A higher Chern-Weil derivation of AKSZ sigma-models
aa r X i v : . [ m a t h - ph ] J un A higher Chern-Weil derivation of AKSZ σ -models Domenico Fiorenza, Christopher L. Rogers, and Urs SchreiberAugust 2, 2018
Abstract
Chern-Weil theory provides for each invariant polynomial on a Lie al-gebra g a map from g -connections to differential cocycles whose volumeholonomy is the corresponding Chern-Simons theory action functional.Kotov and Strobl have observed that this naturally generalizes from Liealgebras to dg-manifolds and dg-bundles and that the Chern-Simons ac-tion functional associated this way to an n -symplectic manifold is theaction functional of the AKSZ σ -model whose target space is the given n -symplectic manifold (examples of this are the Poisson σ -model or theCourant σ -model, including ordinary Chern-Simons theory, or higher di-mensional abelian Chern-Simons theory). Here we show how, within theframework of the higher Chern-Weil theory in smooth ∞ -groupoids, thisresult can be naturally recovered and enhanced to a morphism of higherstacks, the same way as ordinary Chern-Simons theory is enhanced to amorphism from the stack of principal G -bundles with connections to the3-stack of line 3-bundles with connections. Contents σ -Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 L ∞ -algebroids 13 L ∞ -algebroids . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Cocycles, invariant polynomials and Chern-Simons elements . . 153.3 Symplectic Lie n -algebroids . . . . . . . . . . . . . . . . . . . . . 17 σ -model . . . . . . . . . . . . . . . . . . . . . . . 234.1.3 Courant σ -model . . . . . . . . . . . . . . . . . . . . . . . 244.1.4 Higher abelian Chern-Simons theory in d = 4 k + 3 . . . . 26 n -groupoids and nontrivial topology . . . . . . . . . . 346.2 ∞ -Connections on nontrivial a -principal ∞ -bundles: AKSZ in-stantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.3 Twisted AKSZ-structures and higher extensions of symplectic L ∞ -algebroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.4 Relation to higher dimensional supergravity . . . . . . . . . . . . 36 References 38
The class of topological field theories known as
AKSZ σ -models [AKSZ97] con-tains in dimension 3 ordinary Chern-Simons theory (see [Fre] for a comprehen-sive review) as well as its Lie algebroid generalization (the Courant σ -model [Ike03, HoPa04, Royt07]), and in dimension 2 the Poisson σ -model [Ike94,ScSt94a] (see [ScSt94b, CaFe01] for a review). It is therefore clear that theAKSZ construction is some sort of generalized Chern-Simons theory. That thisis indeed true, has been first formalized and rigorously established in [KoSt07],as a particular case of a general Chern-Weil-type construction of characteris-tic classes for Q-bundles (see also [KoSt10] for a review). Here we show how,within the framework of the ∞ -Chern-Weil homomorphism of [FSS10, Sch10],this result can be naturally recovered and enhanced to a morphism of higherstacks, the same way as ordinary Chern-Simons theory is enhanced to a mor-phism from the stack of principal G -bundles with connections to the 3-stack ofline 3-bundles with connections.Our discussion proceeds from the observation that the standard Chern-Simons action functional has a systematic origin in Chern-Weil theory (see forinstance [GHV] for a classical textbook treatment and [HoSi05] for the refine-ment to differential cohomology that we need here):The refined Chern-Weil homomorphism assigns to any invariant polyno-mial h−i : g ⊗ k → R on a Lie algebra g of compact type a map that sends g -connections ∇ on a smooth manifold X to cocycles [ˆ p h−i ( ∇ )] ∈ H k diff ( X ) in ordinary differential cohomology . These differential cocycles refine the curvaturecharacteristic class [ h F ∇ i ] ∈ H kdR ( X ) in de Rham cohomology to a fully fledged line (2 k − -bundle with connection , also known as a bundle (2 k − -gerbe withconnection . And just as an ordinary line bundle (a “line 1-bundle”) with con-nection assigns holonomy to curves, so a line n -bundle with connection assignsholonomy hol ˆ p (Σ) to n -dimensional trajectories Σ → X . For the special casewhere h−i is the Killing form polynomial and X = Σ with dim Σ = 3 one findsthat this volume holonomy map ∇ 7→ hol ˆp h−i ( ∇ ) (Σ) is precisely the standardChern-Simons action functional. Similarly, for h−i any higher invariant poly-nomial this holonomy action functional has as Lagrangian the correspondinghigher Chern-Simons form. In summary, this means that Chern-Simons-typeaction functionals on Lie algebra-valued connections are the images of the re-fined Chern-Weil homomorphism.In previous work [Sch10, FSS10] a generalization of the Chern-Weil homo-morphism to higher (“derived”) differential geometry has been considered. Inthis framework, smooth manifolds are generalized first to orbifolds, then togeneral Lie groupoids, to Lie 2-groupoids and finally to smooth ∞ -groupoids(smooth ∞ -stacks), while Lie algebras are generalized to Lie 2-algebras etc.,up to L ∞ -algebras and more generally to Lie m -algebroids and finally to L ∞ -algebroids. For a any L ∞ -algebroid, one has a natural notion of a -valued ∞ -connections on exp( a )-principal smooth ∞ -bundles (where exp( a ) is a smooth See [KoSt07] for a treatment of the higher Chern-Weil homomorphism from the equivalentpoint of view of Q-manifolds. ∞ -groupoid obtained by Lie integration from a ). By analyzing the abstractly de-fined higher Chern-Weil homomorphism in this context one finds a direct higheranalog of the above situation: there is a notion of invariant polynomials h−i onan L ∞ -algebroid a and these induce maps from a -valued ∞ -connections to line n -bundles with connections as before [SSS09a, FSS10]. The corresponding classof action functionals we call ∞ -Chern-Simons theory .This construction drastically simplifies when one restricts attention to trivial ∞ -bundles with (nontrivial) a -connections. Over a smooth manifold Σ these aresimply given by dg-algebra homomorphisms A : W( a ) → Ω • (Σ) , where W( a ) is the Weil algebra of the L ∞ -algebroid a [KoSt07, SSS09a], andΩ • (Σ) is the de Rham algebra of Σ (which is indeed the Weil algebra of Σthought of as an L ∞ -algebroid concentrated in degree 0). Then for h−i ∈ W( a )an invariant polynomial, the corresponding ∞ -Chern-Weil homomorphism ispresented by a choice of “Chern-Simons element” cs ∈ W( a ), which exhibits the transgression of h−i to an L ∞ -cocycle (the higher analog of a cocycle in Liealgebra cohomology): the dg-morphism A naturally maps the Chern-Simonselement cs of A to a differential form cs( A ) ∈ Ω • (Σ) and its integral is thecorresponding ∞ -Chern-Simons action functional S h−i S h−i : A hol ˆ p h−i ( A ) (Σ) = Z Σ cs h−i ( A ) . Even though trivial ∞ -bundles with a -connections are a very particular sub-case of the general ∞ -Chern-Weil theory, they are rich enough to contain AKSZtheory. Namely, here we show that a symplectic dg-manifold of grade n – whichis the geometrical datum of the target space defining an AKSZ σ -model – isnaturally equivalent to an L ∞ -algebroid P endowed with a quadratic and non-degenerate invariant polynomial ω of grade n . Moreover, under this identi-fication the canonical Hamiltonian π on the symplectic target dg-manifold isidentified as an L ∞ -cocycle on P . Finally, the invariant polynomial ω is natu-rally in transgression with the cocycle π via a Chern-Simons element cs ω thatturns out to be the Lagrangian of the AKSZ σ -model [KoSt07]: Z Σ L AKSZ ( A ) = Z Σ cs ω ( A ) . (An explicit description of L AKSZ is given below in def. 2.13)In summary this means that we find the following dictionary of concepts:
Chern-Weil theory AKSZ theory cocycle π Hamiltoniantransgression element cs Lagrangianinvariant polynomial ω symplectic structureMore precisely, we (explain and then) prove here the following theorem: Theorem 1.1.
For ( P , ω ) an L ∞ -algebroid with a quadratic non-degenerateinvariant polynomial, the corresponding ∞ -Chern-Weil homomorphism ∇ 7→ hol ˆ p ω ( ∇ ) (Σ) sends P -valued ∞ -connections ∇ to their corresponding exponentiated AKSZaction: hol ˆ p ω ( ∇ ) (Σ) = Z Σ L AKSZ ( ∇ ) . The local differential form data involved in this statement is at the focusof attention in this article here and contained in proposition 4.2 below. Weindicate the global aspects of the construction in Section 5. The more abstracthigher Chern-Weil theoretic interpretation of AKSZ σ -models implies variousfurther constructions and generalizations. We close in Section 6 by giving anoutlook on these.After a preliminary version of this article had appeared on arXiv we wereinformed by Alexei Kotov and Thomas Strobl that the main constructionsfrom Sections 2-4 are presented in [KoSt07] in the equivalent framework of Q-manifolds. For the reader’s convenience, here is a basic dictionary between thenotations used in this article and those in [KoSt07]: what Kotov and Strobl calla Q-manifold is by definition a dg-manifold in our terminology; our homologicalvector field v is their Q and our d W( a ) is their total differential (see proposition3.4 below); our A in definition 3.11 is their chain map f ∗ ; our proposition 3.18 istheir lemma 4.5; our proposition 3.20 is their lemma 4.6, with the Chern-Simonselement we denote cs called ˆ α there; finally, our proposition 4.2 and corollary4.3 are their theorem 4.4. Acknowledgement.
We thank Dmitry Roytenberg for helpful conversationat an early stage of this project, Alexei Kotov and Thomas Strobl for havingpointed our attention to [KoSt07], and the Referee for comments and sugges-tions on the first version of this article. CLR acknowledges support by a Ju-nior Research Fellowship from the Erwin Schr¨odinger International Institute forMathematical Physics.
The first half of the seminal article [AKSZ97] presented some key observationson, what from a modern perspective would be called, symplectic derived geome-try [Lu09] in its variant of symplectic dg-geometry [ToVe05]. In its second half,it describes the role of such symplectic dg-geometry in quantum field theory ingeneral, and σ -model theory in particular.In this section we briefly review some basics in order to establish the contextfor our discussion. In higher differential geometry , smooth manifolds are generalized first to orb-ifolds – which are special Lie groupoids – then to higher Lie groupoids: smooth ∞ -groupoids [Sch10]. Moreover, in derived differential geometry , the functionalgebras are generalized to smooth ∞ -algebras [Sp08, Ste10]. All of these ingredi-ents have presentations in terms of compound structures in ordinary differentialgeometry. There is a bit of theory involved in exactly how these presentationsmodel the general abstract theory, but the main statement that we want todiscuss here can be described already in a rather simple-minded setup.Therefore, here we shall be content with the following simple definitionsof what might be called affine smooth graded manifolds and affine smooth dg-manifolds . Despite their simplicity these definitions capture in a precise senseall the relevant structure: namely the local smooth structure. Globalizations ofthese definitions can be obtained, if desired, by general abstract constructions.We give some outlook on this in section 6. Definition 2.1.
The category of affine smooth N -graded manifolds – here called smooth graded manifolds for short – is the full subcategorySmoothGrMfd ⊂ GrAlg op R of the opposite category of N -graded-commutative R -algebras on those isomor-phic to Grassmann algebras of the form ∧ • C ∞ ( X ) Γ( V ∗ ) , where X is an ordinary smooth manifold, V → X is an N -graded smoothvector bundle over X degreewise of finite rank, and Γ( V ∗ ) is the graded C ∞ ( X )-module of smooth sections of the dual bundle.For a smooth graded manifold X ∈ SmoothGrMfd, we write C ∞ ( X ) ∈ cdgAlg R for its corresponding dg-algebra of functions . Remarks. • The full subcategory of these objects is equivalent to that of all objectsisomorphic to one of this form. We may therefore use both points of viewinterchangeably. • Much of the theory works just as well when V is allowed to be Z -graded.This is the case that genuinely corresponds to derived (instead of justhigher) differential geometry. An important class of examples for thiscase are BV-BRST complexes which motivate much of the literature. Forthe purpose of this short note, we shall be content with the N -graded case. • For an N -graded C ∞ ( X )-module Γ( V ∗ ) we have ∧ • C ∞ Γ( V ∗ ) = C ∞ ( X ) ⊕ Γ( V ∗ ) ⊕ (cid:0) Γ( V ∗ ) ∧ C ∞ ( X ) Γ( V ∗ ) ⊕ Γ( V ∗ ) (cid:1) ⊕ · · · , with the leftmost summand in degree 0, the next one in degree 1, and soon. • There is a canonical functorSmoothMfd ֒ → SmthGrMfdwhich identifies an ordinary smooth manifold X with the smooth gradedmanifold whose function algebra is the ordinary algebra of smooth func-tions C ∞ ( X ) := C ∞ ( X ) regarded as a graded algebra concentrated indegree 0. This functor is full and faithful and hence exhibits a full sub-category.All the standard notions of differental geometry apply to differential gradedgeometry. For instance for X ∈ SmoothGrMfd, there is the graded vector spaceΓ(
T X ) of vector fields on X , where a vector field is identified with a graded derivation v : C ∞ ( X ) → C ∞ ( X ). This is naturally a graded (super) Lie algebrawith super Lie bracket the graded commutator of derivations. Notice that for v ∈ Γ( T X ) of odd degree we have [ v, v ] = v ◦ v + v ◦ v = 2 v : C ∞ ( X ) → C ∞ ( X ). Definition 2.2.
The category of (affine, N -graded) smooth differential-gradedmanifolds is the full subcategorySmoothDgMfd ⊂ cdgAlg op R of the opposite of differential graded-commutative R -algebras on those objectswhose underlying graded algebra comes from SmoothGrMfd.This is equivalently the category whose objects are pairs ( X, v ) consistingof a smooth graded manifold X ∈ SmoothGrMfd and a grade 1 vector field v ∈ Γ( T X ), such that [ v, v ] = 0, and whose morphisms ( X , v ) → ( X , v ) aremorphisms f : X → X such that v ◦ f ∗ = f ∗ ◦ v . Remark 2.3.
The dg-algebras appearing here are special in that their degree-0algebra is naturally not just an R -algebra, but a smooth algebra (a “ C ∞ -ring”,see [Ste10] for review and discussion). In a more theoretical account than wewant to present here, we would use the corresponding more general notion of smooth dg-algebras . For our present purposes, this will only briefly play a rolein def. 3.3 below. Definition 2.4.
The de Rham complex functor Ω • ( − ) : SmoothGrMfd → cdgAlg op R sends a dg-manifold X with C ∞ ( X ) ≃ ∧ • C ∞ ( X ) Γ( V ∗ ) to the Grassmann algebraover C ∞ ( X ) on the graded C ∞ ( X )-moduleΓ( T ∗ X ) ⊕ Γ( V ∗ ) ⊕ Γ( V ∗ [ − , where Γ( T ∗ X ) denotes the ordinary smooth 1-form fields on X and where V ∗ [ −
1] is V ∗ with the grades increased by one. This is equipped with thedifferential d defined on generators as follows: • d | C ∞ ( X ) = d dR is the ordinary de Rham differential with values in Γ( T ∗ X ); • d | Γ( V ∗ ) → Γ( V ∗ [ − • and d vanishes on all remaining generators. Definition 2.5.
Observe that Ω • ( − ) evidently factors through the defininginclusion SmoothDgMfd ֒ → cdgAlg R . Write T ( − ) : SmoothGrMfd → SmoothDgMfdfor this factorization.The dg-space T X is often called the shifted tangent bundle of X and denoted T [1] X . Observation 2.6.
For Σ an ordinary smooth manifold and for X a gradedmanifold corresponding to a vector bundle V → X , there is a natural bijection SmoothGrMfd( T Σ , X ) ≃ Ω • (Σ , V ) where on the right we have the set of V -valued smooth differential forms on Σ : tuples consisting of a smooth function φ : Σ → X , and for each n > an ordinary differential n -form φ n ∈ Ω n (Σ , φ ∗ V n − ) with values in the pullbackbundle of V n − along φ . The standard Cartan calculus of differential geometry generalizes directlyto graded smooth manifolds. For instance, given a vector field v ∈ Γ( T X ) on X ∈ SmoothGrMfd, there is the contraction derivation ι v : Ω • ( X ) → Ω • ( X )on the de Rham complex of X , and hence the Lie derivative L v := [ ι v , d ] : Ω • ( X ) → Ω • ( X ) . Definition 2.7.
For X ∈ SmoothGrMfd the
Euler vector field ǫ ∈ Γ( T X ) isdefined over any coordinate patch U → X to be given by the formula ǫ | U := X a deg( x a ) x a ∂∂x a , where { x a } is a basis of generators and deg( x a ) the degree of a generator. The grade of a homogeneous element α in Ω • ( X ) is the unique natural number n ∈ N with L ǫ α = nα . Remarks. • This implies that for x i an element of grade n on U , the 1-form d x i isalso of grade n . This is why we speak of grade (as in “graded manifold”)instead of degree here. • Since coordinate transformations on a graded manifold are grading-preserving,the Euler vector field is indeed well-defined. Note that the degree-0 coor-dinates do not appear in the Euler vector field.The existence of ǫ implies the following useful statement (amplified in [Royt99]),which is a trivial variant of what in grade 0 would be the standard Poincar´elemma. Observation 2.8.
On a graded manifold, every closed differential form ω ofpositive grade n is exact: the form λ := 1 n ι ǫ ω satisfies d λ = ω . Definition 2.9. A symplectic dg-manifold of grade n ∈ N is a dg-manifold( X, v ) equipped with 2-form ω ∈ Ω ( X ) which is • non-degenerate; • closed;as usual for symplectic forms, and in addition • of grade n ; • v -invariant: L v ω = 0.In a local chart U with coordinates { x a } we may find functions { ω ab ∈ C ∞ ( U ) } such that ω | U = 12 d x a ω ab ∧ d x b , where summation of repeated indices is implied. We say that U is a Darbouxchart for (
X, ω ) if the ω ab are constant.0 Observation 2.10.
The function algebra of a symplectic dg-manifold ( X, ω ) ofgrade n is naturally equipped with a Poisson bracket {− , −} : C ∞ ( X ) ⊗ C ∞ ( X ) → C ∞ ( X ) which decreases grade by n . On a local coordinate patch { x a } this is given by { f, g } = f ∂ x a ∂ ω ab ∂g∂x b , where { ω ab } is the inverse matrix to { ω ab } , and where the graded differentiationin the left factor is to be taken from the right, as indicated. Definition 2.11.
For π ∈ C ∞ ( X ) and v ∈ Γ( T X ), we say that π is a Hamil-tonian for v , or equivalently, that v is the Hamiltonian vector field of π if d π = ι v ω . Note that the convention ( − n +1 d π = ι v ω is also frequently used for defin-ing Hamiltonians in the context of graded geometry. Remark 2.12.
In a local coordinate chart { x a } the defining equation d π = ι v ω becomes d x a ∂π∂x a = ω ab v a ∧ d x b = ω ab d x a ∧ v b , implying that ω ab v b = ∂π∂x a . σ -Models We now consider, in definition 2.13 below, for any symplectic dg-manifold (
X, ω )a functional S AKSZ on spaces of maps T Σ → X of smooth graded manifolds, andspecialize this to the explicit formula (2.2.1) in the special case the target man-ifold is endowed with global Darboux coordinates. While only this particularsituation is referred to in the remainder of the article, we begin by indicatinginformally the original motivation of S AKSZ . The reader uncomfortable withthese somewhat vague considerations can take formula (2.2.1) as a definitionand then skip to the next section.Generally, a σ -model field theory is, roughly, one1. whose fields over a space Σ are maps φ : Σ → X to some space X ;2. whose action functional is, apart from a kinetic term, the transgression ofsome kind of cocycle on X to the mapping space Map(Σ , X ).Here the terms “space”, “maps” and “cocycles” are to be made precise in asuitable context. One says that Σ is the worldvolume , X is the target space andthe cocycle is the background gauge field .1For instance, an ordinary charged particle (such as an electron) is describedby a σ -model where Σ = (0 , t ) ⊂ R is the abstract worldline , where X is a(pseudo-)Riemannian smooth manifold (for instance our spacetime), and wherethe background cocycle is a line bundle with connection on X (a degree-2 cocyclein ordinary differential cohomology of X , representing a background electromag-netic field ). Up to a kinetic term, the action functional is the holonomy of theconnection over a given curve φ : Σ → X . A textbook discussion of thesestandard kinds of σ -models is, for instance, in [DM99].The σ -models which we consider here are higher generalizations of this ex-ample, where the background gauge field is a cocycle of higher degree (a higherbundle with connection) and where the worldvolume is accordingly higher di-mensional. In addition, X is allowed to be not just a manifold, but an approx-imation to a higher orbifold (a smooth ∞ -groupoid).More precisely, here we take the category of spaces to be SmoothDgMfdfrom def. 2.2. We take target space to be a symplectic dg-manifold ( X, ω )and the worldvolume to be the shifted tangent bundle T Σ of a compact smoothmanifold Σ. Following [AKSZ97], one may imagine that we can form a smooth Z -graded mapping space Maps( T Σ , X ) of smooth graded manifolds. On thisspace the canonical vector fields v Σ and v X naturally have commuting actionsfrom the left and from the right, respectively, so that their sum v Σ + v X equipsMaps( T Σ , X ) itself with the structure of a differential graded smooth manifold.Next we take the “cocycle” on X (to be made precise in the next section) tobe the Hamiltonian π (def. 2.11) of v X with respect to the symplectic structure ω , according to def. 2.9. One wants to assume that there is a kind of Riemannianstructure on T Σ that allows to form the transgression Z T Σ ev ∗ ω := p ! ev ∗ ω by pull-push through the canonical correspondenceMaps( T Σ , X ) o o p Maps( T Σ , X ) × T Σ ev / / X .
When one succeeds in making this precise, one expects to find that R T Σ ev ∗ ω isin turn a symplectic structure on the mapping space.This implies that the vector field v Σ + v X on mapping space has a Hamilto-nian S ∈ C ∞ (Maps( T Σ , X )) , s.t. dS = ι v Σ + v X Z T Σ ev ∗ ω . Definition 2.13.
The grade-0 component S AKSZ := S | Maps( T Σ ,X ) of the Hamiltonian S constitutes a functional on the space of morphisms ofgraded manifolds φ : T Σ → X . This is the AKSZ action functional defining theAKSZ σ -model with target space X and background field/cocycle ω .2In [AKSZ97], this procedure is indicated only somewhat vaguely. The fo-cus of attention there is on a discussion, from this perspective, of the actionfunctionals of the 2-dimensional σ -models called the A-model and the
B-model .In [Royt07] a more detailed discussion of the general construction is given, in-cluding an explicit formula for S , and hence for S AKSZ , in the case that X admits global Darboux coordinates. That formula is the following: for ( X, ω )a symplectic dg-manifold of grade n with global Darboux coordinates { x a } , Σa smooth compact manifold of dimension ( n + 1) and k ∈ R , the AKSZ actionfunctional S AKSZ : SmoothGrMfd( T Σ , X ) → R is S AKSZ : φ Z Σ (cid:18) ω ab φ a ∧ d dR φ b − φ ∗ π (cid:19) , (2.2.1)where π is the Hamiltonian for v X with respect to ω and where on the right weare interpreting fields as forms on Σ according to prop. 2.6.This formula hence defines an infinite class of σ -models depending on thetarget space structure ( X, ω ). (One can also consider arbitrary relative factorsbetween the first and the second term, but below we shall find that the abovechoice is singled out). In [AKSZ97], it was already noticed that ordinary Chern-Simons theory is a special case of this for ω of grade 2, as is the Poisson σ -modelfor ω of grade 1 (and hence, as shown there, also the A-model and the B-model).The main example in [Royt07] spells out the general case for ω of grade 2, whichis called the Courant σ -model there. (We review and re-derive all these examplesin detail in 4.1 below.)One nice aspect of this construction is that it follows immediately that thefull Hamiltonian S on the mapping space satisfies { S , S } = 0. Moreover, usingthe standard formula for the internal hom of chain complexes, one finds thatthe cohomology of (Maps( T Σ , X ) , v Σ + v X ) in degree 0 is the space of func-tions on those fields that satisfy the Euler-Lagrange equations of S AKSZ . Takentogether, these facts imply that S is a solution of the “master equation” of aBV-BRST complex for the quantum field theory defined by S AKSZ . This is acrucial ingredient for the quantization of the model, and this is what the AKSZconstruction is mostly used for in the literature (for instance [CaFe01]).Here we want to focus on another nice aspect of the AKSZ-construction: ithints at a deeper reason for why the σ -models of this type are special. It is indeedone of the very few proposals for what a general abstract mechanism might bethat picks out among the vast space of all possible local action functionals thosethat seem to be of relevance “in nature”.We now proceed to show that the class of action functionals S AKSZ areprecisely those that higher Chern-Weil theory canonically associates to targetdata (
X, ω ). Since higher Chern-Weil theory in turn is canonically given onvery general abstract grounds [Sch10], this in a sense amounts to a derivationof S AKSZ from “first principles”, and it shows that a wealth of very generaltheory applies to these systems. More on this will be discussed elsewhere, some3indication are in Section 5. Here we shall focus on a concrete computationexhibiting S AKSZ as the image of the higher Chern-Weil homomorphism. L ∞ -algebroids We now discuss the ∞ -Lie theoretic concepts in terms of which we shall re-express the AKSZ σ -model below in 4. L ∞ -algebroids We survey some basics of ∞ -Lie theory that we need later on. The explicit L ∞ -algebraic constructions are from [SSS09a], a more encompassing discussionis in [Sch10].The following definition essentially repeats def. 2.2 with different terminol-ogy. While this may look like a redundancy, it is useful to instead regard it asthe beginning of a useful dictionary between higher Lie theory and dg-geometry.The examples to follow will illustrate this. Definition 3.1.
The category of L ∞ -algebroids is equivalent to that of smoothdg-manifolds from def. 2.2: L ∞ Algd ≃ SmoothDgMfd ֒ → cdgAlg op R . For a ∈ L ∞ Algd we write CE( a ) ∈ cdgAlg R for the corresponding dg-algebraand call it the Chevalley-Eilenberg algebra of a .If the graded algebra underlying CE( a ) has generators of grade at most n ,we say that a is a Lie n -algebroid . Examples. • Any (degreewise finite-dimensional) L ∞ -algebra (and so, in particular, anyLie algebra) g can be seen as a (canonically pointed) L ∞ -algebroid b g overthe point: for D : ∨ • g → ∨ • g the nilpotent derivation on the free gradedcoalgebra over g which defines the k -ary brackts on g by[ − , − , · · · , − ] k := D | ∨ k g : ∨ k g → g , we have CE( b g ) := ( ∧ • g ∗ , d := D ∗ ) . One directly finds that L ∞ -algebroid morphims b g → b h are precisely L ∞ -algebra morphism g → h . This means that there is a full and faithfulinclusion b : L ∞ Alg ֒ → L ∞ Algdof the traditional category of L ∞ -algebras into that of L ∞ -algebroids.We refer to b g as the delooping of g . This notation is the infinitesimalanalog of the notation B G for the one-object Lie groupoid corresponding4to a Lie group G : the loop space object Ω B G is equivalent to G , hencethe name “delooping” given to B G .For g a Lie algebra, the algebra CE( b g ) is the ordinary Chevalley-Eilenbergalgebra of g . • For n ∈ N the delooping of the line Lie n -algebra is the L ∞ -algebroid b n R defined by the fact that CE( b n R ) is generated over R from a singlegenerator in degree n with vanishing differential. • For X a smooth manifold, the tangent Lie algebroid a = T X is defined byCE( T X ) = (Ω • ( X ) , d dR ) ; • For ( X, {− , −} ) a Poisson manifold, the corresponding Poisson Lie alge-broid P ( X ) is defined byCE( P ( X )) = ( ∧ • C ∞ ( X ) Γ( T X ) , { π, −} ) , where π ∈ ∧ C ∞ ( X ) Γ( T X ) is the Poisson tensor and the bracket means thecanonical extension to the tangent bundle: the Schouten bracket.
Remark 3.2.
For a an L ∞ -algebroid and { x i } local coordinates on the cor-responding graded manifold, the vector field v corresponding to the Chevellay-Eilenberg differential d CE( a ) is v (cid:12)(cid:12) U = v i ∂∂x i , with v i := d CE( a ) x i . Definition 3.3.
For a an L ∞ -algebroid, its Weil algebra is that representative ofthe free smooth dg-algebra , remark 2.3, on the underlying word-length-1 complexof a that makes the canonical projection of complexes i ∗ : W ( a ) → CE( a )into a dg-algebra homomorphism. Proposition 3.4.
Explicitly, the Weil algebra W( a ) has • as underlying graded algebra the de Rham complex Ω • ( a ) from def. 2.4,applied to the corresponding graded manifold; i.e., the differential gradedmanifold corresponding to W( a ) is the tangent Lie ∞ -algebroid Ta . Thiscan be equivalently written as W( a ) = CE( Ta ) . • as differential the sum d W( a ) = d + L v , where d is the differential from def. 2.4 and where L v is the Lie derivativealong the vector field v corresponding to the Chevalley-Eilenberg differen-tial. Remark.
Therefore the Weil algebra W( a ) is a twisted de Rham complex onthe graded smooth manifold corresponding to a , where the twist is dictated bythe characterizing morphism i ∗ from def. 3.3. In the abstract theory indicatedbelow in 5 this makes W( a ) part of the construction of a certain homotopicalresolution of the Lie integration of a . This is the deeper reason for the roleplayed by the Weil algebra in higher Lie theory. But for the present purposethe above explicit definition is sufficient. Examples 3.5. • For g a Lie algebra, the definition of W( b g ) reduces tothe ordinary definition of the Weil algebra. • For a = Σ an ordinary smooth manifold, W(Σ) = Ω • (Σ). • For G a Lie group with Lie algebra g acting on a manifold Σ, write Σ // g for the corresponding action Lie algebroid. Then W(Σ // g ) is the Cartan-Weil model for G -equivariant de Rham cohomology on Σ. • For a = b n R the delooping of the line Lie n -algebra, we have that W( b n R )is the free dg-algebra on a single generator c in degree n : this is the gradedalgebra on two generators c and γ , with c in degree n and γ in degree n +1,equipped with a differential defined by d W( b n R ) : c γ . The key technical notion for our main theorem is that of Chern-Simons ele-ments witnessing trangression between invariant polynomials and L ∞ -algebroidcocycles, which is def. 3.10 below. We show in Section 5 how these notions arerelated to the ∞ -Chern-Weil homomorphism for ∞ -bundles with connections. Definition 3.6.
Let a be an L ∞ -algebroid. An L ∞ -cocycle on a is an element µ ∈ CE( a ) which is closed. Examples 3.7. • For a = b g the delooping of an ordinary Lie algebra, L ∞ -cocycles on a are precisely traditional Lie algebra cocycles on g . • For n ∈ N and a = b n R the delooping of the line Lie n -algebra, there is,up to scale, precisely one nontrivial L ∞ -cocycle on a , which is in degree n . Definition 3.8. An invariant polynomial on a is an element h−i in W( a ) whichis 1. closed: d W( a ) h−i = 0;2. horizontal: an element of the subalgebra generated by the shifted elementsin the Weil algebra.6 Examples 3.9. • For a = b g the delooping of an ordinary Lie algebra,one readily checks that the above definition reproduces the traditionaldefinition of invariant polynomials. • For a = Σ a 0-Lie algebroid (a smooth manifold), an invariant polynomialis a closed differential form of positive degree. • For n ∈ N and a = b n R the delooping of the line Lie n -algebra, there is a1-dimensional vector space of invariant polynomials of degree ( n + 1) andevery other homogeneous invariant polynomial is a wedge power of these.In particular for even n all further invariant polynomials vanish. Definition 3.10.
For h−i ∈ W( a ) an invariant polynomial on an L ∞ -algebroid a , we say a cocycle µ ∈ CE( a ) is in transgression with h−i if there exists anelement cs in W( a ) such that1. d W( a ) cs = h−i ;2. i ∗ cs = µ .We say that cs is a transgression element or Chern-Simons element witnessingthis transgression.As we noticed above, if we look at an ordinary smooth manifold Σ as an L ∞ -algebroid, then the Weil algebra of Σ is the de Rham algebra Ω • (Σ). Thismotivates the following definition. Definition 3.11.
For a an L ∞ -algebroid and Σ a smooth manifold, we say amorphism A : W( a ) → Ω • (Σ)is a degree 1 a -valued differential form on Σ. Remark 3.12.
The name “degree 1 a -valued differential forms” given to dgcamorphisms W( a ) → Ω • (Σ) has the following origin: if g is a Lie algebra, then theWeil algebra W( b g ) is the free differential graded commutative algebra generatedby a shifted copy g ∗ [ −
1] of the linear dual of g . Hence a dgca morphism W( b g ) → Ω • (Σ) is precisely the datum of a morphism of graded vector spaces g ∗ [ − → Ω • (Σ), i.e., an element of Ω (Σ; g ).We say that an a -valued differential form A is flat if the morphism A :W( a ) → Ω • (Σ) factors through i ∗ : W( a ) → CE( a ). The curvature of A is theinduced morphism of graded vector spaces given by the compositeΩ • (Σ) o o A W( a ) o o ∧ V [1] : F A , where the morphism on the right is the inclusion of the linear subspace of theshifted generators into the Weil algebra. A is flat precisely if F A = 0.7 Remark 3.13.
For { x a } a coordinate chart of a and A a := A ( x a ) ∈ Ω deg( x a ) (Σ)the differential form assigned to the generator x a by the a -valued form A , wehave the curvature components F aA = A ( d x a ) ∈ Ω deg( x a )+1 (Σ) . Since d W = d CE + d , this can be equivalently written as F aA = A ( d W x a − d CE x a ) , so the curvature of A precisely measures the “lack of flatness” of A . Also noticethat, since A is required to be a dg-algebra homomorphism, we have A ( d W( a ) x a ) = d dR A a , so that A ( d CE( a ) x a ) = d dR A a − F aA . Assume now A is a degree 1 a -valued differential form on the smooth mani-fold Σ, and that cs is a Chern-Simons element transgressing an invariant poly-nomial h−i of a to some cocycle µ . We can then consider the image A (cs) ofthe Chern-Simons element cs in Ω • (Σ). Equivalently, we can look at cs as amap from degree 1 a -valued differential forms on Σ to ordinary (real valued)differential forms on Σ. Definition 3.14.
In the notations above, we writeΩ • (Σ) o o A W( a ) o o cs W( b n +1 R ) : cs( A )for the differential form associated by the Chern-Simons element cs to the degree1 a -valued differential form A , and call this the Chern-Simons differential form associated with A .Similarly, for h−i an invariant polynomial on a , we write h F A i for the eval-uation Ω • closed (Σ) o o A W( a ) o o h−i inv( b n +1 R ) : h F A i . We call this the curvature characteristic form of A with respect to h−i . n -algebroids We now consider L ∞ -algebroids that are equipped with certain natural extrastructure (symplectic structure) and show how this extra structure canonicallyinduces an invariant polynomial and hence by observation 5.15 a σ -model fieldtheory. In the next section we demonstrate that the field theories arising thisway are precisely the AKSZ σ -models.8 Definition 3.15. A symplectic Lie n -algebroid ( P , ω ) is a Lie n -algebroid P equipped with a quadratic non-degenerate invariant polynomial ω ∈ W ( P ) ofdegree n + 2.This means that • on each chart U → X of the base manifold X of P , there is a basis { x a } for CE( a | U ) such that ω = 12 d x a ω ab ∧ d x b with { ω ab ∈ R ֒ → C ∞ ( X ) } and deg( x a ) + deg( x b ) = n ; • the coefficient matrix { ω ab } has an inverse; • we have d W( P ) ω = d CE( P ) ω + d ω = 0 . The following observation essentially goes back to [Sev05] and [Royt99].
Proposition 3.16.
There is a full and faithful embedding of symplectic dg-manifolds of grade n into symplectic Lie n -algebroids.Proof. The dg-manifold itself is identified with an L ∞ -algebroid by def. 3.1.For ω ∈ Ω ( X ) a symplectic form, the conditions d ω = 0 and L v ω = 0 imply( d + L v ) ω = 0 and hence that under the identification Ω • ( X ) ≃ W( a ) this is aninvariant polynomial on a .It remains to observe that the L ∞ -algebroid a is in fact a Lie n -algebroid.This is implied by the fact that ω is of grade n and non-degenerate: the formercondition implies that it has no components in elements of grade > n and thelatter then implies that all such elements vanish. (cid:3) The following characterization may be taken as a definition of Poisson Lie alge-broids and Courant Lie 2-algebroids.
Proposition 3.17.
Symplectic Lie n -algebroids are equivalently: • for n = 0 : ordinary symplectic manifolds; • for n = 1 : Poisson Lie algebroids; • for n = 2 : Courant Lie 2-algebroids. See [Royt99, Sev05] for more discussion.
Proposition 3.18.
Let ( P , ω ) be a symplectic Lie n -algebroid for positive n inthe image of the embedding of proposition 3.16. Then it carries the canonical L ∞ -algebroid cocycle π := 1 n + 1 ι ǫ ι v ω ∈ CE( P ) which moreover is the Hamiltonian, according to definition 2.11, of d CE( P ) . Proof.
Since d ω = L v ω = 0, we have d ι ǫ ι v ω = d ι v ι ǫ ω = ( ι v d − L v ) ι ǫ ω = ι v L ǫ ω − [ L v , ι ǫ ] ω = nι v ω − ι [ v,ǫ ] ω = ( n + 1) ι v ω, where Cartan’s formula [ L v , ι ǫ ] = ι [ v,ǫ ] and the identity [ v, ǫ ] = − [ ǫ, v ] = − v havebeen used. Therefore π := n +1 ι ǫ ι v ω satisfies the defining equation d π = ι v ω from definition 2.11. (cid:3) Remark 3.19.
On a local chart with coordinates { x a } we have π (cid:12)(cid:12) U = 1 n + 1 ω ab deg( x a ) x a ∧ v b . Our central observation now is the following.
Proposition 3.20.
The cocycle n π from prop. 3.18 is in transgression with theinvariant polynomial ω . A Chern-Simons element witnessing the transgressionaccording to def. 3.10 is cs = 1 n ( ι ǫ ω + π ) . Proof.
It is clear that i ∗ cs = n π . So it remains to check that d W( P ) cs = ω .As in the proof of proposition 3.18, we use d ω = L v ω = 0 and Cartan’s identity[ L v , ι ǫ ] = ι [ v,ǫ ] = − ι v . By these, the first summand in d W( P ) ( ι ǫ ω + π ) is d W( P ) ι ǫ ω = ( d + L v ) ι ǫ ω = [ d + L v , ι ǫ ] ω = nω − ι v ω = nω − d π . The second summand is simply d W( P ) π = d π since π is a cocycle. (cid:3) Remark 3.21.
In a coordinate patch { x a } the Chern-Simons element iscs (cid:12)(cid:12) U = 1 n (cid:0) ω ab deg( x a ) x a ∧ d x b + π (cid:1) . d = d W − d CE , and this kind of substitutionwill be crucial for the proof our main statement in proposition 4.2 below. Since d CE x i = v i and using remark 3.19 we find X a ω ab deg( x a ) x a ∧ d CE x b = ( n + 1) π , and hence cs (cid:12)(cid:12) U = 1 n (cid:0) deg( x a ) ω ab x a ∧ d W( P ) x b − nπ (cid:1) . In the following section we show that this transgression element cs is theAKSZ-Lagrangian. We now show how an L ∞ -algebroid a endowed with a triple ( π, cs , ω ) consistingof a Chern-Simons element transgressing an invariant polynomial ω to a cocycle π defines an AKSZ-type σ -model action. The starting point is to take as targetspace the tangent Lie ∞ -algebroid Ta , i.e., to consider as space of fields of thetheory the space of maps Maps( T Σ , Ta ) from the worldsheet Σ to Ta . Dually,this is the space of morphisms of dgcas from W( a ) to Ω • (Σ), i.e., the space ofdegree 1 a -valued differential forms on Σ from definition 3.11. Remark 4.1.
As we noticed in the introduction, in the context of the AKSZ σ -model a degree 1 a -valued differential form on Σ should be thought of as thedatum of a (notrivial) a -valued connection on a trivial principal ∞ -bundle onΣ. We will come back to this point of view in Section 5.Now that we have defined the space of fields, we have to define the action.We have seen in definition 3.14 that a degree 1 a -valued differential form A on Σ maps the Chern-Simons element cs ∈ W( a ) to a differential form cs( A )on Σ. Integrating this differential form on Σ will therefore give an AKSZ-typeaction which, as we will see in Section 5, is naturally interpreted as an higherChern-Simons action functional:Maps( T Σ , Ta ) → R A Z Σ cs( A ) . Theorem 1.1 then reduces to showing that, when { a , ( π, cs , ω ) } is the set of L ∞ -algebroid data arising from a symplectic Lie n -algebroid ( P , ω ), the AKSZ-type action dscribed above is precisely the AKSZ action for ( P , ω ). More pre-cisely, this is stated as follows.1 Proposition 4.2.
For ( P , ω ) a symplectic Lie n -algebroid coming by proposition3.16 from a symplectic dg-manifold of positive grade n with global Darboux chart,the action functional induced by the canonical Chern-Simons element cs ∈ W( P ) from proposition 3.20 is the AKSZ action from formula (2.2.1) : Z Σ cs = Z Σ L AKSZ . In fact the two Lagrangians differ at most by an exact term cs ∼ L AKSZ . Proof.
We have seen in remark 3.21 that in Darboux coordinates { x a } where ω = 12 ω ab d x a ∧ d x b the Chern-Simons element from proposition 3.20 is given bycs = 1 n (cid:0) deg( x a ) ω ab x a ∧ d W( P ) x b − nπ (cid:1) . This means that for Σ an ( n + 1)-dimensional manifold andΩ • (Σ) ← W( P ) : φ a (degree 1) P -valued differential form on Σ we have Z Σ cs( φ ) = 1 n Z Σ X a,b deg( x a ) ω ab φ a ∧ d dR φ b − nπ ( φ ) , where we used φ ( d W( P ) x b ) = d dR φ b , as in remark 3.13. Here the asymmetry inthe coefficients of the first term is only apparent. Using integration by parts ona closed Σ we have Z Σ X a,b deg( x a ) ω ab φ a ∧ d dR φ b = Z Σ X a,b ( − x a ) deg( x a ) ω ab ( d dR φ a ) ∧ φ b = Z Σ X a,b ( − (1+deg( x a ))(1+deg( x b )) deg( x a ) ω ab φ b ∧ ( d dR φ a )= Z Σ X a,b deg( x b ) ω ab φ a ∧ ( d dR φ b ) , where in the last step we switched the indices on ω and used that ω ab =( − (1+deg( x a ))(1+deg( x b )) ω ba . Therefore Z Σ X a,b deg( x a ) ω ab φ a ∧ d dR φ b = 12 Z Σ X a,b deg( x a ) ω ab φ a ∧ d dR φ b + 12 Z Σ X a,b deg( x b ) ω ab φ a ∧ d dR φ b = n Z Σ ω ab φ a ∧ d dR φ b . . Z Σ cs( φ ) = Z Σ (cid:18) ω ab φ a ∧ d dR φ b − π ( φ ) (cid:19) , which is formula (2.2.1) for the action functional. (cid:3) Corollary 4.3.
In the hypothesis of Proposition 4.2, if N is an ( n +2) -dimensionalcompact oriented manifold with ∂N = Σ , then Z Σ L AKSZ = Z N ω ( F φ ) , where ω ( F φ ) is the symplectic form ω , seen as an invariant polynomial, evaluatedon the curvature of φ : W( P ) → Ω • (Σ) . We unwind the general statement of proposition 4.2 and its ingredients inthe central examples of interest, from proposition 3.17: the ordinary Chern-Simons action functional, the Poisson σ -model Lagrangian, the Courant σ -modelLagrangian, and the higher abelian Chern-Simons functional. (The ordinaryChern-Simons model is the special case of the Courant σ -model for P having asbase manifold the point. But since it is the archetype of all models consideredhere, it deserves its own discussion.)By the very content of proposition 4.2 there are no surprises here and thefollowing essentially amounts to a review of the standard formulas for theseexamples. But it may be helpful to see our general ∞ -Lie theoretic derivationof these formulas spelled out in concrete cases, if only to carefully track thevarious signs and prefactors. Let P = b g be a semisimple Lie algebra regarded as an L ∞ -algebroid withbase space the point and let ω := h− , −i ∈ W( b g ) be its Killing form invariantpolynomial. Then ( b g , h− , −i ) is a symplectic Lie 2-algebroid.For { t a } a dual basis for g , being generators of grade 1 in W( g ) we have d W t a = − C abc t a ∧ t b + d t a where C abc := t a ([ t b , t c ]) and ω = 12 P ab d t a ∧ d t b , P ab := h t a , t b i . The Hamiltonian cocycle π from prop. 3.18 is π = 12 + 1 ι v ι ǫ ω = 13 ι v P ab t a ∧ d t b = − P ab C bcd t a ∧ t c ∧ t d =: − C abc t a ∧ t b ∧ t c . Therefore the Chern-Simons element from prop. 3.20 is found to becs = 12 (cid:18) P ab t a ∧ d t b − C abc t a ∧ t b ∧ t c (cid:19) = 12 (cid:18) P ab t a ∧ d W t b + 13 C abc t a ∧ t b ∧ t c (cid:19) . This is indeed, up to an overall factor 1 /
2, the familiar standard choice ofChern-Simons element on a Lie algebra. To see this more explicitly, notice thatevaluated on a g -valued connection formΩ • (Σ) ← W( b g ) : A this is2cs( A ) = h A ∧ F A i − h A ∧ [ A, A ] i = h A ∧ d dR A i + 13 h A ∧ [ A, A ] i . If g is a matrix Lie algebra then the Killing form is proportional to the trace ofthe matrix product: h t a , t b i = tr( t a t b ). In this case we have h A ∧ [ A, A ] i = A a ∧ A b ∧ A c tr( t a ( t b t c − t c t b ))= 2 A a ∧ A b ∧ A c tr( t a t b t c )= 2 tr( A ∧ A ∧ A )and hence2cs( A ) = tr (cid:18) A ∧ F A − A ∧ A ∧ A (cid:19) = tr (cid:18) A ∧ d dR A + 23 A ∧ A ∧ A (cid:19) . σ -model Let ( M, {− , −} ) be a Poisson manifold and let P be the corresponding PoissonLie algebroid. This is a symplectic Lie 1-algebroid. Over a chart for the shiftedcotangent bundle T ∗ [ − X with coordinates { x i } of degree 0 and { ∂ i } of degree1, respectively, we have d W x i = − π ij ∂ j + d x i ; d W ∂ i = 12 ∂π jk ∂x i ∂ j ∧ ∂ k + d ∂ i , π ij := { x i , x j } and ω = d x i ∧ d ∂ i . The Hamiltonian cocycle from prop. 3.18 is π = 12 ι v ι ǫ ω = − π ij ∂ i ∧ ∂ j and the Chern-Simons element from prop. 3.20 iscs = ι ǫ ω + π = ∂ i ∧ d x i − π ij ∂ i ∧ ∂ j . In terms of d W instead of d this iscs = ∂ i ∧ d W x i − π = ∂ i ∧ d W x i + 12 π ij ∂ i ∂ j . So for Σ a 2-manifold and Ω • (Σ) ← W( P ) : ( X, η )a Poisson-Lie algebroid valued differential form on Σ – which in components isa function X : Σ → M and a 1-form η ∈ Ω (Σ , X ∗ T ∗ M ) – the correspondingAKSZ action is Z Σ cs( X, η ) = Z Σ η ∧ d dR X + 12 π ij ( X ) η i ∧ η j . This is the Lagrangian of the Poisson σ -model [Ike94, ScSt94a, CaFe01]. σ -model A Courant algebroid is a symplectic Lie 2-algebroid. By the previous examplethis is a higher analog of a Poisson manifold. Expressed in components inthe language of ordinary differential geometry, a Courant algebroid is a vectorbundle E over a manifold M , equipped with: a non-degenerate bilinear form h· , ·i on the fibers, a bilinear bracket [ · , · ] on sections Γ( E ), and a bundle map(called the anchor) ρ : E → T M , satisfying several compatibility conditions.The bracket [ · , · ] may be required to be skew-symmetric (Def. 2.3.2 in [Royt99]),in which case it gives rise to a Lie 2-algebra structure, or, alternatively, it maybe required to satisfy a Jacobi-like identity (Def. 2.6.1 in [Royt99]), in whichcase it gives a Leibniz algebra structure.It was shown in [Royt99] that Courant algebroids E → M in this com-ponent form are in 1-1 correspondance with (non-negatively graded) grade 2symplectic dg-manifolds ( M, v ). Via this correspondance, M is obtained asa particular symplectic submanifold of T ∗ [2] E [1] equipped with its canonicalsymplectic structure.5Let ( M, v ) be a Courant algebroid as above. In Darboux coordinates, thesymplectic structure is ω = d p i ∧ d q i + 12 g ab d ξ a ∧ d ξ b , with deg q i = 0 , deg ξ a = 1 , deg p i = 2 , and g ab are constants. The Chevalley-Eilenberg differential corresponds to thevector field: v = P ia ξ a ∂∂q i + g ab (cid:0) P ib p i − T bcd ξ c ξ d (cid:1) ∂∂ξ a + (cid:18) − ∂P ja ∂q i ξ a p j + 16 ∂T abc ∂q i ξ a ξ b ξ c (cid:19) ∂∂p i . Here P ia = P ia ( q ) and T abc = T abc ( q ) are particular degree zero functions encod-ing the Courant algebroid structure. Hence, the differential on the Weil algebrais: d W q i = P ia ξ a + d q i d W ξ a = g ab (cid:0) P ib p i − T bcd ξ c ξ d (cid:1) + d ξ a d W p i = − ∂P ja ∂q i ξ a p j + 16 ∂T abc ∂q i ξ a ξ b ξ c + d p i . Following remark. 3.19, we construct the corresponding Hamiltonian cocyclefrom prop. 3.18: π = 1 n + 1 ω ab deg( x a ) x a ∧ v b = 13 (cid:0) p i ∧ v ( q i ) + g ab ξ a ∧ v ( ξ b ) (cid:1) = 13 (cid:0) p i P ia ξ a + ξ a P ia p i − T abc ξ a ξ b ξ c (cid:1) = P ia ξ a p i − T abc ξ a ξ b ξ c . The Chern-Simons element from prop. 3.20 is:cs = 12 X ab deg( x a ) ω ab x a ∧ d W x b − π ! = p i d W q i + 12 g ab ξ a d W ξ b − π = p i d W q i + 12 g ab ξ a d W ξ b − P ia ξ a p i + 16 T abc ξ a ξ b ξ c . So for a Courant Lie 2-algebroid valued differential form datumΩ • (Σ) ← W( P ) : ( X, A, F )6on a closed 3-manifold Σ, we have Z Σ cs( X, A, F ) = Z Σ F i ∧ d dR X i + 12 g ab A a d dR A b − P ia A a F i + 16 T abc A a A b A c . This is the AKSZ action for the Courant algebroid σ -model from [Ike03, HoPa04,Royt07]. d = 4 k + 3For k ∈ N , let a be the delooping of the line Lie 2 k -algebra: a = b k +1 R . Byexamples 3.9 there is, up to scale, a unique binary invariant polynomial on b k +1 R , and this is the wedge product of the unique generating unary invariantpolynomial γ in degree 2 k + 2 with itself: ω := γ ∧ γ ∈ W( b k +4 R ) . This invariant polynomial is clearly non-degenerate: for c the canonical gener-ator of CE( b k +1 R ) we have ω = d c ∧ d c . Therefore ( b k +1 R , ω ) induces an AKSZ σ -model in dimension n + 1 = 4 k + 3.(On the other hand, on b k R there is only the 0 binary invariant polynomial, sothat no AKSZ- σ -models are induced from b k R .)The Hamiltonian cocycle from proposition 3.18 vanishes π = 0because the differential in the Chevalley-Eilenberg algebra CE( b k +1 R ) is trivial.The Chern-Simons element from proposition 3.20 iscs = c ∧ d c . A field configuration (definition 3.11)Ω • (Σ) ← W( b k +1 ) : C of this σ -model over a (4 k +3)-dimensional manifold Σ is simply a (2 k +1)-form.The AKSZ action functional in this case is S AKSZ : C Z Σ C ∧ d dR C .
The simplicity of this discussion is deceptive. In terms of the outlook in Section6 below, it results from the fact that in AKSZ theory we are only looking at ∞ -Chern-Simons theory for universal Lie integrations and for topologically trivial ∞ -bundles. The ∞ -Chern-Simons theory for a = b k +1 R becomes nontrivialand rich when one drops these restrictions. Then its configuration space is thatof circle (2 k + 1) -bundles with connection on Σ (abelian 2 k -gerbes), classifiedby ordinary differential cohomology in degree 2 k + 2, and the action functional7is given by the fiber integration along the projection to a point of the Beilinson-Deligne cup product in differential cohomology, which is locally given by theabove formula, but contains global twists. This is discussed in depth in [HoSi05].Once globalized this way, the above action functional is the action functionalof higher U (1)-Chern-Simons theory in dimention 4 k +3. In dimension 3 ( k = 0)this is discussed for instance in [GT08] (notice that U (1) is not simply connected,whence even in this dimension there is a refinement of the standard story). Indimension 7 ( k = 1) this higher Chern-Simons theory is the system whoseholographic boundary theory encodes the self-dual 2-form gauge theory on thefivebrane [Wi97]. Generally, for every k the (4 k + 3)-dimensional abelian Chern-Simons theory induces a self-dual higher gauge theory holographically on itsboundary, see [BeMo06]. We indicate now the roots of the construction of the higher Chern-Simons actionfunctionals discussed above in a more encompassing general theory. We referthe reader to [Sch10, FSS10] for details on this section.The first step in this identification involves the
Lie integration of an L ∞ -algebroid a to a smooth ∞ -groupoid exp( a ) in analogy to how a Lie algebraintegrates to a Lie group. This in turn involves two aspects: the notion of a bare ∞ -groupoid on the one hand, and its smooth structure on the other.Bare ∞ -groupoids are presented by Kan complexes : simplicial sets such thatfor all adjacent k -cells there exists a composite k -cell, and such that every k -cellhas an inverse, up to ( k + 1)-cells, under this composition. For instance for G any ordinary groupoid there is such a Kan complex whose 0-cells are the objectsof the groupoid, and whose k -cells are the sequences of composable k -tuples ofmorphisms of the groupoid.These bare ∞ -groupoids are equipped with geometric structure by providinga rule for what the geometric families of k -cells in the ∞ -groupoid are supposedto be. In this sense a smooth structure on an ∞ -groupoid A is given by declar-ing for each Cartesian space U = R n a set A k ( U ) of smooth U -parameterizedfamilies of k -morphisms in A , for k, n ∈ N . Collecting this data for all k and all U produces a functor A : CartSp op → sSet U ([ k ] A k ( U ))from the opposite of the category of Cartesian spaces to the category of simplicialsets – a simplicial presheaf – and this functor encodes the structure of a smooth ∞ -groupoid.For instance if A = ( A / / / / A ) is an ordinary Lie groupoid, with asmooth manifold of objects A and a smooth manifold of morphisms A , thisassignment is given in the two lowest degrees by sending U to the set of smoothfunctions from U to the spaces of objects and morphisms: A : U C ∞ ( U, A k ).8 Definition 5.1. A smooth ∞ -groupoid is a simplicial presheaf on the categoryof Cartesian spaces and smooth functions between them, A : CartSp op → sSet , such that for each U ∈ CartSp, the simplicial set A ( U ) is a Kan complex.A morphism f : A → A of smooth ∞ -groupoids is a morphism of theunderlying simplicial presheaves (a natural transformation of functors). A mor-phism is an equivalence of smooth ∞ -groupoids if it is stalkwise a weak homotopyequivalence of Kan complexes. Remark 5.2.
Here the category of Cartesian spaces is just the simplest ofmany possible choices. It can be varied at will, corresponding to which kindof geometric structure the ∞ -groupoids are to be equipped with. For instancewe can equivalently take instead the full category of smooth manifolds, withoutchanging the notion of smooth ∞ -groupoid, up to equivalence. We could alsotake richer categories, such as that of smooth dg-manifolds. For non-positively -graded dg-manifolds we would speak of derived smooth ∞ -groupoids in thiscase. These are necessary for discussion of the Lie integration of the full AKSZBV-action, as opposed to just the grade-0 functional that we concentrate onhere. Remark 5.3.
It turns out that under the above notion of equivalence, every simplicial presheaf is equivalent to one that is objectwise a Kan complex. In amore abstract discussion than we want to get at here, we would more naturallysay that : the ∞ -category of smooth ∞ -groupoids is the simplicial localization L W [CartSp op , sSet] at the stalkwise weak equivalences ([Sch10]).When regarded as simplicial presheaves on smooth test spaces, smooth ∞ -groupoids have a canonical construction from L ∞ -algebroids by what is a pa-rameterized version of the classical Sullivan construction in rational homotopytheory: the original construction [Sul77] sends – in our ∞ -Lie theoretic language– an L ∞ -algebroid a to the simplicial setexp( a )( ∗ ) : [ k ] Hom cdgAlg R (CE( a ) , Ω • (∆ k )) , whose k -cells are the flat a -valued differential forms on the k -simplex (recalldefinition 3.11). It was noticed in [Hin97, Get09] (for the special case of L ∞ -algebras) that this construction deserves to be understood as forming the dis-crete ∞ -groupoid that underlies the Lie integration of a . In [Henr08] the objectexp( a ), still for the case that a = b g comes from an L ∞ -algebra, is observed tobe naturally equipped with a Banach manifold structure. Moreover, a detaileddiscussion is given showing that the truncations τ n exp( a ) (the decategorifica-tion of the ∞ -groupoid to an n -groupoid) corresponds to the Lie integrationto an n -group. For instance for a = b g coming from an ordinary Lie algebra, τ exp( b g ) is B G : the one-object groupoid corresponding to the classical simplyconnected Lie group integrating g . A detailed discussion of the smooth structureof τ exp( a ) for the case that a is a Lie 1-algebroid was given in [CrFe03]. There9it is found that a certain cohomological obstruction has to vanish in order thatthis is a genuine Lie groupoid coming from a simplicial smooth manifold. In[TsZh06] it was pointed out that however for Lie 1-algebroids a the 2-truncation τ exp( a ) is always a simplicial manifold.In [FSS10] it was observed that without any assumption on a and the trun-cation degree, the construction always naturally – and usefully – extends tosmooth structure as encoded by presheaves on Cartesian test-spaces, simplyby declaring the U -parameterized families of k -cells in exp( a ) to be given by U -parameterized families of flat a -valued connections: Definition 5.4.
For a an L ∞ -algebroid, the functorexp( a ) : CartSp op → sSetto the category of simplicial sets is defined by setting, for U ∈ CartSp and k ∈ N , exp( a ) : ( U, [ k ]) (cid:26) Ω • vert , si ( U × ∆ k ) o o A vert CE( a ) (cid:27) , where ∆ k is the standard realization of the k -simplex as a smooth manifoldwith boundary and corners, and where Ω • vert , si ( U × ∆ k ) is the dg-algebra of vertical differential forms on U × ∆ k → U , that have sitting instants towardsthe boundary faces of the simplex (see [FSS10] for details).We say that this simplicial presheaf presents the universal Lie integration of a .This can be understood as saying that the Lie integration of a always exists asa diffeological ∞ -groupoid [BaHo11]. Remark 5.5.
The simplicial presheaf exp( a ) can naturally be thought of asthe presheaf of U -points of the simplicial set [ k ] Hom cdgAlg R (CE( a ) , Ω • (∆ k ))described above. Indeed, the dg-algebra of vertical differential forms on U × ∆ k isnaturally isomorphic to CE( U × T ∆ k ). Also note that this is in turn isomorphicto the (completed) tensor product CE( U ) ˆ ⊗ CE( T ∆ k ) = C ∞ ( U ) ˆ ⊗ Ω • (∆ k ).Indeed, the simplicial presheaf given in definition 5.4 is a Lie ∞ -groupoid inthe sense of definition 5.1. Proposition 5.6.
For a an L ∞ -algebroid, the simplicial presheaf exp( a ) is aLie ∞ -groupoid (is objectwise a Kan complex).Proof. Since the differential forms in the above definition are required tohave sitting instants, they can be smoothly pulled back along the standard con-tinuous retract projections ∆ n → Λ ni of the n -horns, because these are smoothaway from the boundary. This provides horn fillers in the standard way. Seealso the proof of Proposition 4.2.10 in [FSS10]. Remark 5.7.
While it can be useful in specific computations to know thatexp( a ) is degreewise a smooth manifold, if indeed it is, no general concept insmooth higher geometry requires this assumption. On the other hand, onecan show [Sch10], that every smooth ∞ -groupoid is equivalent to a simplicialpresheaf that is degreewise a disjoint union of smooth manifolds, even to onethat is degreewise a disjoint union of Cartesian spaces.0 Remark 5.8.
A category with weak equivalences, such as that of smooth ∞ -groupoids, is canonically equipped with a derived hom-functor , which to smooth ∞ -groupoids X and exp( a ) assigns an ∞ -groupoid RHom ( X, exp( a )). Onefinds that the objects of this ∞ -groupoid are ˇCech cocycles for principal ∞ -bundles P → X that are modeled on a in higher analogy of how an ordinarysmooth principal bundle is “modeled on” the Lie algebra of its structure group.The 1-morphisms in RHom ( X, exp( a )) are the gauge transformations of theseprincipal ∞ -bundles, and so on [NSS, Sch10].Note that in the definition of exp( a ) only the Chevalley-Eilenberg algebra of a is relevant. The Weil algebra W( a ) is then introduced in order to describe adifferential refinement of exp( a ). Definition 5.9.
For a an L ∞ -algebroid write exp( a ) diff for the simplicial presheafgiven by exp( a ) diff : ( U, [ k ]) Ω • vert , si ( U × ∆ k ) vert o o A vert CE( a )Ω • ( U × ∆ k ) O O o o ( A,F A ) W( a ) O O , where on the right we have the set of horizontal dg-algebra homomorphims thatmake the square commute, as indicated.Notice that by definition 3.11 the bottom horizontal morphisms on the rightare a -valued differential forms on U × ∆ k . Proposition 5.10.
The canonical projection morphism exp( a ) diff → exp( a ) to the Lie integration from definition 5.4 is an equivalence of smooth ∞ -groupoids. Remark 5.11.
It is via this property that the Weil algebra serves in ∞ -Chern-Weil theory as part of a resolution of exp( a ) from which curvature characteristicsare built.Let now h−i be an invariant polynomial on a ( definition 3.8). Evaluating h−i on the curvature F A of an a -connection A gives a closed differential form h F A i on U × ∆ k , according to definition 3.14. This differential form, however, willin general not descend to the base space U . This naturally leads to consideringthe following definition, which picks the universal subobject of exp( a ) diff thatmakes all curvature characteristic forms h F A i descend to base space.1 Definition 5.12.
Define the simplicial presheafexp( a ) conn : ( U, [ k ]) Ω • vert , si ( U × ∆ k ) o o A vert CE( a )Ω • ( U × ∆ k ) O O o o A W( a ) O O Ω • ( U ) closed O O o o h F A i inv( a ) O O , where on the right we have the set of a -valued forms A on U × ∆ k that makethe diagram commute as indicated.This has the following interpretation (first considered in [SSS09a]):1. The commutativity of the top diagram say that the a -valued differentialform A on U × ∆ k is vertically flat with respect to the trivial simplexbundle U × ∆ → U . This is an analogue of the verticality condition foran ordinary Ehresmann connection .2. The commutativity of the lower diagram says that all curvature forms F A transform covariantly along the simplices in such a way as to makeall the curvature characteristic form h F A i descent to base space. Thisis the analogue of the horizonatlity condition on an ordinary Ehresmannconnection.One finds therefore that for X a smooth manifold, an element in RHom ( X, exp( a ) conn )is 1. a choice of good open cover { U i → X } ;2. on each patch U i differential form data A i with values in a ;3. on each double intersection a choice of 1-parameter gauge transformationbetween the corresponding differential form data;4. on each triple intersection a choice of 2-parameter gauge-of-gauge trans-formation;5. and so on.Such a differential ˇCech cocycle is essentially what defines an ∞ -connection ona principal ∞ -bundle. This is discussed in detail in [FSS10, Sch10].2 Remark 5.13.
Since for the discussion of the simple case of AKSZ σ -modelswe can assume that the underlying ∞ -bundle is trivial , only a single 0-simplex C ∞ (Σ) o o A vert CE( a )Ω • (Σ) O O o o A W( a ) O O Ω • (Σ) closed O O o o h F A i inv( a ) O O is involved in the description of the AKSZ σ -model.With these concepts in hand, we can now explain how the datum of a triple( µ, cs , h−i ) consisting of a Chern-Simons element witnessing the transgressionbetween an invariant polynomial and a cocyle (definition 3.10) serves to presenta differential characteristic class in terms of a morphism of smooth ∞ -groupoids.To see this, recall that the line delooping L ∞ -algebroid b n +1 R of the Lie line( n + 1)-algebra is defined by the fact that CE( b n +1 R ) is generated over R froma single generator in degree n + 1 with vanishing differential. As an immediateconsequence, an ( n + 1)-cocylce µ on an L ∞ -algebroid a is the same thing as adg-algebra morphism µ : CE( b n +1 R ) → CE( a ) . Similarly, a triple ( µ, cs , h−i ) is naturally identified with a commutative diagramof dg-algebras: CE( a ) o o µ CE( b n +1 R )W( a ) o o cs O O W( b n +1 R ) O O inv( a ) o o h−i O O inv( b n +1 R ) O O . Pasting this diagram to the one above defining exp( a ) conn leads to the followingobservation, discussed in [FSS10]. Proposition 5.14.
Every triple ( µ, cs , h−i ) induces a morphism exp( a ) conn → exp( b n +1 R ) conn . This morphism is in fact the presentation of the ∞ -Chern-Weil homomorphisminduced by the invariant polynomial h−i . This means that for( ∇ : X → exp( a ) conn ) ∈ RHom ( X, exp( a ) conn )3an a -valued ∞ -connection, the composite X ∇ −→ exp( a ) conn exp(cs) −−−−→ exp( b n +1 R ) conn is a representative of the curvature ( n + 1)-form on X that the ∞ -Chern-Weilhomomorphism induced by h−i assigns to ∇ .We can now formalize the observation mentioned in the introduction, thatthe Chern-Weil homomorphism plays the role of action functional for σ -modelquantum field theories. Indeed, in view of the above constructions, the AKSZ σ -model Lagrangian corresponds to forming the following commutative diagram: C ∞ (Σ) CE( P ) φ vert o o CE( b n +1 R ) n π o o : n π ( φ vert )Ω • (Σ) O O W( P ) O O φ o o W( b n +1 R ) cs o o O O : cs( φ )Ω • (Σ) closed O O inv( P ) F φ o o O O inv( b n +1 R ) O O ω o o : ω ( F φ ) . In other words, under the identification of the AKSZ action functional as aninstance of the ∞ -Chern-Weil homomorphism we indeed translate concepts asshown in the table in the introduction: the symplectic form is the invariantpolynomial that induces the Chern-Weil homomorphism, the Hamiltonian is thecocycle that it transgresses to, and the Chern-Simons element that witnesses thetransgression is the Lagrangian.This suggests the following general definition of a higher Chern-Simons fieldtheory. Definition 5.15.
Let Σ be an ( n + 1)-dimensional compact smooth manifoldand a an L ∞ -algebroid equipped with an invariant polynomial h−i . Let cs bea Chern-Simons element witnessing its transgression to a cocycle µ . Then waymay say • A morphism φ : Σ → exp( a ) conn is a field configuration on Σ with valuesin a . • The assignment φ cs( φ ) ∈ Ω n +1 (Σ)is the Lagrangian defined by cs; • The assignment φ Z Σ cs( φ ) ∈ R is the action functional defined by cs.The collection of these notions we call the higher Chern-Simons field theory defined by cs.4 The identification of the AKSZ action functionals as a special case of the generalabstract Chern-Weil homomorphism allows to tranfer various insights about thegeneral theory and about its other special cases to AKSZ theory. We closethis article by briefly indicating a few. More detailed discussion shall be givenelsewhere. n -groupoids and nontrivial topology The smooth ∞ -groupoid exp( P ) that integrates a symplectic Lie n -algebroid(as discussed in 5) is the “higher universal” Lie integration of P . One finds(see [Sch10] for the discussion in the case of smooth ∞ -groupoids, following thediscussion of Banach- ∞ -groupoids in [Henr08]) that its geometric realization intopological spaces is ∞ -connected (hence: contractible) in analogy to how theclassical universal Lie integration of a Lie algebra is 1-connected (hence: simplyconnected).As we have shown here, this is sufficient for the traditional description ofAKSZ σ -models. But more generally one will be interested in the universal inte-gration to the smooth n -groupoid P := τ n exp( P ) obtained as the n -truncation (where one retains only equivalence classes of n -cells in exp( P )).For instance for P = b g the delooping of a semisimple Lie algebra g (the caseof the Courant Lie 2-algebroid over the point) we have that τ exp( b g ) ≃ B G isthe one-object Lie groupoid obtained from the simply-connected Lie group thatintegrates g , while the untruncated exp( b g ) is some higher extension of B G byhigher abelian ∞ -groups.This truncation, however, also affects the coefficient object of the ∞ -Chern-Weil homomorphism (discussed in detail in [FSS10]): notably the untruncatedAKSZ action functionalexp(cs ω ) : exp( P ) conn → exp( b n +1 R ) conn ≃ B n +1 R conn descends to the truncation only up to a quotient by the group K ⊂ R of periods of the hamiltonian cocycle π :exp(cs ω ) : τ n exp( P ) conn → B n +1 R /K conn . Typically we have K ≃ Z and hence R /K ≃ U (1). This way the properly trun-cated AKSZ action functional indeed takes values in circle n + 1-bundles. Thisbecomes a further quantization condition for the field configurations, discussedin the next item. ∞ -Connections on nontrivial a -principal ∞ -bundles:AKSZ instantons As we have shown in this article, the fields of AKSZ σ -model theories may beunderstood as ∞ -connections on trivial exp( P )-principal ∞ -bundles or, by theprevious paragraph, on trivial τ n exp( P )-principal ∞ -bundles.5The general theory of [FSS10, Sch10] provides also a description of ∞ -connections on non-trivial such ∞ -bundles and there is no reason to restrictattention to the trivial ones. Alternatively, as propagated by Kotov and Strobl,one can use the notion of (possibly non-trivial) Q-bundles, in which connectionsturn out to be sections in the category of graded manifolds [KoSt07]. The re-lation of our ∞ -bundles to their Q-bundles generalizes the one of an ordinaryprincipal bundle to its associated Atiyah algebroid. Such fields with non-trivialunderlying principal ∞ -bundles correspond to what in the analog situation ofYang-Mills theory are called instanton field configurations. These are of impor-tance in a comprehensive discussion of the quantum theory.This issue plays only a minor role in low dimensions. For instance the reasonthat the fields of Chern-Simons theory are and can be taken to be connectionson trivial G -principal bundles on Σ is that for simply connected Lie groups G the classifying space BG has its first non-trivial homotopy group in degree 4, sothat all G -principal bundles on a 3-dimensional Σ are necessarily trivializable.But by the same argument there are inevitably AKSZ instanton contribu-tions from fields that are connections on non-trivial ∞ -bundles as soon as wepass to 4-dimensional AKSZ models and beyond. L ∞ -algebroids For any differential characteristic classˆ c : A conn → B n +1 R /K conn such as obtained from Lie integration of a Chern-Simons element:exp(cs) : τ n exp( a ) conn → B n +1 R /K it is of interst to study the homotopy fibers that this induces on cocycle ∞ -groupoids over a given base space X . In [SSS09b, FSS10] the ∞ -groupoidˆ c Struc( X ) of twisted ˆ c -structures is introduced as the homotopy pullbackˆ c Struc tw ( X ) tw / / (cid:15) (cid:15) H n diff ( X, K ) (cid:15) (cid:15) RHom ( X, A conn ) ˆ c / / RHom ( X, B n +1 K conn ) , where the right vertical morphisms – unique up to equivalence – picks onecocycle representative in each cohomology class. The morphism tw sends agiven twisted differential cocycle to its twist . The fibers over the trivial twist areprecisely the b A -principal ∞ -bundles with connection, where b A is the extensionof A classified by c , wich is characterized by the fact that it sits in a fibersequence B n K → b A → A . c = p is a smoothrefinement of the first fractional Pontryagin class and for the case c = p of the fractional second Pontryagin class. In these cases the extension ˆ A isthe delooping, respectively, of the smooth string 2-group and of the smooth fivebrane 6-group . The corresponding twisted differential string-structures and twisted differential fivebrane structures are shown there (following [SSS09b])to encode the Green-Schwarz mechanism in heterotic string theory and dualheterotic string theory, respectively.By our discussion here, all these constructions have their natural analogsfor AKSZ σ -models, too. In particular, every symplectic Lie n -algebroid ( P , ω )with Hamiltonian π ∈ CE( P ) has a canonical (“string-like”) extension b n R → b P → P classified by π regarded as an L ∞ -cocycle π : P → b n +1 R .This extension is easy to describe: the Chevalley-Eilenberg algebra CE( b P )is that of P with a single generator b in degree n adjoined, and the differentialextended to this generator by the formula d CE( b P ) : b π . A twisted differential exp( P ) -structure is accordingly an exp( P )- ∞ -connection φ (an AKSZ σ -model field) equipped with an equivalence of its curvature char-acteristic ω ( φ ) to a presribed “twisting class”. When the twisting class is trivial,then these are equivalently exp( b P )-principal ∞ -connections.Notice that these considerations are relevant only over a base space of dimen-sion at least ( n + 2). Compare this again to the familiar case of Chern-Simonstheory: in Chern-Simons theory itself the G -principal bundle may always betaken to be trivial, since the base space Σ is taken to be 3-dimensional. Butall the constructions of Chern-Simons theory make sense also over arbitrary X . Generally, the Chern-Weil homomorphism assigns a Chern-Simons circle3-bundle to every suitabe G -principal bundle on X , and its volume holonomy isthe corresponding Chern-Simons functional for this situation. Analogously onecan consider AKSZ theory over higher dimensional base spaces. AKSZ theory is not the only class of field theories where it was noticed thatfield configurations have an interpretation in terms of morphisms of dg-algebras.Almost two decades earlier originates the observation that (higher dimensional) supergravity (see for instance [DM99] for standard itroductions) has a ratherbeautiful description in such terms. A detailed exposition of this dg-algebraicapproach to supergravity is in the textbook [CaDAFr91].The authors there speak of “free differential algebras” (“FDAs”) where theywould mean what in the mathematical literature are called “quasi-free dg-algebras” or “semi-free dg-algebras” – those whose underlying graded algebra is7free, as in our definition 3.1 of Chevalley-Eilenberg algebras. Moreover, what wehere observe are morphisms out of the Weil algebra (definition 3.11) they call“soft group manifolds”. But apart from these purely terminological differencesone finds that the observations that drive the developments there are preciselythe following, here reformulated in our ∞ -Lie theoretic language (see section 4of [Sch10]):First of all it is a standard fact that in “first order formulation” the fieldconfigurations of gravity in d +1 dimensions are naturally presented by iso ( d, iso ( d,
1) = R d +1 ⋉ so ( d,
1) is the Poincar´e-Liealgebra. This perspective is inevitable in the context of supergravity, wherethe first-order formulation is required by the coupling to fermions. There is anevident super-algebra generalization of the Poincar´e Lie algebra to the super-Poincar´e -Lie algebra siso ( d,
1) and a field configuration of supergravity is an siso ( d, graviton field and its superpartner field, the gravitino field, but not yet the higher bosonicform fields generically present in higher supergravity theories. The first centralobservation of [DAFr82] is (in our words) that these naturally appear afterpassage to L ∞ -extensions of siso ( d, L ∞ -algebroids are generalized to super L ∞ -algebroids as Lie algebrasare generalized to super Lie algebras (see section 3.5 of [Sch10]).It turns out that super-Lie algebra cohomology of siso ( d,
1) contains a certainfinite number of exceptional cocycles µ : siso ( d, → b n +1 R for certain valuesof d , whose existence is naturally understood from the existence of the fournormed division algebras ([Hu11]). In particular, for d = 10 there is a 4-cocycle µ : siso (10 , → b R . The Lie 3-algebra extension that it classifies b R → sugra → siso (10 , supergravity Lie 3-algebra in [SSS09a]. It turns out ([DAFr82])that this carries, in turn, a 7-cocycle µ : sugra → b R . The original obser-vation of [DAFr82] was (not in these words, though, but easily translated into ∞ -Lie theoretic terms using our discussion here) that 11-dimensional supergrav-ity, including its higher form field degrees of freedom, is naturally understoodas a theory of ∞ -connections with values in the corresponding supergravity Lie6-algebra b R → \ sugra → sugra and that the construction of its action functional is governed by the higher Lietheory of this object.While 11-dimensional supergravity is not entirely a higher Chern-Simons-theory, it crucially does involve Chern-Simons terms in its action functionals.Indeed, one can see that one of the characterizing conditons on a supergravityaction functional – the one called the cosmo-cocycle condition in [CaDAFr91] –is the defining condition on a Chern-Simons element in our def. 3.10, but solvedonly up to first order in the curvature terms. It may be noteworthy in this8context that there are various speculations (see [BaTrZa96] for discussion andreview) that higher dimensional supergravity should be thought of as a limitingtheory of a genuine higher Chern-Simons theory.This shows that there is a close conceptual relation between AKSZ σ -models,higher Chern-Simons theories and higher dimensional supergravity, mediated byabstract higher Chern-Weil theory. The various implications of this observationshall be explored elsewhere. References [AKSZ97] M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky,
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