A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

Abstract

The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented C ∞ -manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points. A notion of de Rham chains, which is a certain hybrid of the notions of singular chains and differential forms, is a key ingredient in our construction. Combined with a generalization of cyclic Deligne's conjecture, this dg operad produces a chain model of the free loop space which admits an action of a chain model of the framed little disks operad, recovering the string topology BV algebra structure on the homology level.2010 Mathematics Subject Classification. 55P50, 57N75, 18D50. 1 well-known fact: the cup product on (singular) cohomology can be defined at the chain level, and (part of) extra information is used to define the classical Massey products; see [22] Section 9.4.5. For (a), what is called Deligne's conjecture claims that a certain chain model of the little disks operad acts on the Hochschild cochain complex. Various affirmative solutions to this conjecture and its variations are known; see [23] Part I Section 1.19, [22] Section 13.3.15, and the references therein. The aim of this paper is to propose a chain level algebraic structure which lifts (b) (the Batalin-Vilkovisky (BV) algebra structure in string topology), and compare it with a solution to Deligne's conjecture via a chain map which is a chain level lift of (c).Our construction is based on generalizations of Deligne's conjecture in operadic contexts. More specifically, we use results in a recent paper [28], which we describe here very briefly. For any (nonsymmetric) dg operad O = (O(k)) k≥0 and a Maurer-Cartan element ζ = (ζ k ) k≥2 of O, one can assign a chain complex (Õ, ∂ ζ ) (here we use our notation in Section 2.5, which is different from the notation in [28]). Then Theorem A in [28] asserts that the chain complexÕ admits an action of a certain chain model of the little disks operad. Moreover, if O has a unital cyclic structure and ζ is cyclically invariant, Theorem B in [28] asserts that this action extends to an action of a certain chain model of the framed little disks operad (strictly speaking, in Theorem B one has to consider a quasi-isomorphic subcomplexÕ nm ⊂Õ which consists of "normalized chains"). Actually, in this paper we consider only the simple case that ζ k = 0 for every k = 2. A Maurer-Cartan element satisfying this condition is equivalent to a "multiplication" of the operad (see Definition 2.6). Let us briefly describe our main result. For any closed oriented C ∞ -manifold M , we define a nonsymmetric cyclic dg operad O M with a multiplication and a unit. Applying Theorem B in [28], the associated chain complex O M nm admits an action of a chain model of the framed little disks operad. In particular, the homology H * ( O M nm ) ∼ = H * ( O M ) has a BV algebra structure. Then we show that there exists an isomor...

Chain level loop bracket and pseudo‐holomorphic disks

“…In Sections 4–6, we reduce Theorem to Theorem (see Section 6), which will be proved in Sections 7–9 using the pseudo‐holomorphic curve theory. In Section 4 we introduce the space of Moore loops with marked points, and the notion of de Rham chains on these spaces, following with minor modifications. Then we define the chain complex of de Rham chains and study its basic properties.…”

“…We call this chain complex the de Rham chain complex of , and denote its homology by . Remark Here are slight differences between the presentation in this section and that in . The sign for the boundary operator in is different from that in . In the definition of we only require that is a submersion.…”

“… The sign for the boundary operator in is different from that in . In the definition of we only require that is a submersion. On the other hand, in [, Section 7.2], we consider maps such that are submersions for all . However, the resulting chain complexes are quasi‐isomorphic. …”

“…The construction of the chain model in uses the idea of ‘de Rham chains’, which was already sketched in but modified in . Actually, the definition of ‘approximate de Rham chains’ in [, Section 6] has some troubles (see Remark ) and we emphasize that the modification which we gave in is nontrivial. Another key idea in is to systematically use (Moore) loops with arbitrarily many marked points, which enables us to define the chain model of the free loop space as a geometric analogue of the Hochschild complex of the space of differential forms.…”

“…We also recall a few applications in symplectic topology from . In Sections 4–6, we introduce our setup in string topology, following with minor modifications, and reduce Theorem to a chain‐level statement (Theorem ). Further details will be explained in the last paragraph of Section 3.…”