Effective Batalin–Vilkovisky Theories, Equivariant Configuration Spaces and Cyclic Chains

Abstract

Dedicated to Murray Gerstenhaber and Jim Stasheff

Kontsevich’s formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifoldMisL _∞ -quasi-isomorphic to its cohomology. The construction of theL _∞ -map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of${\mathbb{R}}^{d}$with standard volume form, we obtain anL _∞ -morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin–Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.

Mathematics Subject Classification 2010: Primary 53D55; Secondary 81T40, 81T45, 81T70

1 Introduction

The Hochschild complex of any algebra with unit carries a differential graded Lie algebra structure introduced by Gerstenhaber [14]. In the case of the algebra of smooth functions on a manifold, one has a differential graded Lie subalgebra${\mathfrak{g}}_{G}$of multidifferential operators, whose cohomology is the graded Lie algebra${\mathfrak{g}}_{S}$of multivector fields with Schouten–Nijenhuis bracket.¹Kontsevich [17] showed that${\mathfrak{g}}_{G}$and${\mathfrak{g}}_{S}$are quasi-isomorphic asL _∞ -algebras, a notion introduced by Stasheff as the Lie version ofA _∞ -algebras [28], see [24,20]. A striking application of this result is the classification of formal associative deformations of the product of functions in terms of Poisson structures. Kontsevich’sL _∞ -quasi-isomorphism is given in terms of integrals over configuration spaces of points in the upper half-plane. As shown in [3], these are Feynman amplitudes of a topological quantum field theory known as the Poisson sigma model [16,23].

In this paper we consider the case of a manifoldMendowed with a volume formΩ. In this case${\mathfrak{g}}_{S}$comes with a differential, the divergence operator div _Ω of degree − 1. One considers then the differential graded Lie algebra${\mathfrak{g}}_{S}^{\Omega } = {\mathfrak{g}}_{S}[v]$wherevis an indeterminate of degree 2, the bracket is extended byv-linearity and the differential isvdiv _Ω . The relevant topological quantum field theory is a BF theory (or Poisson sigma model with trivial Poisson structure) on a disk whose differential is the Cartan differential onS ¹-equivariant differential forms. This theory is described in Section2. The new feature, compared to the original setting of Kontsevich’s formality theorem, is that zero modes are present. We use recent ideas of Losev, Costello and Mnev to treat them in the Batalin–Vilkovisky quantization scheme. This gives the physical setting from which the Feynman amplitudes are derived. In the remaining sections of this paper, which can be read independently of Section2, we give a purely mathematical treatment of the same objects. The basic result is the construction for$M = {\mathbb{R}}^{d}$of anL _∞ -morphism of${\mathfrak{g}}_{S}^{\Omega }$-modules from the module of negative cyclic chains (C _{− ∙}(A)[u],b+uB) to the trivial module (Γ( ∧^{− ∙} TM), div _Ω ). We also check that thisL _∞ -morphism has properties needed to extend the result to general manifolds.

As in the case of Kontsevich’s theorem, the coefficients of theL _∞ -morphism are integrals of differential forms on configuration spaces. Whereas Kontsevich considers the spaces ofn-tuples of points in the upper half-plane modulo the action of the group of dilations and horizontal translations, we consider the space ofn-tuples of points in the unit disk and work equivariantly with respect to the action of the rotation group. The quadratic identities defining theL _∞ -relations are then proved by means of an equivariant version of the Stokes theorem.

As a first application we construct traces in deformation quantization associated with unimodular Poisson structures. Our construction can also be extended to the case of supermanifolds; the trace is then replaced by a nondegenerate cyclic cocycle (Calabi–Yau structure, see [18], Section 10.2, and [10]) for theA _∞ -algebra obtained by deformation quantization in [5]. Further applications will be studied in a separate publication [6]. In particular we will derive the existence of anL _∞ -quasi-isomorphism of${\mathfrak{g}}_{S}^{\Omega }$-modules from the complex${\mathfrak{g}}_{S}^{\Omega }$with the adjoint action to the complex of cyclic cochains with a suitable module structure. This is a module version of the Kontsevich–Shoikhet formality conjecture for cyclic cochains [26].

1.1 Notations and conventions

All vector spaces are over$\mathbb{R}$. We denote byS _n the group of permutations ofnletters and by ε:S _n → { ± 1} the sign character. We write | α | for the degree of a homogeneous element α of a$\mathbb{Z}$-graded vector space. The sign rules for tensor products of graded vector spaces hold: iffandgare linear maps on graded vector spaces, (f⊗g)(v⊗w) = ( − 1)^{|g| ⋅ |v|} f(v) ⊗g(w). The graded vector spaceV[n] isVshifted byn:V[n]ⁱ=V ⁿ⁺ⁱ. There is a canonical map (the identity)s ⁿ:V[n] →Vof degreen. The graded symmetric algebraS ^∙ V= ⊕_{n≥ 0} S ⁿ Vof a graded vector spaceVis the algebra generated byVwith relationsa⋅b= ( − 1)^{|a| ⋅ |b|} b⋅a,a,b∈V; the degree of a product of generators is the sum of the degrees. If σ ∈S _n is a permutation and${a}_{1},\ldots,{a}_{n} \in V$, thena _σ(1)⋯a _σ(n)= εa ₁⋯a _n ; we call$\epsilon = \epsilon (\sigma ;{a}_{1},\ldots,{a}_{n})$the Koszul sign of σ anda _i . The exterior algebra ∧Vis defined by the relationsa∧b= − ( − 1)^{|a| ⋅ |b|} b∧aon generators. We have a linear isomorphismS ⁿ(V[1]) → ( ∧ⁿ V)[n] given by${v}_{1}\cdots {v}_{n}\mapsto {s}^{-n}{(-1)}^{\sum (n-j)(\vert {v}_{j}\vert -1)}s{v}_{ 1} \wedge \cdots \wedge s{v}_{n}$,v _j ∈V[1].

2 BV formalism and zero modes

This section provides the interested reader with some “physical” motivation for the constructions in this paper. It may be safely skipped by the reader who is only interested in the construction and not in its motivation.

The basic idea is to use the Batalin–Vilkovisky (BV) formalism in order to deal with theories with symmetries (like the Poisson sigma model). What is interesting for this paper is the case when “zero modes” are present.

It is well known in algebraic topology that structures may be induced on subcomplexes (in particular, on an embedding of the cohomology) like, e.g., induced differentials in spectral sequences or Massey products. It is also well known in physics that low-energy effective field theories may be induced by integrating out high-energy degrees of freedom. As observed by Losev [21] (and further developed by Mnev [22] and Costello [9]), the two things are actually related in terms of the BV approach to (topological) field theories. We are interested in the limiting case when the low-energy fields are just the zero modes, i.e., the critical points of the action functional modulo its symmetries.

Let$\mathcal{M}$be an SP-manifold, i.e., a graded manifold endowed with a symplectic form of degree − 1 and a compatible Berezinian [25]. LetΔbe the corresponding BV-Laplace operator. The compatibility amounts to saying thatΔsquares to zero and that it generates the BV bracket (, ) (i.e., the Poisson bracket of degree 1 determined by the symplectic structure of degree − 1): namely,

$$\Delta (AB) = (\Delta A)B + {(-1)}^{\vert A\vert }A\Delta B - {(-1)}^{\vert A\vert }(A,B). $$

(1)

Assume now that$\mathcal{M}$is actually a product of SP-manifolds${\mathcal{M}}_{1}$and${\mathcal{M}}_{2}$, with BV-Laplace operatorsΔ ₁andΔ ₂,Δ=Δ ₁+Δ ₂. The central observation is that for every Lagrangian submanifold$\mathcal{L}$of${\mathcal{M}}_{2}$and any functionFon$\mathcal{M}$– for which the integral makes sense – one has

$${\Delta }_{1}{ \int }_{\mathcal{L}}F ={ \int }_{\mathcal{L}}\Delta F. $$

(2)

In infinite dimensions, where we would really like to work, this formula is very formal as both the integration andΔare ill-defined. In finite dimensions, on the other hand, this is just a simple generalization of the fact that, for any differential form α on the Cartesian product of two manifoldsM ₁andM ₂and any closed submanifoldSofM ₂on which the integral of α converges, we have

$$\mathrm{d}{\int }_{S}\alpha = \pm {\int }_{S}\mathrm{d}\alpha,$$

where integration onSyields a differential form onM ₁. The correspondence with the BV language is obtained by taking${\mathcal{M}}_{1,2} := {T}^{{_\ast}}[-1]{M}_{1,2}$and$\mathcal{L} := {N}^{{_\ast}}[-1]S$(whereN ^∗denotes the conormal bundle). The Berezinian on$\mathcal{M}$is determined by a volume formv=v ₁∧v ₂onM: =M ₁×M ₂, withv _i a volume form onM _i . Finally,Δis ϕ _v ^{− 1}∘ d ∘ ϕ _v , with ϕ _v :Γ( ∧^∙ TM) →Ω ^{dimM− ∙}(M),X↦ϕ _v (X) : = ι _X v. The generalization consists in the fact that there are Lagrangian submanifolds of${\mathcal{M}}_{2}$not of the form of a conormal bundle; however, by a result of Schwarz [25], they can always be brought to this form by a symplectomorphism so that formula (2) holds in general.

In the application we have in mind,${\mathcal{M}}_{2}$(and so$\mathcal{M}$) is infinite-dimensional, but${\mathcal{M}}_{1}$is not. Thus, we have a well-defined BV-Laplace operatorΔ ₁and try to make sense ofΔby imposing (2), following ideas of [21,22] and, in particular, [9]. More precisely, we consider “BF-like” theories. Namely, let$(\mathcal{V},\delta )$and$(\tilde{\mathcal{V}},\delta )$be complexes with a nondegenerate pairing, of degree − 1 which relates the two differentials:

$$\left \langle \,\mathsf{B}\,,\,\delta \mathsf{A}\,\right \rangle = \left \langle \,\delta \mathsf{B}\,,\,\mathsf{A}\,\right \rangle,\qquad \forall \mathsf{A} \in \mathcal{V},\ \mathsf{B} \in \tilde{\mathcal{V}}. $$

(3)

We set$\mathcal{M} = \mathcal{V}\oplus \tilde{\mathcal{V}}$and define$S \in {C}^{\infty }(\mathcal{M})$as

$$S(\mathsf{A},\mathsf{B}) := \left \langle \,\mathsf{B}\,,\,\delta \mathsf{A}\,\right \rangle. $$

(4)

The pairing defines a symplectic structure of degree − 1 on$\mathcal{M}$and the BV bracket withSis δ. In particular,

$$(S,S) = 0. $$

(5)

We denote by$\mathcal{H}$($\tilde{\mathcal{H}}$) the δ-cohomology of$\mathcal{V}$($\tilde{\mathcal{V}})$. Then we choose an embedding of${\mathcal{M}}_{1} := \mathcal{H}\oplus \tilde{\mathcal{H}}$into$\mathcal{M}$and a complement${\mathcal{M}}_{2}$.

Example 1.

Take$\mathcal{V} = \Omega (\Sigma )[1]$and$\tilde{\mathcal{V}} = \Omega (\Sigma )[s - 2]$, withΣa closed, compacts-manifold, and δ = d, the de Rham differential, on$\mathcal{V}$; up to a sign, δ on$\tilde{\mathcal{V}}$is also the de Rham differential if the pairing is defined by integration:B,A: = ∫ _Σ B∧A,$\mathsf{A} \in \mathcal{V}$,$\mathsf{B} \in \tilde{\mathcal{V}}$. In this case${\mathcal{M}}_{1} = H(\Sigma )[1] \oplus H(\Sigma )[s - 2]$, withH(Σ) the usual de Rham cohomology. A slightly more general situation occurs whenΣhas a boundary; in this case, appropriate boundary conditions have to be chosen so that δ has an adjoint as in (3). Let∂Σ=∂ ₁ Σ⊔∂ ₂ Σ(each of the boundary components∂ _{1, 2} Σmay be empty). We then choose$\mathcal{V} = \Omega (\Sigma,{\partial }_{1}\Sigma )[1]$and$\tilde{\mathcal{V}} = \Omega (\Sigma,{\partial }_{2}\Sigma )[s - 2]$, whereΩ(Σ,∂ _i Σ) denotes differential forms whose restrictions to∂ _i Σvanish. In this case,${\mathcal{M}}_{1} = H(\Sigma,{\partial }_{1}\Sigma )[1] \oplus H(\Sigma,{\partial }_{2}\Sigma )[s - 2]$.

Example 2.

Suppose thatS ¹acts onΣ(and that the∂ _i Σs are invariant). Let$\vec{v}$denote the vector field onΣgenerating the infinitesimal action. Let${\Omega }_{{S}^{1}}(\Sigma,\partial \Sigma ) := \Omega {(\Sigma,\partial \Sigma )}^{{S}^{1} }[u]$denote the Cartan complex with differential$\mathrm{{d}}_{{S}^{1}} =\mathrm{ d} - u{\iota }_{\vec{v}}$, whereuis an indeterminate of degree 2. Then we may generalize Example1replacingΩ(Σ,∂Σ) with${\Omega }_{{S}^{1}}(\Sigma,\partial \Sigma )$.

Now suppose that$\mathcal{H}$(and so$\tilde{\mathcal{H}}$) is finite-dimensional, as in the examples above. In this case it is always possible to choose a BV-LaplacianΔ ₁on${\mathcal{M}}_{1}$. Once and for all we also choose a Lagrangian submanifold$\mathcal{L}$on which the infinite-dimensional integral makes sense in perturbation theory. AssumingΔS= 0, the first consequence of (2) and (5) is that the partition function

$${Z}_{0} ={ \int }_{\mathcal{L}}\mathrm{{e}}^{ \frac{\mathrm{i}} {\hslash } S}$$

isΔ ₁-closed. Actually, in the case at hand,Z ₀is constant on${\mathcal{M}}_{1}$.

For every functional$\wr$on$\mathcal{M}$for which integration on$\mathcal{L}$makes sense, we define the expectation value

$${\left \langle \;\wr \;\right \rangle }_{0} := \frac{{\int }_{\mathcal{L}}\mathrm{{e}}^{ \frac{\mathrm{i}} {\hslash } S}\,\wr } {{Z}_{0}}.$$

The second consequence of (2), and of the fact thatZ ₀is constant on${\mathcal{M}}_{1}$, is the Ward identity

$${\Delta }_{1}{\left \langle \;\wr \;\right \rangle }_{0} ={ \left \langle \;\Delta \wr - \frac{\mathrm{i}} {\hslash }\delta \wr \;\right \rangle }_{0}, $$

(6)

where we have also used (1).

To interpret the Ward identity for$\wr = \mathsf{B} \otimes \mathsf{A}$, we denote by {θ^μ} a linear coordinate system on$\mathcal{H}$and by {ζ_μ} a linear coordinate system on$\tilde{\mathcal{H}}$, such that their union is a Darboux system for the symplectic form on${\mathcal{M}}_{1}$with${\Delta }_{1} = \frac{\partial } {\partial {\theta }^{\mu }} \frac{\partial } {\partial {\zeta }_{\mu }}$. We next writeA= α_μθ^μ+aandB= β^μζ_μ+bwith$\mathsf{a} \oplus \mathsf{b} \in {\mathcal{M}}_{2}$. The left-hand side of the Ward identity is now simply${\Delta }_{1}{\left \langle \;\mathsf{B} \otimes \mathsf{A}\;\right \rangle }_{0} ={ \sum }_{\mu }{(-1)}^{\vert {\beta }^{\mu }\vert }{\beta }^{\mu } \otimes {\alpha }_{\mu } =: \phi$. On the assumption that the ill-defined BV-LaplacianΔshould be a second-order differential operator, the first term  Δ(B⊗A) ₀on the right-hand side is ill-defined but constant on${\mathcal{M}}_{1}$; we denote it byK. Since δ vanishes in cohomology and, as a differential operator, it can be extracted from the expectation value, (6) yields a constraint for the propagator

$$\omega := \frac{\mathrm{i}} {\hslash }{\left \langle \;\mathsf{b} \otimes \mathsf{a}\;\right \rangle }_{0}; $$

(7)

namely,

$$\delta \omega = K - \phi.$$

From now on we assume that$\mathcal{M}$is defined in terms of differential forms as in Examples1and2. In this case, ω is a distributional (s− 1)-form onΣ×Σwhile ϕ is a representative of the Poincaré dual of the diagonalD _Σ inΣ×Σ. By the usual naive definition ofΔ,Kis equal to the delta distribution onD _Σ . Thus, the restriction of ω to the configuration spaceC ₂(Σ) : =Σ×Σ∖D _Σ is a smooth (m− 1)-form satisfying dω = ϕ. IfΣhas a boundary, ω satisfies in addition the conditions ι₁ ^∗ω = ι₂ ^∗ω = 0 with ι₁the inclusion ofΣ×∂ ₁ ΣintoΣ×Σand ι₂the inclusion of∂ ₂ Σ×ΣintoΣ×Σ. Denoting by π_{1, 2}the two projectionsΣ×Σ→Σand by π_∗ ^{1, 2}the corresponding fiber-integrations, we may defineP:Ω(Σ,∂ ₁ Σ) →Ω(Σ,∂ ₁ Σ) and$\tilde{P}: \Omega (\Sigma,{\partial }_{2}\Sigma ) \rightarrow \Omega (\Sigma,{\partial }_{2}\Sigma )$byP(σ) = π_∗ ²(ω ∧ π₁ ^∗σ) and$\tilde{P}(\sigma ) = {\pi }_{{_\ast}}^{1}(\omega \wedge {\pi }_{2}^{{_\ast}}\sigma )$. Then the equation for ω implies thatPand$\tilde{P}$are parametrices for the complexesΩ(Σ,∂ ₁ Σ) andΩ(Σ,∂ ₂ Σ); namely, dP+Pd = 1 − ϖ and$\mathrm{d}\tilde{P} +\tilde{ P}\mathrm{d} = 1 -\tilde{ \varpi }$, where ϖ and$\tilde{\varpi }$denote the projections onto cohomology.

This characterization of the propagator of a “BF-like” theory also appears in [9]. Even though not justified in terms of the BV formalism, this choice of propagator was done before in [2] for Chern–Simons theory out of purely topological reasons, and later extended toBFtheories in [7]. A propagator with these properties also appears in [13] for the Poisson sigma model on the interior of a polygon.

The quadratic action (4) is usually the starting point for a perturbative expansion. The first singularity that may occur comes from evaluating ω onD _Σ (“tadpole”). A mild form of renormalization consists in removing tadpoles or, in other words, in imposing that ω should vanish onD _Σ . By consistency, one has then to setKequal to the restriction of ϕ toD _Σ . In other words, one has to impose

$$\Delta (\mathsf{B}(x)\mathsf{A}(x)) = \psi (x) :={ \sum }_{\mu }{(-1)}^{\vert {\beta }^{\mu }\vert }{\beta }^{\mu }(x){\alpha }_{ \mu }(x),\qquad \forall x \in \Sigma. $$

(8)

Observe that ψ is a representative of the Euler class ofΣ. By (1) and (8) one then obtains a well-defined version ofΔon the algebra${C}^{\infty }(\mathcal{M}){\prime}$generated by local functionals. This may be regarded as an asymptotic version (for the energy scale going to zero) of Costello’s regularized BV-Laplacian [9]. Actually,

Lemma 1.

$({C}^{\infty }(\mathcal{M}){\prime},\Delta )$ is a BV algebra.

We now restrict ourselves to the setting of the Poisson sigma model [16,23]. Namely, we assumeΣto be two-dimensional and take$\mathcal{V} = \Omega (\Sigma,{\partial }_{1}\Sigma )[1] \otimes {({\mathbb{R}}^{m})}^{{_\ast}}$and$\tilde{\mathcal{V}} = \Omega (\Sigma,{\partial }_{2}\Sigma ) \otimes {\mathbb{R}}^{m}$. Here (ℝ ^m)^∗×ℝ ^mis a local patch of the cotangent bundle of anm-dimensional target manifoldM. (Whatever we say here and in the following may be globalized by taking$\mathcal{M}$to be the graded submanifold of Map (T[1]Σ,T ^∗[1]M) defined by the given boundary conditions.) There is a Lie algebra morphism from the graded Lie algebra${\mathfrak{g}}_{S} = \Gamma ({\wedge }^{\bullet +1}\mathit{TM})$of multivector fields onMto${C}^{\infty }(\mathcal{M}){\prime}$endowed with the BV bracket [4]: to γ ∈Γ( ∧^k TM) it associates the local functional

$${S}_{\gamma } ={ \int }_{\Sigma }{\gamma }^{{i}_{1},\ldots,{i}_{k} }(\mathsf{B})\,{\mathsf{A}}_{{i}_{1}}\cdots {\mathsf{A}}_{{i}_{k}}.$$

Moreover, fork> 0, (S,S _γ) = 0. With the regularized version of the BV-Laplacian, we get

$$\Delta {S}_{\gamma } ={ \int }_{\Sigma }\psi \,{({\mathrm{div} }_{\Omega }\gamma )}^{{i}_{1},\ldots,{i}_{k-1} }(\mathsf{B})\,{\mathsf{A}}_{{i}_{1}}\cdots {\mathsf{A}}_{{i}_{k-1}},$$

where div _Ω is the divergence with respect to the constant volume formΩonℝ ⁿ. To account for this systematically, we introduce the differential graded Lie algebra${\mathfrak{g}}_{S}^{\Omega } := {\mathfrak{g}}_{S}[v]$, wherevis an indeterminate of degree two and the differential δ _Ω is defined asvdiv _Ω (and the Lie bracket is extended byv-linearity). To γ ∈Γ( ∧^k TM)v ^lwe associate the local functional

$${S}_{\gamma } = {(-\mathrm{i}\hslash )}^{l}{ \int }_{\Sigma }{\psi }^{l}\,{\gamma }^{{i}_{1},\ldots,{i}_{k} }(\mathsf{B})\,{\mathsf{A}}_{{i}_{1}}\cdots {\mathsf{A}}_{{i}_{k}}.$$

It is now not difficult to prove the following

Lemma 2.

The map γ↦S _γ is a morphism of differential graded Lie algebras from $({\mathfrak{g}}_{S}^{\Omega },[\,\ ],{\delta }_{\Omega })$ to $({C}^{\infty }(\mathcal{M}){\prime},(\,\ ),-\mathrm{i}\hslash \Delta )$ . Moreover, for every γ ∈ Γ(∧ ^k TM) v^lwith k or l strictly positive, we have (S,S_γ) = 0. If ∂Σ = ∅, the last statement holds also for k = l = 0.

Observe that ψ²= 0 by dimensional reasons. However, in the generalization to the equivariant setting of Example2, higher powers of ψ survive.

A first application of this formalism is the Poisson sigma model onΣ. If π is a Poisson bivector field (i.e., π ∈Γ( ∧² TM), [π, π] = 0), thenS _π: =S+S _πsatisfies the master equation (S _π,S _π) = 0 but in general not the quantum master equation$\frac{1} {2}({\mathsf{S}}_{\pi },{\mathsf{S}}_{\pi }) + \mathrm{i}\hslash \Delta {\mathsf{S}}_{\pi } = 0$, which by (1) is equivalent to$\Delta \mathrm{{e}}^{ \frac{\mathrm{i}} {\hslash }{ \mathsf{S}}_{\pi }} = 0$. Unless ψ is trivial²(which is, e.g., the case forΣthe upper half plane, as in [3], or the torus), this actually happens only if π is divergence free. More generally, if π is unimodular [19], by definition we may find a functionfsuch that div _Ω π = [π,f]. So$\tilde{\pi } := \pi + vf$is a Maurer–Cartan element in${\mathfrak{g}}_{S}^{\Omega }$(i.e.,${\delta }_{\Omega }\tilde{\pi } -\frac{1} {2}[\tilde{\pi },\tilde{\pi }] = 0$). Hence${\mathsf{S}}_{\tilde{\pi }} := S + {S}_{\tilde{\pi }}$satisfies the quantum master equation. It is not difficult to check that, for ψ nontrivial, the unimodularity of π is a necessary and sufficient condition for having a solution of the quantum master equation of the formS+S _π+O(ℏ). ForΣthe sphere this was already observed in [1] though using slightly different arguments.

We will now restrict ourselves to the case of interest for the rest of the paper: namely,Σthe disk and∂ ₂ Σ=∅. In this caseH(Σ) is one-dimensional and concentrated in degree 0 whileH(Σ,∂Σ) is one-dimensional and concentrated in degree two. Thus,$\mathcal{H} = {({\mathbb{R}}^{m})}^{{_\ast}}[-1]$and$\mathcal{H} = {\mathbb{R}}^{m}$which implies${\mathcal{M}}_{1} = {T}^{{_\ast}}[-1]M$. Functions on${\mathcal{M}}_{1}$are then multivector fields onMbut with reversed degree and the operatorΔ ₁turns out to be the usual divergence operator div _Ω (which is now of degree + 1) for the constant volume form. A first simple application is the expectation value

$$\mathrm{tr} g := \frac{{\int }_{\mathcal{L}}\mathrm{{e}}^{ \frac{\mathrm{i}} {\hslash }{ \mathsf{S}}_{\tilde{\pi }}}\,{\wr }_{g}} {{Z}_{0}} ={ \left \langle \;\mathrm{{e}}^{ \frac{\mathrm{i}} {\hslash } {S}_{\tilde{\pi }}}\,{\wr }_{g}\;\right \rangle }_{0},\qquad g \in {C}^{\infty }(M),$$

where$\tilde{\pi }$is a Maurer–Cartan element corresponding to a unimodular Poisson structure andO _g (A,B) : =g(B(1)), with 1 in∂Σwhich we identify with the unit circle. Consider now${\mathrm{tr} }_{2}(g,h) :={ \left \langle \;\mathrm{{e}}^{ \frac{\mathrm{i}} {\hslash } {S}_{\tilde{\pi }}}\,{\wr }_{g,h}\;\right \rangle }_{0}$, with${\wr }_{g,h} := g(\mathsf{B}(1)){\int }_{\partial \Sigma \setminus \{1\}}h(\mathsf{B})$. By (1), we then have${\Delta }_{1}{ \mathrm{tr} }_{2}(g,h) ={ \left \langle \;\mathrm{{e}}^{ \frac{\mathrm{i}} {\hslash } {S}_{\tilde{\pi }}}\,\delta {\wr }_{g,h}\;\right \rangle }_{0}$. Arguing as in [3], we see that the right-hand side corresponds to moving the two functionsgandhclose to each other (in the two possible ways) and by “bubbling” the disk around them; so we get

$${\Delta }_{1}{ \mathrm{tr} }_{2}(g,h) = \mathrm{tr} g \star h -\mathrm{tr} h \star g,$$

where ⋆ is Kontsevich’s star product [17] which corresponds to the Poisson sigma model on the upper half plane [3]. SinceΔ ₁is just the divergence operator with respect to the constant volume formΩ, for compactly supported functions we have the trace

$$\mathrm{Tr} g :={ \int }_{M} \mathrm{tr} g\;\Omega.$$

More generally, we may work out the Ward identities relative to the quadratic action (4) (there is also an equivariant version forS ¹acting by rotations onΣ). Given${a}_{0},{a}_{1},\ldots,{a}_{p}$inC ^∞(M) (or inC ^∞(M)[u] for the equivariant version), we define

$${\wr }_{{a}_{0},\ldots,{a}_{p}} := {a}_{0}(\mathsf{B}(1)){\int }_{{t}_{1}<{t}_{2}<\cdots<{t}_{p}\in \partial \Sigma \setminus \{1\}}{a}_{1}(\mathsf{B})\cdots {a}_{p}(\mathsf{B})$$

and

$${G}_{n}({\gamma }_{1},\ldots,{\gamma }_{n};{a}_{0},\ldots,{a}_{p}) :={ \left \langle \;{S}_{{\gamma }_{1}}\ldots {S}_{{\gamma }_{n}}\,{\wr }_{{a}_{0},\ldots,{a}_{p}}\;\right \rangle }_{0},$$

${\gamma }_{i} \in {\mathfrak{g}}_{S}^{\Omega }$,$i = 1,\ldots,n$. By (6) we then have

$$\begin{array}{rcl} -\mathrm{i}\hslash {\Delta }_{1}{G}_{n}({\gamma }_{1},\ldots,{\gamma }_{n};{a}_{0},\ldots,{a}_{p})& =& -\mathrm{i}\hslash {\left \langle \;\Delta ({S}_{{\gamma }_{1}}\ldots {S}_{{\gamma }_{n}}\,{\wr }_{{a}_{0},\ldots,{a}_{p}})\;\right \rangle }_{0} + \\ & & +{\left \langle \;\delta ({S}_{{\gamma }_{1}}\ldots {S}_{{\gamma }_{n}}\,{\wr }_{{a}_{0},\ldots,{a}_{p}})\;\right \rangle }_{0}.\end{array}$$

The left-hand side is just ( − iℏ) times the divergence operator applied to the multivector fieldG _n . The first term on the right-hand side may then be computed as

$$\begin{array}{rcl} & & -\mathrm{i}\hslash {\left \langle \;\Delta ({S}_{{\gamma }_{1}}\ldots {S}_{{\gamma }_{n}}\,{\wr }_{{a}_{0},\ldots,{a}_{p}})\;\right \rangle }_{0} = \\ & & \quad ={ \sum }_{i=1}^{n}{(-1)}^{{\sigma }_{i} }{G}_{n}({\gamma }_{1},\ldots,{\delta }_{\Omega }{\gamma }_{i},\ldots,{\gamma }_{n};{a}_{0},\ldots,{a}_{p}) + \\ & & \quad -\mathrm{i}\hslash { \sum }_{1\leq i<j\leq n}{(-1)}^{{\sigma }_{ij} }{G}_{n-1}([{\gamma }_{i},{\gamma }_{j}],{\gamma }_{1},\ldots,\hat{{\gamma }}_{i},\ldots,\hat{{\gamma }}_{j},\ldots,{\gamma }_{n};{a}_{0},\ldots,{a}_{p}), \\ \end{array}$$

where the caret denotes omission and

$$\begin{array}{rcl} {\sigma }_{i}& :=& {\sum }_{c=1}^{i-1}\vert {\gamma }_{ c}\vert, \\ {\sigma }_{ij}& :=& \vert {\gamma }_{i}\vert {\sum }_{c=1}^{i-1}\vert {\gamma }_{ c}\vert + \vert {\gamma }_{j}\vert {\sum }_{c=1,\,c\not =i}^{j-1}\vert {\gamma }_{ c}\vert + \vert {\gamma }_{i}\vert + 1, \\ \end{array}$$

with | γ | =kfor γ ∈Γ( ∧^k TM)[v]. The second term on the right-hand side is a boundary contribution (in the equivariant sense if$\delta =\mathrm{ {d}}_{{S}^{1}} =\mathrm{ d} - u{\iota }_{\vec{v}}$). By bubbling as in [3], some of the γ _i s collapse together with some of the consecutivea _k s and the result – which is Kontsevich’s formality map – is put back intoG. The whole formula can then be interpreted as anL _∞ -morphism from the cyclic Hochschild complex to the complex of multivector fields regarded asL _∞ -modules over${\mathfrak{g}}_{S}^{\Omega }$, as we are going to explain in the rest of the paper.

The only final remark is that iℏoccurs in this formula only as a book-keeping device. We defineF _n by formally setting iℏ= 1 inG _n .

3 Hochschild chains and cochains of algebras of smooth functions

Kontsevich’s theorem states that there is anL _∞ -quasi-isomorphism from the graded Lie algebra${\mathfrak{g}}_{S} = \Gamma ({\wedge }^{\bullet +1}\mathit{TM})$of multivector fields on a smooth manifoldM, with the Schouten–Nijenhuis bracket and trivial differential, to the differential graded Lie algebra${\mathfrak{g}}_{G}$of multidifferential operators onMwith Gerstenhaber bracket and Hochschild differential. Through Kontsevich’s morphism the Hochschild and cyclic chains become a module over${\mathfrak{g}}_{S}$. In this section we review these notions as well as results and conjectures about them.

3.1 Multivector fields and multidifferential operators

Let${\mathfrak{g}}_{S}$be the graded vector space${\mathfrak{g}}_{S} = {\oplus }_{j\geq -1}{\mathfrak{g}}_{S}^{j}$of multivector fields:${\mathfrak{g}}_{S}^{-1} = {C}^{\infty }(M),{\mathfrak{g}}_{S}^{0} = \Gamma (\mathit{TM}),{\mathfrak{g}}_{S}^{1} = \Gamma ({\wedge }^{2}\mathit{TM})$, and so on. The Schouten–Nijenhuis bracket of multivector fields is defined to be the usual Lie bracket on vector fields and is extended to arbitrary multivector field by the Leibniz rule: [α ∧ β, γ] = α ∧ [β, γ] + ( − 1)^{| γ | ⋅( | β | + 1)}[α, γ] ∧ β,$\alpha,\beta,\gamma \in {\mathfrak{g}}_{S}$. The graded Lie algebra${\mathfrak{g}}_{S}$is considered here as a differential graded Lie algebra with trivial differential.

The differential graded Lie algebra${\mathfrak{g}}_{G}$of multidifferential operators is, as a complex, the subcomplex of the shifted Hochschild complex Hom(A ^{⊗ ( ∙ + 1)},A) of the algebraA=C ^∞(M) of smooth functions, consisting of multilinear maps that are differential operators in each argument. The Gerstenhaber bracket [14] on${\mathfrak{g}}_{G}$is the graded Lie bracket [ϕ, ψ] = ϕ ∙ _G ψ − ( − 1)^{| ϕ | ⋅ | ψ |}ψ ∙ _G ϕ with Gerstenhaber product³

$$\phi {\bullet }_{G}\psi ={ \sum }_{k=0}^{n}{(-1)}^{\vert \psi \vert (\vert \phi \vert -k)}\phi \circ (\mathrm{{id}}^{\otimes k} \otimes \psi \otimes \mathrm{{ id}}^{\otimes \vert \phi \vert -k}). $$

(9)

The Hochschild differential can be written in terms of the bracket as [μ, ⋅], where$\mu \in {\mathfrak{g}}_{G}^{1} =\mathrm{ Hom}(A \otimes A,A)$is the multiplication inA.

The Hochschild–Kostant–Rosenberg map${\mathfrak{g}}_{S}^{\bullet }\rightarrow {\mathfrak{g}}_{G}^{\bullet }$induces an isomorphism of graded Lie algebras on cohomology. It is the identity on${\mathfrak{g}}_{S}^{-1} = {C}^{\infty }(M) = {\mathfrak{g}}_{G}^{-1}$and, for any vector fields${\xi }_{1},\ldots,{\xi }_{n}$, it sends the multivector field${\xi }_{1} \wedge \cdots \wedge {\xi }_{n}$to the multidifferential operator

$${f}_{1} \otimes \cdots \otimes {f}_{n}\mapsto \frac{1} {n!}{\sum }_{\sigma \in {S}_{n}}\epsilon (\sigma ){\xi }_{\sigma (1)}({f}_{1})\cdots {\xi }_{\sigma (n)}({f}_{n}),\quad {f}_{i} \in A.$$

Although the HKR map is a chain map inducing a Lie algebra isomorphism on cohomology, it does not respect the Lie bracket at the level of complexes. The correct point of view on this problem was provided by Kontsevich in his formality conjecture, which he then proved in [17]. The differential graded Lie algebras${\mathfrak{g}}_{S}$,${\mathfrak{g}}_{G}$should be considered asL _∞ -algebras and the HKR map is the first component of anL _∞ -morphism. Let us recall the definitions.

3.2 L _∞ -algebras

For any graded vector spaceVletS ⁺ V= ⊕_{j= 1} ^∞ S ^j Vbe the free coalgebra without counit cogenerated byV. The coproduct isΔ(a ₁⋯a _n ) = ∑_{p= 1} ^{n− 1}∑_σ±a _σ(1)⋯a _σ(p)⊗a _{σ(p+ 1)}⋯a _σ(n), with summation over shuffle permutations with Koszul signs. A coderivation of a coalgebra is an endomorphismDobeyingΔ∘D= (D⊗ id + id ⊗D) ∘Δ. Coderivations with the commutator bracket form a Lie algebra. What is special about the free coalgebraS ⁺ Vis that for any linear mapD:S ⁺ V→Vthere is a unique coderivation$\tilde{D}$such that$D = \pi \circ \tilde{ D}$, where π is the projection ontoV=S ¹ V. By definition anL _∞ -algebra is a graded vector space$\mathfrak{g}$together with a coderivationDof degree 1 of${S}^{+}(\mathfrak{g}[1])$obeying [D,D] = 0. A coderivation is thus given by a sequence of maps (the Taylor components)${D}_{n}: {S}^{n}\mathfrak{g}[1] \rightarrow \mathfrak{g}[2]$(or${\wedge }^{n}\mathfrak{g} \rightarrow \mathfrak{g}[2 - n]$),$n = 1,2,\ldots$, obeying quadratic relations. In particularD ₁is a differential andD ₂is a chain map obeying the Jacobi identity up to a homotopyD ₃. It follows thatD ₂induces a Lie bracket on theD ₁-cohomology. Differential graded Lie algebras areL _∞ -algebras withD ₃=D ₄= ⋯ = 0. AnL _∞ -morphism$(\mathfrak{g},D) \rightsquigarrow (\mathfrak{g}{\prime},D{\prime})$is a homomorphism$U : {S}^{+}\mathfrak{g}[1] \rightarrow {S}^{+}\mathfrak{g}{\prime}[1]$of graded coalgebras such thatU∘D=D′∘U. Homomorphisms of free coalgebras are uniquely defined by their composition with the projection$\pi {\prime}: {S}^{+}\mathfrak{g}{\prime}[1] \rightarrow \mathfrak{g}{\prime}[1]$; thusUis uniquely determined by its Taylor components${U}_{n}: {S}^{n}\mathfrak{g}[1] \rightarrow \mathfrak{g}{\prime}[1]$(or${\wedge }^{n}\mathfrak{g} \rightarrow \mathfrak{g}{\prime}[1 - n]$):U _n is the restriction to${S}^{n}\mathfrak{g}[1]$of π′∘U. Conversely, any such sequenceU _n comes from a coalgebra homomorphism. The first relation betweenD,D′andUis thatU ₁is a chain map.

Theorem 1.

(Kontsevich[17])There is an L _∞ -morphism ${\mathfrak{g}}_{S}(M) \rightsquigarrow {\mathfrak{g}}_{G}(M)$ whose first Taylor component U ₁ is the Hochschild–Kostant–Rosenberg map.

IfMis an open subset of${\mathbb{R}}^{d}$the formula for the Taylor componentsU _n is explicitly given in [17] as a sum over Feynman graphs.

3.3 Multivector fields and differential forms

The algebraΩ ^∙(M) of differential forms on a manifoldMis a module over the differential graded Lie algebra${\mathfrak{g}}_{S}(M)$of multivector fields: a multivector field γ ∈Γ( ∧^{p+ 1} TM) acts on forms as${\mathcal{L}}_{\gamma }\omega =\mathrm{ d}{\iota }_{\gamma } + {(-1)}^{p}{\iota }_{\gamma }d$generalizing Cartan’s formula for Lie derivatives of vector fields. Here d is the de Rham differential and the interior multiplication ι_γis the usual multiplication if γ is a function and is the composition of interior multiplications of vector fields ξ _j if γ = ξ₁∧ ⋯ ∧ ξ _k . Moreover the action of${\mathfrak{g}}_{S}(M)$onΩ ^∙(M) commutes with the de Rham differential and induces the trivial action on cohomology.

3.4 Hochschild cochains and cyclic chains

The algebrasΩ ^∙(M) andH ^∙(M) are cohomologies of the complexes of the Hochschild and of the periodic cyclic chains ofC ^∞(M). The normalized Hochschild chain complex of a unital algebraAis${C}_{\bullet }(A) = A \otimes \bar{ {A}}^{\otimes \bullet }$, where$\bar{A} = A/\mathbb{R}1$. If we denote by$({a}_{0},{a}_{1},\ldots,{a}_{p})$the class ofa ₀⊗ ⋯ ⊗a _p inC _p (A), the Hochschild differential is

$$\begin{array}{rcl} b({a}_{0},\ldots,{a}_{p})& =& {\sum }_{i=0}^{p-1}{(-1)}^{i}({a}_{ 0},\ldots,{a}_{i}{a}_{i+1},\ldots,{a}_{p}) \\ & & +{(-1)}^{p}({a}_{ p}{a}_{0},{a}_{1},\ldots,{a}_{p-1}).\end{array}$$

We setC _p (A) = 0 forp< 0. There is an HKR mapC _∙(A) →Ω ^∙(M) given by

$$({a}_{0},\ldots,{a}_{p})\mapsto \frac{1} {p!}{a}_{0}\mathrm{d}{a}_{1}\cdots \mathrm{d}{a}_{p}. $$

(10)

It is a chain map if we consider differential forms as a complex withtrivialdifferential. The HKR map induces an isomorphism on homology, provided we take a suitable completion of the tensor productC ^∞(M)^{⊗ (p+ 1)}, for example the jets at the diagonal of smooth maps${M}^{p+1} \rightarrow \mathbb{R}$. On the Hochschild chain complex there is a second differentialBof degree 1 and anticommuting withb, see [8]:

$$B({a}_{0},\ldots,{a}_{p}) ={ \sum }_{i=0}^{p}{(-1)}^{ip}(1,{a}_{ i},\ldots,{a}_{p},{a}_{0},\ldots {a}_{i-1}).$$

The negative cyclic complex, in the formulation of [15], isCC _{− ∙} ⁻(A) =C _{− ∙}(A)[u] with differentialb+uB, whereuis of degree 2. The extension of the HKR map by$\mathbb{R}[u]$-linearity defines a quasi-isomorphism

$$(C{C}_{-\bullet }^{-}(A),b + uB) \rightarrow ({\Omega }^{-\bullet }(M)[u],u\,d).$$

Now bothC(A) andCC ⁻(A) are differential graded modules over the Lie algebra${\mathfrak{g}}_{G}$of multidifferential operators. The action is the restriction of the action of cochains on chainsC ^k(A) ⊗C _p (A) →C _{p−k+ 1}(A), ϕ ⊗a↦ϕ ⋅a, defined for any associative algebra with unit as

$$\begin{array}{rcl}{ (-1)}^{(k-1)(p+1)}\phi \cdot ({a}_{ 0},\ldots,{a}_{p})&& \\ & =& {\sum }_{i=0}^{p-k+1}{(-1)}^{i(k-1)}({a}_{ 0},\ldots,{a}_{i-1},\phi ({a}_{i},\ldots,{a}_{i+k-1}),{a}_{i+k},\ldots,{a}_{p}) \\ & & +{\sum }_{i=p-k+2}^{p}{(-1)}^{ip}(\phi ({a}_{ i},\ldots,{a}_{p},{a}_{0},\ldots,{a}_{i+k-p-2}),{a}_{i+k-p-1},\ldots,{a}_{i-1}).\end{array}$$

This action extends by$\mathbb{R}[u]$-linearity to an action on the negative cyclic complex.

3.5 L _∞ -modules

Let$(\mathfrak{g},D)$be anL _∞ -algebra. The free${S}^{+}\mathfrak{g}[1]$-comodule generated by a vector spaceVis$\hat{V } = S\mathfrak{g}[1] \otimes V$with coaction${\Delta }^{V }: \hat{V } \rightarrow {S}^{+}\mathfrak{g}[1] \otimes \hat{ V }$defined as

$${\Delta }^{V }({\gamma }_{ 1}\cdots {\gamma }_{n} \otimes v) ={ \sum }_{p=1}^{n}{ \sum }_{\sigma \in {S}_{p,n-p}} \pm {\gamma }_{\sigma (1)}\cdots {\gamma }_{\sigma (p)} \otimes ({\gamma }_{\sigma (p+1)}\cdots {\gamma }_{\sigma (n)} \otimes v).$$

A coderivation of theL _∞ -moduleVis then an endomorphismD ^Vof$\hat{V }$obeyingΔ ^V∘D ^V= (D⊗ id + id ⊗D ^V)Δ ^V. AnL _∞ -module is a coderivationD ^Vof degree 1 of$\hat{V }$obeyingD ^V∘D ^V= 0. A coderivation is uniquely determined by its composition with the projection$\hat{V } \rightarrow V$onto the first direct summand and is thus given by its Taylor components${D}_{n}^{V }: {S}^{n}\mathfrak{g}[1] \otimes V \rightarrow V [1]$. The lowest componentD ₀ ^Vis then a differential onVandD ₁ ^Va chain map inducing an honest action of the Lie algebra$H(\mathfrak{g},{D}_{1})$on the cohomologyH(V,D ₀ ^V). A morphism ofL _∞ -modulesV ₁→V ₂over$\mathfrak{g}$is a degree 0 morphism of${S}^{+}\mathfrak{g}[1]$-comodules$F : \hat{{V }}_{1} \rightarrow \hat{ {V }}_{2}$intertwining the coderivations. The composition with the projection$\hat{{V }}_{2} \rightarrow {V }_{2}$gives rise to Taylor components

$${F}_{n}: {S}^{n}\mathfrak{g}[1] \otimes {V }_{ 1} \rightarrow {V }_{2},\qquad n = 0,1,2,\ldots $$

that determineFcompletely. The lowest componentF ₀is then a chain map inducing a morphism of$H(\mathfrak{g},{D}_{1})$-modules on cohomology.

3.6 Tsygan and Kontsevich conjectures [30], [26]

Conjecture 1.

There exists a quasi-isomorphism ofL _∞ -modules

$$F : {C}_{-\bullet }({C}^{\infty }(M)) \rightsquigarrow ({\Omega }^{-\bullet }(M),0)$$

such thatF ₀is the HKR map.

Conjecture 2.

There exists a natural$\mathbb{C}[[u]]$-linear quasi-isomorphism ofL _∞ -modules

$$F : C{C}_{-\bullet }^{-}({C}^{\infty }(M)) \rightsquigarrow ({\Omega }^{-\bullet }(M)[[u]],ud)$$

such thatF ₀is the Connes quasi-isomorphism [8], given by theu-linear extension of the HKR map (10).

Conjecture1is now a theorem. Different proofs for$M = {\mathbb{R}}^{d}$were given in [29] and [27]. Shoikhet’s proof [27] gives an explicit formula for the Taylor components ofFin terms of integrals over configuration spaces on the disk and extends to general manifolds, as shown in [11].

Let us turn to Kontsevich’s formality conjecture for cyclic cochains, as quoted in [26]. Recall that a volume formΩ∈Ω ^d(M) on ad-dimensional manifold defines an isomorphismΓ( ∧^k TM) →Ω ^d−k(M), γ↦ι_γ Ω. The de Rham differential d onΩ ^∙(M) translates to a differential div _Ω , the divergence operator of degree − 1. The divergence operator is a derivation of the bracket on${\mathfrak{g}}_{S} = \Gamma ({\wedge }^{\bullet +1}\mathit{TM})$of degree − 1. Let us introduce the differential graded Lie algebra${\mathfrak{g}}_{S}^{\Omega } = ({\mathfrak{g}}_{S}[v],{\delta }_{\Omega }),$wherevis a formal variable of degree 2. The bracket is the Schouten–Nijenhuis bracket and the differential is δ _Ω =vdiv _Ω . The cyclic analogue of${\mathfrak{g}}_{G}$is the differential graded Lie algebra

$${\mathfrak{g}}_{G}^{\mathrm{cycl}} = \left \{\varphi \in {\mathfrak{g}}_{ G},\,{\int }_{M}{a}_{0}\varphi ({a}_{1},\ldots,{a}_{p})\Omega = {(-1)}^{p}{ \int }_{M}{a}_{p}\varphi ({a}_{0},\ldots,{a}_{p-1})\Omega \right \}.$$

Conjecture 3.

For each volume formΩ∈Ω ^d(M) there exists anL _∞ -quasi-isomorphism ofL _∞ -algebras$F : {\mathfrak{g}}_{S}^{\Omega } \rightsquigarrow {\mathfrak{g}}_{G}^{\mathrm{cycl}}.$

Shoikhet [26] constructed a quasi-isomorphism of complexes${C}_{1} : {\mathfrak{g}}_{S}^{\Omega } \rightarrow {\mathfrak{g}}_{G}^{\mathrm{cycl}}$and conjectural formulae for anL _∞ -morphism whose first component isC ₁in terms of integrals over configuration spaces. One consequence of the conjecture is the construction of cyclically-invariant star-products from divergenceless Poisson bivector fields. Such star-products were then constructed independently of the conjecture, see [12].

4 AnL _∞ -morphism for cyclic chains

4.1 The main results

LetΩbe volume form on a manifoldMand${\mathfrak{g}}_{S}^{\Omega }$be the differential graded Lie algebra${\mathfrak{g}}_{S}[v]$with Schouten bracket and differential δ _Ω =vdiv _Ω , see Section3.6. The KontsevichL _∞ -morphism composed with the canonical projection${\mathfrak{g}}_{S}^{\Omega } \rightarrow {\mathfrak{g}}_{S} = {\mathfrak{g}}_{S}^{\Omega }/v{\mathfrak{g}}_{S}^{\Omega }$is anL _∞ -morphism${\mathfrak{g}}_{S}^{\Omega } \rightsquigarrow {\mathfrak{g}}_{G}$. Through this morphism the differential graded${\mathfrak{g}}_{G}$-moduleCC _∙ ⁻(A) of negative cyclic chains ofA=C ^∞(M) becomes anL _∞ -module over${\mathfrak{g}}_{S}^{\Omega }$.

Theorem 2.

Let M be an open subset of ${\mathbb{R}}^{d}$ with coordinates ${x}_{1},\ldots,{x}_{n}$ and volume form Ω =dx ₁ ⋯dx _d . Let A = C ^∞ (M). Let Γ(∧ ^−∙ TM) be the differential graded module over${\mathfrak{g}}_{S}^{\Omega }$with differential div_Ω and trivial ${\mathfrak{g}}_{S}^{\Omega }$ -action. Then there exists an $\mathbb{R}[u]$ -linear morphism of L _∞ -modules over ${\mathfrak{g}}_{S}^{\Omega }$

$$F : C{C}_{-\,\bullet }^{-}(A) \rightsquigarrow \Gamma ({\wedge }^{-\bullet }\mathit{TM})[u],$$

such that

• (i) The component F ₀ of F vanishes on CC _p (A), p > 0 and for f ∈ A ⊂ CC ₀ ⁻ (A), F ₀ (f) = f.

• (ii) For γ ∈ Γ(∧ ^k TM),$\mathcal{l} = 0,1,2,\ldots$,$a = ({a}_{0},\ldots,{a}_{p}) \in C{C}_{p}^{-}(A)$,

  $${F}_{1}(\gamma {v}^{\mathcal{l}};a) = \left \{\begin{array}{ll} {(-1)}^{p}{u}^{s}\gamma \lrcorner H(a),&\mbox{ if $k \geq p$ and $s = k + \mathcal{l} - p - 1 \geq 0$,} \\ 0, &\mbox{ otherwise.}\end{array} \right.$$

  Here ⌟: Γ(∧ ^k TM) ⊗ Ω^p(M) → Γ(∧^k−p TM) is the contraction map and H is the HKR map(10).

• (iii) The maps F _n are equivariant under linear coordinate transformations and F _n (γ ₁ ⋯γ _n ;a) = γ ₁ ∧ F _n−1 (γ ₂ ⋯γ _n ;a) whenever ${\gamma }_{1} = \sum ({c}_{k}^{i}{x}_{k} + {d}^{i}){\partial }_{i} \in {\mathfrak{g}}_{S} \subset {\mathfrak{g}}_{S}^{\Omega }$ is an affine vector field and ${\gamma }_{2},\ldots,{\gamma }_{n} \in {\mathfrak{g}}_{S}^{\Omega }$.

The proof of this Theorem is deferred to Section6.3.

In explicit terms,Fis given by a sequence of$\mathbb{R}[u]$-linear maps${F}_{n}: {S}^{n}{\mathfrak{g}}_{S}^{\Omega }[1] \otimes C{C}^{-}(A) \rightarrow \Gamma ({\wedge }^{n}\mathit{TM})$, γ ⊗a↦F _n (γ;a),n≥ 0, obeying the following relations. For any$\gamma = {\gamma }_{1}\cdots {\gamma }_{n} \in {S}^{n}{\mathfrak{g}}_{S}^{\Omega }[1]$,a∈CC _p ⁻(A).

$$\begin{array}{rcl} & & {F}_{n}({\delta }_{\Omega }\gamma ;a) + {(-1)}^{\vert \gamma \vert +p}{F}_{ n}(\gamma ;(b + uB)a)\end{array}$$

(11)

$$\begin{array}{rcl} & +& {\sum }_{k=0}^{n-1}{ \sum }_{\sigma \in {S}_{k,n-k}}{(-1)}^{\vert \gamma \vert -1}\epsilon (\sigma ;\gamma ){F}_{ k}({\gamma }_{\sigma (1)}\cdots {\gamma }_{\sigma (k)};{U}_{n-k}(\bar{{\gamma }}_{\sigma (k+1)}\cdots \bar{{\gamma }}_{\sigma (n)}) \cdot a) \\ & +& {\sum }_{i<j}{\epsilon }_{ij}{F}_{n-1}({(-1)}^{\vert {\gamma }_{i}\vert -1}[{\gamma }_{ i},{\gamma }_{j}] \cdot {\gamma }_{1}\cdots \hat{{\gamma }}_{i}\cdots \hat{{\gamma }}_{j}\cdots {\gamma }_{n};a) ={ \mathrm{div} }_{\Omega }\,{F}_{n}(\gamma ;a).\end{array}$$

Here $\bar{{\gamma }}_{i}$denotes the projection of γ _i to${\mathfrak{g}}_{S}[1] = {\mathfrak{g}}_{S}^{\Omega }[1]/v{\mathfrak{g}}_{S}^{\Omega }[1]$;S _p,q⊂S _p+qis the set of (p,q)-shuffles and the signs ε(σ; γ), ε _ij are the Koszul signs coming from the permutation of the${\gamma }_{i} \in {\mathfrak{g}}_{S}[1]$; | γ | = ∑ _i | γ _i | ; the differential δ _Ω is extended to a degree 1 derivation of the algebra$S{\mathfrak{g}}_{S}^{\Omega }[1]$; the maps${U}_{k}: {S}^{k}{\mathfrak{g}}_{S}[1] \rightarrow {\mathfrak{g}}_{G}[1]$are the Taylor components of the KontsevichL _∞ -morphism of Theorem1.

We give the explicit expressions of the mapsF _n in Section5. Before that we explore some consequences.

4.2 Maurer–Cartan elements

An element of degree 1 in${\mathfrak{g}}_{S}^{\Omega }$has the form$\tilde{\pi } = \pi + vh$where π is a bivector field andhis a function. The Maurer–Cartan equation${\delta }_{\Omega }\tilde{\pi } -\frac{1} {2}[\tilde{\pi },\tilde{\pi }] = 0$translates to

$$[\pi,\pi ] = 0,\qquad {\mathrm{div} }_{\Omega }\,\pi - [h,\pi ] = 0.$$

Thus π is a Poisson bivector field whose divergence is a Hamiltonian vector field with Hamiltonian h. Such Poisson structures are called unimodular [19]. As explained in [17], Poisson bivector fields in$\epsilon {\mathfrak{g}}_{S}[[\epsilon ]]$are mapped to solution of the Maurer–Cartan equations in$\epsilon {\mathfrak{g}}_{G}[[\epsilon ]]$, which are star-products, i.e., formal associative deformations of the pointwise product inC ^∞(M):

$$f \star g = fg +{ \sum }_{n=1}^{\infty }\frac{{\epsilon }^{n}} {n!}{U}_{n}(\pi,\ldots,\pi )(f \otimes g).$$

Here the function part of$\tilde{\pi }$does not contribute as it is projected away in theL _∞ -morphism${\mathfrak{g}}_{S}^{\Omega } \rightsquigarrow {\mathfrak{g}}_{G}$.

If$\tilde{\pi } = \pi + vh$is a Maurer–Cartan element in${\mathfrak{g}}_{S}^{\Omega }$then$\tilde{{\pi }}_{\epsilon } = \epsilon \pi + vh$is a Maurer–Cartan element in${\mathfrak{g}}_{S}^{\Omega }[[\epsilon ]]$. The twist ofFby$\tilde{\pi }$then gives a chain map from the negative cyclic complex of the algebraA _ε= (C ^∞(M)[[ε]], ⋆ ) toΓ( ∧TM)[u][[ε]]. In particular, we get a trace

$$f\mapsto \tau (f) ={ \sum }_{n=0}^{\infty } \frac{1} {n!}{\int }_{M}{F}_{n}(\tilde{{\pi }}_{\epsilon },\ldots,\tilde{{\pi }}_{\epsilon };f)\Omega, $$

(12)

on the subalgebra ofA _εconsisting of functions with compact support. Here there is a question of convergence since there are infinitely many terms contributing to each fixed power of ε. The point is that these infinitely many terms combine to exponential functions. More precisely we have the following result.

Proposition 1.

The trace(12) can be written as

$$\tau (f) ={ \sum }_{n=0}^{\infty }\frac{{\epsilon }^{n}} {n!}{\int }_{M}{H}_{n}(\pi,h,f)\mathrm{{e}}^{h}\Omega ={ \int }_{M}f\mathrm{{e}}^{h}\Omega + O(\epsilon )$$

where H _n is a differential polynomial in π,h,f.

The proof is based on the expression ofF _n in terms of graphs. We postpone it to Section5.5, after we introduce this formalism.

5 Feynman graph expansion of the L _∞ -morphism

In this section we construct the morphism ofL _∞ -modules of Theorem2. The Taylor components have the form

$${F}_{n}(\gamma ;a) ={ \sum }_{\Gamma \in {\mathcal{G}}_{\mathbf{k},m}}{w}_{\Gamma }{F}_{\Gamma }(\gamma ;a).$$

Here γ = γ₁⋯γ _n , with${\gamma }_{i} \in \Gamma ({\wedge }^{{k}_{i}}\mathit{TM})[v]$,$\mathbf{k} = ({k}_{1},\ldots,{k}_{n})$and$a = ({a}_{0},\ldots,{a}_{m}) \in {C}_{m}(A)$. The sum is over a finite set${\mathcal{G}}_{\mathbf{k},m}$of directed graphs with some additional structure. To each graph a weight${w}_{\Gamma } \in \mathbb{R}[u]$, defined as an integral over a configuration space of points in the unit disk, is assigned.

We turn to the descriptions of the graphs and weights.

5.1 Graphs

Let$m,n \in {\mathbb{Z}}_{\geq 0}$,$\mathbf{k} = ({k}_{1},\ldots,{k}_{n}) \in {\mathbb{Z}}_{\geq 0}^{n}$. We consider directed graphsΓwithn+mvertices with additional data obeying a set of rules. The data are a partition of the vertex set into three totally ordered subsetsV(Γ) =V ₁(Γ) ⊔V ₂(Γ) ⊔V _w (Γ), a total ordering of the edges originating at each vertex and the assignment of a nonnegative integer, thedegree, to each vertex inV ₁(Γ). The rules are:

• 1. There arenvertices inV ₁(Γ). There are exactlyk _i edges originating at theith vertex ofV ₁(Γ).

• 2. There aremvertices inV ₂(Γ). There are no edges originating at these vertices.

• 3. There is exactly one edge pointing at each vertex inV _w (Γ) and no edge originating from it.

• 4. There are no edges starting and ending at the same vertex.

• 5. For each pair of verticesi,jthere is at most one edge fromitoj.

The last rule is superfluous, but since all graphs with multiple edges will have vanishing weight we may just as well exclude them from the start. This has the notational advantage that we may think of the edge setE(Γ) as a subset ofV(Γ) ×V(Γ).

Two graphs are called equivalent if there is a graph isomorphism between them that respects the partition and the orderings. The set of equivalence classes is denoted${\mathcal{G}}_{\mathbf{k},m}$.

The vertices inV ₁(Γ) are called vertices of the first type, those inV ₂(Γ) of the second type. The vertices inV _b (Γ) =V ₁(Γ) ∪V ₂(Γ) are called black, those inV _w (Γ) are called white. We denote byE _b (Γ) the subset ofE(Γ) consisting of edges whose endpoints are black.

To each$\Gamma \in {\mathcal{G}}_{\mathbf{k},m}$there corresponds a multivector fieldF _Γ (γ;a) whose coefficients are differential polynomials in the components of γ _i ,a _i . The rules are the same as in [17] except for the additional white vertices, representing uncontracted indices and the degreesd _i , that select the powerd _i ofvin γ _i . Let us consider for example the graph of Fig.1and suppose that the degrees of the two vertices of the first type arekandℓ. The algebra of multivector fields on$M \subset {\mathbb{R}}^{d}$is generated byC ^∞(M) and anticommuting generators θ_ν=∂∕∂x _ν. Thus γ ∈Γ( ∧^k TM) has the form

$$\gamma = \frac{1} {k!}{\sum }_{{\nu }_{1},\ldots,{\nu }_{k}}{\gamma }^{{\nu }_{1}\ldots {\nu }_{k} }{\theta }_{{\nu }_{1}}\cdots {\theta }_{{\nu }_{k}}.$$

The components${\gamma }^{{\nu }_{1}\ldots {\nu }_{k}} \in {C}^{\infty }(M)$are skew-symmetric under permutation of the indices ν _i . The graph of Fig.1, with the convention that the edges originating at each vertex are ordered counterclockwise, gives then

$${F}_{\Gamma }({\gamma }_{1}{v}^{k},{\gamma }_{ 2}{v}^{\mathcal{l}};{a}_{ 0},{a}_{1},{a}_{2}) = \sum {\gamma }_{1}^{ij}{\partial }_{ j}{\gamma }_{2}^{pqr}{\partial }_{ i}{a}_{0}{\partial }_{p}{a}_{1}{\partial }_{q}{a}_{2}{\theta }_{r},$$

and is zero on other monomials inv.

Fig. 1

A graph in${\mathcal{G}}_{(2,3),3}$with two vertices inV ₁of valencies (2, 3), three inV ₂and one white vertex. The degrees of the vertices of the first type arekandℓ

5.2 Equivariant differential forms on configuration spaces

LetΣbe a manifold with an action of the circle${S}^{1} = \mathbb{R}/\mathbb{Z}$. The infinitesimal action$\mathrm{Lie}({S}^{1}) = \mathbb{R} \frac{\mathrm{d}} {\mathrm{d}t} \rightarrow \Gamma (T\Sigma )$is generated by a vector field$\vec{v} \in \Gamma (T\Sigma )$, the image of$\frac{\mathrm{d}} {\mathrm{d}t}$. The Cartan complex ofS ¹-equivariant forms, computing the equivariant cohomology with real coefficients, is the differential graded algebra

$${\Omega }_{{S}^{1}}^{\bullet }(\Sigma ) = {\Omega }^{\bullet }{(\Sigma )}^{{S}^{1} }[u],$$

of polynomials in an undetermineduof degree 2 with coefficients in theS ¹-invariant smooth differential forms. The differential is$\mathrm{{d}}_{{S}^{1}} =\mathrm{ d} - u{\iota }_{\vec{v}}$, where d is the de Rham differential and${\iota }_{\vec{v}}$denotes interior multiplication by$\vec{v}$, extended by$\mathbb{R}[u]$-linearity. IfΣhas anS ¹-invariant boundary∂Σandj:∂Σ→Σdenotes the inclusion map, then the relative equivariant complex is

$${\Omega }_{{S}^{1}}^{\bullet }(\Sigma,\partial \Sigma ) =\mathrm{ Ker}({j}^{{_\ast}}: {\Omega }_{{ S}^{1}}^{\bullet }(\Sigma ) \rightarrow {\Omega }_{{ S}^{1}}^{\bullet }(\partial \Sigma )).$$

In the case of the unit disk we have:

Lemma 3.

Let $\bar{D} =\{ z \in \mathbb{C},\,\vert z\vert \leq 1\}$ be the closed unit disk.

(i):

The equivariant cohomology ${H}_{{S}^{1}}^{\bullet }(\bar{D})$ of $\bar{D}$ is the free $\mathbb{R}[u]$ -module generated by the class of $1 \in {\Omega }^{0}(\bar{D})$.

(ii):

The relative equivariant cohomology ${H}_{{S}^{1}}^{\bullet }(\bar{D},\partial \bar{D})$ of $(\bar{D},\partial \bar{D})$ is the free $\mathbb{R}[u]$ -module generated by the class of

$$\phi (z,u) = \frac{\mathrm{i}} {2\pi }\mathrm{d}z \wedge \mathrm{ d}\bar{z} + u(1 -\vert z{\vert }^{2}). $$

(13)

5.3 The propagator

The integrals over configuration spaces defining theL _∞ -morphism are constructed out of a propagator, a differential 1-form ω on$\bar{D} \times \bar{ D} \ \Delta$, with a simple pole on the diagonal$\Delta =\{ (z,z),\,z \in \bar{ D}\}$and defining the integral kernel of a homotopy contracting equivariant differential forms to a space of representatives of the cohomology. The explicit formula of the propagator associated to the choice of cocycles in Lemma3is given by

$$\omega (z,w) = \frac{1} {4\pi \mathrm{i}}\left (\mathrm{d}\ln \frac{(z - w)(1 - z\bar{w})} {(\bar{z} -\bar{ w})(1 -\bar{ z}w)} + z\,\mathrm{d}\bar{z} -\bar{ z}\,\mathrm{d}z\right ). $$

(14)

Lemma 4.

Let ${p}_{i}: \bar{D} \times \bar{ D} \rightarrow \bar{ D}$ be the projection to the i-th factor, i = 1,2. The differential form $\omega \in {\Omega }_{{S}^{1}}^{1}(\bar{D} \times \bar{ D} \ \Delta )$ has the following properties:

(i):

Let $j : \partial \bar{D} \times \bar{ D} \rightarrow \bar{ D} \times D$ be the inclusion map. Then j ^∗ ω = 0.

(ii):

$\mathrm{{d}}_{{S}^{1}}\omega = -{p}_{1}^{{_\ast}}\phi$.

(iii):

As z → w, ω(z,w) = (2π) ⁻¹d arg(z − w)+ smooth.

(iv):

As z and w approach a boundary point, ω(z,w) converges to the Kontsevich propagator ${\omega }_{K}(x,y) = {(2\pi )}^{-1}(\mathrm{d}\arg (x - y) -\mathrm{ d}\arg (\bar{x} - y))$ on the upper half-plane H ₊ from[17]. More precisely, for small t > 0 let φ_t (x) = z ₀e ^{i tx} be the inclusion of a neighbourhood of 0 ∈ H ₊ into a neighbourhood of z ₀ ∈ ∂D in D. Thenlim _t→0 (φ _t × φ _t)^∗ ω = ω _K.

The proof is a simple computation left to the reader.

5.4 Weights

The weights are integrals of differential forms over configuration spacesC _n,m ⁰(D) ofnpoints in the unit disk$D =\{ z \in \mathbb{C}\,,\,\vert z\vert< 1\}$andm+ 1 cyclically ordered points on its boundary$\partial \bar{D}$, the first of which is at 1:

$$\begin{array}{rcl}{ C}_{n,m}^{0}(D)& =& \{(z,t) \in {D}^{n} \times {(\partial \bar{D})}^{m}\,,\,{z}_{ i}\neq {z}_{j},(i\neq j), \\ & & 0<\arg ({t}_{1})< \cdots<\arg ({t}_{m})< 2\pi \}.\end{array}$$

The differential forms are obtained from the propagator ω, see (14), and the form ϕ, see (13). Let$\Gamma \in {\mathcal{G}}_{({k}_{1},\ldots,{k}_{n}),m}$. The weightw _Γ ofΓis

$${w}_{\Gamma } = \frac{1} {{\prod }_{i=1}^{n}{k}_{i}!}{\int }_{{C}_{n,m}^{0}(D)}{\omega }_{\Gamma }$$

where ω ∈Ω ^∙(C _n,m ⁰(D))[u] is the differential form

$${\omega }_{\Gamma } ={ \prod }_{i\in {V }_{1}(\Gamma )}{ \prod }_{(i,j)\in {E}_{b}(\Gamma )}\omega ({z}_{i},{z}_{j}){\prod }_{i\in {V }_{1}(\Gamma )}\phi {({z}_{i},u)}^{{r}_{i} }.$$

Herez _i is the coordinate ofz∈C _n,m ⁰(D) assigned to the vertexiofΓ: to the vertices of the first type we assign the points in the unit disk and to the vertices of the second type the points on the boundary. The assignment is uniquely specified by the ordering of the vertices inΓ. The numberr _i is the degree of the vertexiplus the number of white vertices connected to it. The product over (i,j) is over the edges connecting black vertices to black vertices. For example, a point ofC _{2, 3} ⁰(D) is given by coordinates (z ₁,z ₂, 1,t ₁,t ₂) withz _i ∈Dandt _i ∈S ¹. The differential form associated to the graph of Fig.1, with degree assignmentsk,ℓ, is

$$\pm {\omega }_{\Gamma } = \omega ({z}_{1},1)\omega ({z}_{1},{z}_{2})\omega ({z}_{2},{t}_{1})\omega ({z}_{2},{t}_{2})\phi {({z}_{1},u)}^{k}\phi {({z}_{ 2},u)}^{\mathcal{l}+1}.$$

The signs are tricky. A consistent set of signs may be obtained by the following procedure. View a multivector field$\gamma \in {\mathfrak{g}}_{S}[v]$as a polynomial γ(x, θ,v) whose coefficients are functions on${T}^{{_\ast}}[1]M = M \times {\mathbb{R}}^{d}[1]$. Build a function in${C}^{\infty }({({T}^{{_\ast}}[1]M)}^{n+m})[{v}_{1},\ldots,{v}_{n}]$:

$$\begin{array}{rcl} g({x}^{(1)},{\theta }^{(1)},{v}_{ 1},\ldots,{x}^{(\bar{m})})&& \\ & & = {\gamma }_{1}({x}^{(1)},{\theta }^{(1)},{v}_{ 1})\cdots {\gamma }_{n}({x}^{(n)},{\theta }^{(n)},{v}_{ n}){a}_{0}({x}^{(\bar{0})})\cdots {a}_{ m}({x}^{(\bar{m})}).\end{array}$$

Then

$$\begin{array}{rcl}{ F}_{n}(\gamma ;a)& =& {(-1)}^{\vert \gamma \vert m}{ \int }_{{C}_{n,m}^{0}(D)}{i}_{\Delta }^{{_\ast}}\circ \exp ({\Phi }_{ n})(g){\vert }_{{v}_{1}=\cdots ={v}_{n}=0}, \\ {\Phi }_{n}& =& {\sum }_{i\neq k}\omega ({z}_{i},{z}_{k}){\sum }_{\nu =1}^{d} \frac{{\partial }^{2}} {\partial {\theta }_{\nu }^{(i)}\partial {x}_{\nu }^{(k)}} +{ \sum }_{i}\phi ({z}_{i},u)\left ({\sum }_{\nu =1}^{d}{\theta }_{ \nu } \frac{\partial } {\partial {\theta }_{\nu }^{(i)}} + \frac{\partial } {\partial {v}_{i}}\right ).\end{array}$$

The sums overiare from 1 tonand the sum overkis over the set$\{1,\ldots,n,\bar{1},\ldots,\bar{m}\}$, with the understanding that${z}_{\bar{j}} = {t}_{j}$. The mapi _Δ ^∗is the restriction to the diagonal: its effect is to set allx ⁽ⁱ⁾to be equal toxand all θ⁽ⁱ⁾to be equal to θ. The integrand is then an element of the tensor product of graded commutative algebrasΩ(C _n,m ⁰(D)) ⊗C ^∞(T ^∗[1]M)[u]. The integral is defined as ∫(α ⊗ γ) = ( ∫α)γ and the expansion of the exponential functions gives rise to a finite sum over graphs.

5.5 Proof of Proposition 1 on page39

A vertex of a directed graph is called disconnected if there is no edge originating or ending at it.

Lemma 5.

Let $\tilde{{F}}_{n}$ be defined as F _n except that the sum over graphs is restricted to the graphs without disconnected vertices of the first type. Then

$${F}_{k+n}({(hv)}^{k} \cdot {\pi }^{n};f) ={ \sum }_{s=0}^{k}<mfenced-6 separators="" open="(" close=")"> <mfrac-1 linethickness="0"> <mrow>k</mrow><mrow>s</mrow> </mfrac> </mfenced>{h}^{s}\tilde{{F}}_{ k-s+n}({(hv)}^{k-s} \cdot {\pi }^{n};f).$$

Proof.

For each fixed graphΓ ₀without disconnected vertices of the first type, we consider all graphsΓcontributing toF _k+nthat reduce toΓ ₀after removing all disconnected vertices of the first type. The contribution toF _k+nof such a graphΓish ^stimes the contribution ofΓ ₀, wheresis the number of disconnected vertices of the first type. Indeed, each disconnected vertex in a graphΓgives a factorhtoF _Γ and a factor ∫ _D ϕ = 1 to the weightw _Γ . The proof of the lemma is complete. □

Let

$${H}_{n}(\pi,h,f) ={ \sum }_{r=0}^{\infty }\frac{1} {r!}\tilde{{F}}_{n+r}({(hv)}^{r} \cdot {\pi }^{n};f).$$

In this sum there are finitely many terms since in the absence of disconnected vertices only derivatives ofhcan appear and the number of derivatives is bounded (by 2n). ThereforeH _n (π,h,f) is a differential polynomial in π,h,f. We conclude that

$$\begin{array}{rcl} {\sum }_{n=0}^{\infty } \frac{1} {n!}{F}_{n}(\hat{{\pi }}^{n};f) ={ \sum }_{n,k=0}^{\infty } \frac{{\epsilon }^{n}} {k!n!}{F}_{k+n}({(hv)}^{k}{\pi }^{n};f)&& \\ & & ={ \sum }_{n,r,s=0}^{\infty } \frac{{\epsilon }^{n}} {r!s!n!}{h}^{s}\tilde{{F}}_{ n+r}({(hv)}^{r}{\pi }^{n};f) =\mathrm{ {e}}^{h}{ \sum }_{n=0}^{\infty }\frac{{\epsilon }^{n}} {n!}{H}_{n}(\pi,h,f).\end{array}$$

This concludes the proof of Proposition1.

6 Equivariant differential forms on configuration spaces and Stokes theorem

6.1 Configuration spaces and their compactifications

We consider three types of configuration spaces of points, the first two appearing in [17].

(i)Configuration spaces of points in the plane. Let$\mathrm{{Conf}}_{n}(\mathbb{C}) =\{ z \in {\mathbb{C}}^{n}\,,\,{z}_{i}\neq {z}_{j},(i\neq j)\}$,n≥ 2. The three-dimensional real Lie groupG ₃of affine transformations$w\mapsto aw + b,a >0,b \in \mathbb{C}$acts freely on the manifold$\mathrm{{Conf}}_{n}(\mathbb{C})$. We set${C}_{n}(\mathbb{C}) =\mathrm{{ Conf}}_{n}(\mathbb{C})/{G}_{3}$(n≥ 2). It is a smooth manifold of dimension 2n− 3. We fix the orientation defined by the volume form dφ₂∧ ∧_{j≥ 3}dRe(z _j ) ∧ dIm(z _j ), with the choice of representatives with${z}_{1} = 0,{z}_{2} =\mathrm{ {e}}^{\mathrm{i}{\varphi }_{2}}$.

(ii)Configuration spaces of points in the upper half-plane. Let${H}_{+} =\{ z \in \mathbb{C}\,,\,\mathrm{Im}(z) >0\}$be the upper half-plane. Let$\mathrm{{Conf}}_{n,m}({H}_{+}) =\{ (z,x) \in {H}_{+}^{n} \times {\mathbb{R}}^{m},\,{z}_{i}\neq {z}_{j},(i\neq j),{t}_{1}< \cdots< {t}_{m}\}$, 2n+m≥ 2. The two-dimensional real Lie groupG ₂of affine transformations$w\mapsto aw + b,a >0,b \in \mathbb{R}$acts freely on the manifold Conf_n,m(H ₊). We setC _n,m(H ₊) = Conf_n,m(H ₊) ∕G ₂(2n+m≥ 2). It is a smooth manifold of dimension 2n+m− 2. Ifn≥ 1, we fix the orientation by choosing representatives withz ₁=iand taking the volume form dt ₁∧ ⋯ ∧ dt _m ∧ ∧_{j≥ 2}dRe(z _j ) ∧ dIm(z _j ). Ifm≥ 2, we fix the orientation defined by the volume form${(-1)}^{m}\mathrm{d}{t}_{2} \wedge \cdots \wedge \mathrm{ d}{t}_{m-1} \wedge {\wedge }_{j\geq 1}\mathrm{d}\mathrm{Re}({z}_{j}) \wedge \mathrm{ d}\mathrm{Im}({z}_{j})$, with the choice of representatives witht ₁= 0,t _m = 1. Ifm≥ 2 andn≥ 1, it is easy to check that the two orientations coincide.

(iii)Configuration spaces of points in the disk. Let$D =\{ z \in \mathbb{C}\,,\,\vert z\vert< 1\}$be the unit disk,${S}^{1} = \partial \bar{D}$the unit circle. LetC _{n,m+ 1}(D) = { (z,x) ∈D ⁿ×(S ¹)^{m+ 1},z _i ≠z _j , (i≠j), arg(t ₀) < ⋯ < arg(t _m ) < arg(t ₀) + 2π},m≥ 0. The circle group acts freely onC _{n,m+ 1}(D) by rotations. We do not take a quotient here, since the differential forms we will introduce are not basic, and work equivariantly instead. Instead of the quotient we consider the sectionC _n,m ⁰(D) = { (z,x) ∈C _{n,m+ 1}(D),t ₀= 1}, (m≥ 1). It is a smooth manifold of dimension 2n+m. The orientation ofC _{n,m+ 1}(D) is defined by darg(t ₀) ∧ ⋯ ∧ darg(t _m ) ∧ ∧_{j= 1} ⁿdRe(z _j ) ∧ dIm(z _j ). The orientation ofC _n,m ⁰(D) is defined by darg(t ₁) ∧ ⋯ ∧ darg(t _m ) ∧ ∧_{j= 1} ⁿdRe(z _j ) ∧ dIm(z _j ).

As in [17], compactifications$\bar{{C}}_{n}(\mathbb{C})$,$\bar{{C}}_{n,m}({H}_{+})$,$\bar{{C}}_{n,m+1}(D)$,$\bar{{C}}_{n,m}^{0}(D)$of these spaces as manifolds with corners are important. Their construction is the same as in [17]. Roughly speaking, one adds strata of codimension 1 corresponding to limiting configurations in which a group of points collapses to a point, possibly on the boundary, in such a way that within the group the relative position after rescaling remains fixed. Higher codimension strata correspond to collapses of several groups of points possibly within each other. The main point is that the Stokes theorem applies for smooth top differential forms on manifold with corners, and for this only codimension 1 strata are important.

Let us describe the codimension 1 strata ofC _n,m ⁰(D).

Strata of type I. These are strata where a subsetAofn′≥ 2 out ofnpointsz _i in the interior of the disk collapse at a point in the interior of the disk, the relative position of the collapsing points is described by a configuration on the plane and the remaining points and the point of collapse are given by a configuration on the disk. This stratum is thus

$$\begin{array}{rcl}{ \partial }_{A}\bar{{C}}_{n,m}^{0}(D) \simeq \bar{ {C}}_{ n{\prime}}(\mathbb{C}) \times \bar{ {C}}_{n-n{\prime}+1,m}^{0}(D).& &\end{array}$$

(15)

Strata of type II. These are strata where a subsetAofn′out ofnpointsz _i and a subsetBof thempointst _i collapse at a point on the boundary of the disk (2n′+m′≥ 2). The relative position of the collapsing points is described by a configuration on the upper half-plane and the remaining points and the point of collapse are given by a configuration on the disk. This stratum is thus

$$\begin{array}{rcl}{ \partial }_{A,B}\bar{{C}}_{n,m}^{0}(D) \simeq \bar{ {C}}_{ n{\prime},m{\prime}}({H}_{+}) \times \bar{ {C}}_{n-n{\prime},m-m{\prime}+1}^{0}(D).& &\end{array}$$

(16)

6.2 Forgetting the base point and cyclic shifts

Letj ₀:C _n,m ⁰(D) →C _n,m(D) be the map$(z,1,{t}_{1},\ldots,{t}_{m})\mapsto (z,{t}_{1},\ldots,{t}_{m})$forgetting the base pointt ₀= 1. It is an orientation preserving open embedding.

The cyclic shift λ:C _n,m ⁰(D) →C _n,m ⁰(D) is the map

$$\begin{array}{rcl} \lambda : ({z}_{1},\ldots,{z}_{n},1,{t}_{1},\ldots,{t}_{m})\mapsto ({z}_{1},\ldots,{z}_{n},1,{t}_{m},{t}_{1},\ldots,{t}_{m-1}).& & \\ \end{array}$$

It is a diffeomorphism preserving the orientation ifmis odd and reversing the orientation ifmis even. The following fact is then easily checked.

Lemma 6.

The collection of maps j _k = j ₀ ∘ λ ^∘k,$k = 0,\ldots,m - 1$ , defines an embedding j : C _n,m ⁰ (D) ⊔⋯ ⊔ C _n,m ⁰ (D) → C _n,m (D) with dense image. The restriction of j to the kth copy of C _n,m (D) multiplies the orientation by (−1) ^(m−1)k.

6.3 Proof of Theorem 2 on page34

The proof uses the Stokes theorem as in [17]. The new features are: (i) the differential forms in the integrand are not closed and (ii) an equivariant version of the Stokes theorem is used.

We first compute the differential of the differential form associated to a graphΓ.

Lemma 7.

Let ∂ _e Γ be the graph obtained from Γ by adding a new white vertex ∗ and replacing the black-to-black edge e ∈ E _b (Γ) by an edge originating at the same vertex as e but ending at ∗. Then

$$\mathrm{{d}}_{{S}^{1}}{\omega }_{\Gamma } ={ \sum }_{e\in {E}_{b}(\Gamma )}{(-1)}^{\sharp e}{\omega }_{{ \partial }_{e}\Gamma },$$

where ♯e = k if e = e _k and ${e}_{1},\ldots,{e}_{N}$ are the edges of Γ in the ordering specified by the ordering of the vertices and of the edges at each vertex.

Proof.

This follows from the fact that$\mathrm{{d}}_{{S}^{1}}$is a derivation of degree 1 of the algebra of equivariant forms and Lemma4, (ii). □

The next lemma is an equivariant version of the Stokes theorem.

Lemma 8.

Let $\omega \in {\Omega }_{{S}^{1}}^{\bullet }(\bar{{C}}_{n,m+1}(D))$ . Denote also by ω its restriction to $\bar{{C}}_{n,m}^{0}(D) \subset \bar{ {C}}_{n,m+1}(D)$ embedded as the subspace where t ₀ = 1 and to the codimension 1 strata ∂ _i C _n,m ⁰ (D) of C _n,m ⁰ (D). Then

$${\int }_{\bar{{C}}_{n,m}^{0}(D)}\mathrm{{d}}_{{S}^{1}}\omega ={ \sum }_{i}{ \int }_{{\partial }_{i}\bar{{C}}_{n,m}^{0}(D)}\omega - u{\int }_{\bar{{C}}_{n,m+1}(D)}\omega.$$

Proof.

Write$\mathrm{{d}}_{{S}^{1}} =\mathrm{ d} - u{\iota }_{\vec{v}}$. Foru= 0 the claim is just the Stokes theorem for manifolds with corners. Let us compare the coefficients ofu. The action map restricts to a diffeomorphism$f : {S}^{1} \times \bar{ {C}}_{n,m}^{0}(D) \rightarrow \bar{ {C}}_{n,m+1}(D)$. Since ω isS ¹-invariant,${\iota }_{\vec{v}}\omega$is also invariant and we have${f}^{{_\ast}}\omega = 1 \otimes \omega +\mathrm{ d}t \otimes {\iota }_{\vec{v}}\omega \in \Omega ({S}^{1}) \otimes \Omega (\bar{{C}}_{n,m}^{0}(D)) \subset \Omega ({S}^{1} \times \bar{ {C}}_{n,m}^{0}(D))$, wheretis the coordinate on the circle${S}^{1} = \mathbb{R}/\mathbb{Z}$. Thus

$$\begin{array}{rcl} {\int }_{\bar{{C}}_{n,m+1}(D)}\omega ={ \int }_{{S}^{1}\times \bar{{C}}_{n,m}^{0}(D)}\mathrm{d}t \otimes {\iota }_{\vec{v}}\omega ={ \int }_{\bar{{C}}_{n,m}^{0}(D)}{\iota }_{\vec{v}}\omega.& & \\ \end{array}$$

□

Finally we use Lemma6to reduce the integral over$\bar{{C}}_{n,m+1}(D)$to integrals over$\bar{{C}}_{n,m+1}^{0}(D)$. We obtain:

Lemma 9.

Let $\omega \in {\Omega }_{{S}^{1}}^{\bullet }(\bar{{C}}_{n,m+1}(D))$ and let j _k be the maps defined in Lemma 6.Then

$${\int }_{\bar{{C}}_{n,m+1}(D)}\omega ={ \sum }_{k=0}^{m}{(-1)}^{mk}{ \int }_{\bar{{C}}_{n,m+1}^{0}(D)}{j}_{k}^{{_\ast}}\omega.$$

We can now complete the proof of Theorem2. We first prove the identity (11), starting from the right-hand side. Suppose that$a = ({a}_{0},\ldots,{a}_{m}) \in {C}_{-m}(A)$, γ = γ₁⋯γ _n , with${\gamma }_{i} \in \Gamma ({\wedge }^{{k}_{i}}\mathit{TM})$. It is convenient to identifyΓ( ∧TM) with${C}^{\infty }(M)[{\theta }_{1},\ldots,{\theta }_{n}]$where θ _i are anticommuting variables, so that div _Ω = ∑∂ ²∕∂t _i ∂θ _i . It follows that for any$\Gamma \in {\mathcal{G}}_{\mathbf{k},m}$, div _Ω F _Γ (γ;a) can be written as a sum (with signs) of termsF _Γ′ (γ;a), whereΓ′is obtained fromΓby identifying a white vertex with a black vertex and coloring it black. Some of these graphsΓ′have an edge connecting a vertex to itself and contribute toF _n (δ _Ω γ;a). The remaining ones yield, in the notation of Lemma7:

$${\mathrm{div} }_{\Omega }{F}_{n}(\gamma ;a) - {F}_{n}({\delta }_{\Omega }\gamma ;a) ={ \sum }_{(\Gamma,e)}{(-1)}^{\sharp e}{w}_{{ \partial }_{e}\Gamma }{F}_{\Gamma }(\gamma ;a).$$

The summation is over pairs (Γ,e) where$\Gamma \in {\mathcal{G}}_{\mathbf{k},m}$ande∈E _b (Γ) is a black-to-black edge. By Lemmas8and9,

$${\sum }_{e\in {E}_{b}(\Gamma )}{(-1)}^{\sharp e}{w}_{{ \partial }_{e}\Gamma } ={ \sum }_{i}{ \int }_{{\partial }_{i}{C}_{n,m}^{0}(D)}{\omega }_{\Gamma }-u{\sum }_{k=0}^{m}{(-1)}^{km}{ \int }_{\bar{{C}}_{n,m+1}^{0}}{j}_{k}^{{_\ast}}{\omega }_{ \Gamma }.$$

The second term on the right-hand side, containing the sum over cyclic permutations, gives rise toF _{n+ 1}(γ;Ba). The first term is treated as in [17]: the strata of type I (see Section6.1) give zero by Kontsevich’s lemma (see [17], Theorem 6.5) unless the numbern′of collapsing interior points is 2. The sum over graphs contributes then to the term with the Schouten bracket [γ _i , γ _j ] in (11). The strata of type II such thatn−k> 0 interior points approach the boundary give rise to the term containing the components of the KontsevichL _∞ -morphismU _n−k. Finally the strata of type II in which only boundary points collapse give the term with Hochschild differentialF _{n− 1}(γ;ba). This proves (11).

Property (i) is clear:F ₀is a sum over graphs with vertices of the second type only. These graphs have no edges. Thus the only case for which the weight does not vanish is when the configuration space is 0-dimensional, namely, when there is only one vertex. Property (ii) is checked by an explicit calculation of the weight. The only graphs with a nontrivial weight have edges connecting the vertex of the first type with white vertices or to vertices of the second type. There must be at leastpedges otherwise the weight vanishes for dimensional reasons. In this case, i.e., ifk≥p, the integral computing the weight is

$${w}_{\Gamma } = \int \phi {(z,u)}^{\mathcal{l}+k-p}\omega (z,{t}_{ 1})\cdots \omega (z,{t}_{p}), $$

(17)

with integration overz∈D,t _i ∈S ¹, 0 < arg(t ₁) < ⋯ < arg(t _p ) < 2π. The integral of the product of the 1-forms ω is a function ofzthat is independent ofz, as is easily checked by differentiating with respect toz, using the Stokes theorem and the boundary conditions of ω. Thus it can be computed forz= 0. Since$\omega (0,{t}_{i}) = \frac{1} {2\pi }\mathrm{d}\arg ({t}_{i})$the integral is 1 ∕p!. The remaining integral overzcan then be performed. Setℓ+k−p=s+ 1. This power must be positive otherwise the integral vanishes for dimensional reasons.

$${\int }_{D}\phi {(z,u)}^{s+1} = \frac{\mathrm{i}} {2\pi }(s + 1){u}^{s}{ \int }_{D}{(1 -\vert z{\vert }^{2})}^{s}\mathrm{d}z \wedge \mathrm{ d}\bar{z} = {u}^{s},\quad s \geq 0,$$

and we obtainw _Γ =u ^s∕p!. We turn to Property (iii). The equivariance under linear coordinate transformations is implicit in the construction. The graphs contributing toF _n (γ₁⋯ ;a) for linear γ₁are of two types: either the vertex associated with γ₁has exactly one ingoing and one outgoing edge or it has an outgoing edge pointing to a white vertex and there are no incoming edges. The graphs of the second type contribute to γ₁∧F _{n− 1}(⋯ ;a), since their weight factorize as 1 = ∫ _D ϕ times the weight of the graphs obtained by omitting the vertex associated to γ₁and the white vertex connected to it. The claim then follows from the following vanishing lemma.

Lemma 10.

• (i) For all $z,z{\prime} \in \bar{ D}$ , ∫ _w∈D ω(z,w)ω(w,z′) = 0.

• (ii) For all $z \in \bar{ D}$ , ∫ _w∈D ω(z,w)ϕ(w,u) = 0.

Proof.

(i) We reduce the first claim to the second: consider the integral

$$I(z,z{\prime}) ={ \int }_{{w}_{1},{w}_{2}\in D}\mathrm{d}(\omega (z,{w}_{1})\omega ({w}_{1},{w}_{2})\omega ({w}_{2},z{\prime})).$$

On the one hand,I(z,z′) can be evaluated by using Stokes’s theorem, giving three terms all equal up to sign to the integral appearing in (i). On the other hand, the differential can be evaluated explicitly giving

$$I(z,z{\prime}) = -{\int}_{{w}_{1},{w}_{2}\in D}\omega (z,{w}_{1})\omega ({w}_{1},{w}_{2})\phi ({w}_{2},0).$$

The integral overw ₂vanishes if (ii) holds. The proof of (ii) is an elementary computation that uses the explicit expression of ω and ϕ. Alternatively, one shows that ∫_w∈Dω(z,w)ϕ(w,u) is a closed 1-form on the disk that vanishes on the boundary, is invariant under rotations and odd under diameter reflections. Therefore it vanishes. We leave the details to the reader. □

6.4 Acknowledgments

We are grateful to Francesco Bonechi, Damien Calaque, Kevin Costello, Florian Schätz, Carlo Rossi, Jim Stasheff, Thomas Willwacher and Marco Zambon for useful comments. This work been partially supported by SNF Grants 20-113439 and 200020-105450, by the European Union through the FP6 Marie Curie RTN ENIGMA (contract number MRTN-CT-2004-5652), and by the European Science Foundation through the MISGAM program. The first author is grateful to the Erwin Schrödinger Institute, where part of this work was done, for hospitality.