...ally from the physics of open strings. After considering the constant ones in subsection 4.1, we shall first define general nonconstant ones in subsection 4.2. Such geometry is discussed by A. Schwarz=-=[82]-=- in case of graded commutative supermanifolds, where the existence of the Darboux theorem is known. In [82] the body (even coordinates parts) of the supermanifolds are treated global. However in nonco... |

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Citation Context ...ch k ≥ 1. ω(ei1 ,mk(ei2 , · · · ,eik+1 )) = (−1)ei2ω(ei2 ,mk(ei3 , · · · ,eik+1 ,ei1 )) A∞-algebras with cyclic symmetry as above are considered in the context of mathematical physics for instance in =-=[54, 118, 25]-=-. See also [71]. Remark 2.4 Let us define a collection of degree zero multilinear map S := {Vk : H⊗· · ·⊗H → C}k≥2 by Vk+1(ei1 , · · · ,eik+1 ) := (−1)ei1ω(ei1 ,mk(ei2 , · · · ,eik+1 )) . The cyclicit... |

87 |
Strongly Homotopy Lie Algebras
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Citation Context ... field is written in terms of the graded commutative fields φi pq ,1 ≤ p,q ≤ N. Graded symmetrizing the corresponding coefficients, for instance in eq.(1.14), then leads a L∞structure(see for example =-=[66]-=-). Another choice of the structure groups leads another L∞-algebra. Namely, when one fixes a structure group, one loses a part of the informations which the open string theory has. A more familiar exa... |

81 |
Algebraic structures and differential geometry
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Citation Context ...nman graphs appeared above. This implies that the collection of on-shell correlation functions of open string theory forms a minimal cyclic A∞-algebra. This statement is essentially already known. In =-=[116]-=- it is shown that the tree closed string theory has the structure of the L∞-algebra (and it is extended to quantum case [110]). Thus, the minimal model theorem implies on a fixed conformal background ... |

74 |
Infinite loop spaces
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Citation Context ...ghly speaking, H-spaces are group-like topological spaces. A typical example is a based loop space. Let Y = ΩX be the space of based loops in X. For a based point x0 ∈ X, an element of Y is a map x : =-=[0,1]-=- → X where x(0) = x(1) = x0 (Figure 1 (a)). We have a product as a group-like 2x X t = (• •) • 0 1/4 1/2 1 t = 0 t = 1 (a) x0 K3 t = 0 1/2 3/4 1 • (• •) Figure 1: (a). An element in Y . (b). A homoto... |

73 | Oriented Open–Closed String Theory Revisited - Zwiebach - 1998 |

70 |
A∞-algebras and the cyclic bar complex
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Citation Context ...oduced, the degree given in Definition 2.2 is simpler for A∞-algebras. For this reason, we use this convention in the present paper. The precise relation between these two conventions can be found in =-=[26]-=-. Definition 2.5 (A∞-morphism) Given two weak A∞-algebras (H,m) and (H ′ ,m ′ ), a weak A∞-morphism F : (H,m) → (H ′ ,m ′ ) is a cohomomorphism from C(H) to C(H ′ ) satisfying Fm = m ′ F . (2.8) In pa... |

70 | Modular operads - Getzler, Kapranov - 1998 |

68 |
homotopy, A ∞ -category, and Floer homologies
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(Show Context)
Citation Context ...as A∞-algebras. Such notions are applied to mathematical physics in many ways. One of the application is the homological mirror conjecture[55] which states various equivalences between an A∞-category =-=[21]-=- on Calabi-Yau manifolds M (A-model side in physical terms) and the category of coherent sheaves on the mirror dual manifold ˆ M (B-model side). This conjecture implies that both sides, that is, not o... |

66 | Perturbation theory in differential homological algebra
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(Show Context)
Citation Context ...c to the original A∞-algebra H [44]. This fact is called the minimal model theorem. Similar issues have been investigated as important subjects in algebraic topology by many authors (for example, see =-=[43, 36, 34, 73]-=-). The minimal model theorem implies, for the case of dga of differential forms, that one can recover the rational homotopy type of M by considering the A∞-structure on the deRham cohomology instead o... |

66 | Introduction to A-infinity algebras and modules
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(Show Context)
Citation Context ... the higher Massey-Yoneda products. One may also consider the (complex of) modules over M and Ext between them. Correspondingly, there are the notion of A∞-modules over M and an A∞-category on M (see =-=[51]-=-). It is then known that the stories stated above hold in a similar way as A∞-algebras. Such notions are applied to mathematical physics in many ways. One of the application is the homological mirror ... |

63 |
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(Show Context)
Citation Context ...d open strings, respectively. We use the term ‘classical’ (resp. quantum) for theory without loop (resp. with loops). There exists an abstract standard way for constructing these string field theories=-=[68, 117]-=-. We shall review it briefly in the case of classical open string field theories below. The essence is the same for the other ones. Let {ei} be open string states. They are given canonically for a fix... |

59 | Strongly homotopy algebras of a Kähler manifold
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(Show Context)
Citation Context ...c to the original A∞-algebra H [44]. This fact is called the minimal model theorem. Similar issues have been investigated as important subjects in algebraic topology by many authors (for example, see =-=[43, 36, 34, 73]-=-). The minimal model theorem implies, for the case of dga of differential forms, that one can recover the rational homotopy type of M by considering the A∞-structure on the deRham cohomology instead o... |

58 |
Consistent couplings between fields with a gauge freedom and deformations of the master equation, Phys
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Citation Context ...aints for the determination of the higher interaction terms. String field theory as reviewed in subsection 1.2 is just the latter case. A similar application to topological field theories is given by =-=[8]-=- and developed for example in [3, 42, 78, 19, 9]. 61In any case the action in the BV-formalism is, by power series of fields, written as S = 1 2 Vi1i2φi2 ∑ i1 1 φ + k Vi1···ikφik i1 · · · φ , Vi1···i... |

51 |
Rational homotopy theory and differential forms. Birkhäuser
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Citation Context ...mology. This paper deals with this algebra side, some ‘deRham rings up to homotopy’. Such algebraic treatments of homotopy theory were developed in rational homotopy theory by Quillen[80] and Sullivan=-=[105, 33]-=-. In particular [105] deals with differential forms on a manifold M, which form a differential graded algebra (dga). It is then shown that the dga of differential forms on M has the information of the... |

50 | Small models for chain algebras - Huebschmann, Kadeishvili - 1991 |

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Citation Context ... cyclic symmetry, a cyclic operad[27, 28]. By taking its representation, one obtains an algebra over the operad, where H is the operad algebra. For the relevance of operads in string field theory see =-=[52, 99, 111]-=-. As seen in the next subsection, the action which has cyclic vertices and satisfies the classical BV-master equation has an A∞-structure. The A∞-algebra in addition possesses the odd 11symplectic in... |

44 | Infinitesimal computations in topology. Inst. Hautes études - Sullivan |

43 |
Homotopy algebras are homotopy algebras, math.AT/9907138
- Markl
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Citation Context ...A∞-algebras In this section we shall discuss some classical properties of A∞-algebras which has their origin from homotopy theory itself. There are some literatures which discuss similar problem (see =-=[70]-=- and reference therein). A characteristic of our approach is to apply the decomposition theorem (Theorem 5.2) in section 5. In subsection 7.1 we shall define homotopy equivalence between A∞morphisms. ... |

41 | Deformation theory and the Batalin-Vilkovisky master equation,” q-alg/9702012
- Stasheff
(Show Context)
Citation Context ...replaced to a L∞-algebra, too. In the cyclic case, these properties are closely related to the BV-formalism [10, 11, 32, 40]. To exploring these relations should be very interesting (see for instance =-=[103, 102]-=-). 83Acknowledgments I am very grateful to A. Kato for helpful discussions, precise advice and encouragement. Also, I would like to thank T. Asakawa, H. Hata, Y. Kazama, T. Kugo, H. Kunitomo and T. N... |