infinity-Chern-Simons theory

This is a page with notes, produced and being produced together with various collaborators, on ”\infty-Chern-Simons theory” in the context of differential cohomology in a cohesive topos.

See below for Surveys, expositions and lecture notes and for Related projects.



We observe that in higher geometry and specifically in higher differential geometry the traditional notion of Chern-Simons theory generalizes naturally to a very general class of field theories (prequantum field theories) that on the one hand are of profound conceptual nature – they are precisely the universal characteristic classes in differential cohomology over higher moduli stacks – and on the other hand subsume a fairly encompassing list of fundamental field theories, not all of which are traditionally recognized as being of Chern-Simons type. This is true in particular if all boundary field theories and defects field theories for \infty-Chern-Simons theories are also taken into account, which then subsumes also Wess-Zumino-Witten theory, Wilson line-theories and their higher and \infty- analogs.

In fact, \infty-Chern-Simons theories turn out to be themselves precisely the boundary field theories for a single prequantum field theory which we call the “universal higher topological Yang-Mills theory”. This is further discussed at Higher Chern-Simons local prequantum field theory.

More in technical detail:

The action functional of ordinary Chern-Simons theory for a simple Lie group GG may be understood as being the volume holonomy of the Chern-Simons 2-gerbe with connection that the refined Chern-Weil homomorphism assigns to any connection on a GG-principal bundle.

We observe that all the ingredients of this statement have their general abstract analogs in any cohesive (∞,1)-topos H\mathbf{H}: for any cohesive ∞-group GG and any representative c:BGB nA\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n A of a characteristic class for GG there is canonically the induced ∞-Chern-Weil homomorphism L c:H conn(,BG)H diff n()L_{\mathbf{c}} : \mathbf{H}_{conn}(-,\mathbf{B}G) \to \mathbf{H}_{diff}^n(-) that sends intrinsic G-connections to cocycles in intrinsic differential cohomology with coefficients in AA. This may be thought of as the Lagrangian of the \infty-Chern-Simons theory induced by c\mathbf{c}.

In the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids we give a natural general abstract procedure for integration of L cL_{\mathbf{c}} over an nn-dimensional parameter space ΣH\Sigma \in \mathbf{H}. The resulting smooth function exp(S c):[Σ,BG conn]U(1)\exp(S_{\mathbf{c}}) : [\Sigma,\mathbf{B}G_{conn}] \to U(1) is the exponentiated action functional of \infty-Chern-Simons theory on the smooth ∞-groupoid of field configurations. It may be regarded itself as a degree-0 characteristic class on the space of field configurations. As such, its differential refinement dexp(S c):[Σ,BG conn] dRBU(1)d \exp(S_{\mathbf{c}}) : [\Sigma, \mathbf{B}G_{conn}] \to \mathbf{\flat}_{dR} \mathbf{B}U(1) is the Euler-Lagrange equation of the theory. Its homotopy fiber is the smooth ∞-groupoid of classical solutions: the derived phase space.

We show that this construction subsumes the action functional of ordinary Chern-Simons theory, which is the one induced for c\mathbf{c} the Lie integration of the canonical Lie algebra 3-cocycle on GG. We observe that there is a general abstract notion of ∞-Lie algebroid cohomology, each cocycle in which induces after Lie integration one type of \infty-Chern-Simons theory, by the above. Among the action functionals obtained this way are those of BF-theory coupled to topological Yang-Mills theory, of all versions of AKSZ theory including the Poisson sigma-model and the Courant sigma model in lowest degree as well as Chern-Simons supergravity.

More on the general context of \infty-Chern-Simons theory is at higher category theory and physics.

Introduction and overview

We give here a leisurely introduction to the ideas of \infty-Chern-Simons theory and survey some of the constructions and results that are discussed in detail below.

  1. Ordinary Chern-Simons theory from refined Chern-Weil theory

    A brief review of ordinary Chern-Weil theory and its relation to ordinary Chern-Simons theory.

  2. Deficiencies of ordinary Chern-Weil theory

    A brief indication of what classical Chern-Weil theory fails to accomplish, as a motivation for the generalization to follow.

  3. General abstract theory

    A survey of the general abstract formulation and generalization of Chern Weil theory in terms of cohesive (∞,1)-topos theory.

  4. Construction by Lie integration of L-∞-Chern-Simons elements

    Key ingredients in the explicit realization of the general abstract construction in terms of Lie integration of L-∞ algebra cohomology.

  5. Examples

    A brief list of classes of gauge theory Lagrangians that are examples of \infty-Chern-Simons theory, and some of their properties.

Ordinary Chern-Simons theory from refined Chern-Weil theory

What is called Chern-Weil theory is a theory of refining characteristic classes of principal bundles from ordinary cohomology to ordinary differential cohomology.

Concretely, the Chern-Weil homomorphism is presented by the following simple construction:


we get for each smooth manifold XX an assignment

[c]:GBund(X) H n+1(X,) [c] : G Bund(X)_\sim \to H^{n+1}(X,\mathbb{Z})

of integral cohomology classes of base space to equivalence classes of GG-principal bundles by sending a bundle classified by a map f:XBGf : X \to B G to the class [f *c][f^* c].

Let [c] H n+1(BG,)[c]_\mathbb{R} \in H^{n+1}(B G, \mathbb{R}) be the image of [c][c] in real cohomology, induced by the evident inclusion of coefficients \mathbb{Z} \hookrightarrow \mathbb{R}.

The first main statement of Chern-Weil theory is that there is an invariant polynomial

:=ϕ 1[c] \langle- \rangle := \phi^{-1} [c]_{\mathbb{R}}

on the Lie algebra 𝔤\mathfrak{g} of GG associated to [c] [c]_{\mathbb{R}}, given by an isomorphism (of real graded vector space)s

ϕ:inv(𝔤)H (BG,). \phi : inv(\mathfrak{g}) \stackrel{\simeq}{\to} H^\bullet(B G, \mathbb{R}) \,.

The second main statement is that this invariant polynomial serves to provide a differential (Lie integration) construction of [c] [c]_{\mathbb{R}}:

for any choice of connection \nabla on a GG-principal bundle PXP \to X we have the curvature 2-form F Ω 2(P,𝔤)F_\nabla \in \Omega^2(P, \mathfrak{g}) and fed into the invariant polynomial this yields an (n+1)(n+1)-form

F F Ω n+1(X). \langle F_\nabla \wedge \cdots \wedge F_\nabla \rangle \in \Omega^{n+1}(X) \,.

The statement is that under the de Rham theorem-isomorphism H dR (X)H (X,)H^\bullet_{dR}(X) \simeq H^\bullet(X, \mathbb{R}) this presents the class [c] [c]_{\mathbb{R}}.

The third main statement, says that this construction may be refined by combining integral cohomology and de Rham cohomology to ordinary differential cohomology: the (n+1)(n+1)-form F F \langle F_\nabla \wedge \cdots F_\nabla\rangle may be realized itself as the curvature (n+1)(n+1)-form of a circle n-bundle with connection c diff\mathbf{c}_{diff} – one speaks of a secondary characteristic class.

[c diff]:GBund conn(X) U(1)nBund conn(X) H diff n+1(X). [\mathbf{c}_{diff}] : G Bund_{conn}(X)_\sim \to U(1) n Bund_{conn}(X)_\sim \simeq H^{n+1}_{diff}(X) \,.

Notice that this map goes from differential nonabelian cohomology to abelian differential cohomology .

In summary this yields the following picture:

[c diff] [c] [F F ] [c] H diff n+1(X) H n+1(X,) H dR n+1(X) H n+1(X,). \array{ && [\mathbf{c}_{diff}] \\ & \swarrow && \searrow \\ [c] &&&& [\langle F_\nabla \wedge \cdots F_\nabla\rangle] \\ & \searrow && \swarrow \\ && [c]_{\mathbb{R}} } \;\;\;\;\;\;\;\;\; \in \;\;\;\;\;\;\;\;\; \array{ && H_{diff}^{n+1}(X) \\ & \swarrow && \searrow \\ H^{n+1}(X,\mathbb{Z}) && && H_{dR}^{n+1}(X) \\ & \searrow && \swarrow \\ && H^{n+1}(X, \mathbb{R}) } \,.

A central implication of the last step is that with the refinement from curvatures in de Rham cohomology to circle n-bundles with connection in differential cohomology is that these come with a notion of higher parallel transport and higher holonomy:


So the refined Chern-Weil homomorphism provides a large family of gauge quantum field theories of Chern-Simons type in odd dimensions whose field configurations are always connections on principal bundles and whose Lagrangians are Chern-Simons elements on a Lie algebra.

But the notion of invariant polynomials and Chern-Simons elements naturally exists much more generally for L-∞ algebras, and even more generally for L-∞ algebroids. We claim here that in this fully general case there is still a natural analog of the Chern-Weil homomorphism – which we call the ∞-Chern-Weil homomorphism . Accordingly this gives rise to a wide class of action functionals for gauge quantum field theories, which we call \infty-Chern-Simons theories.

Deficiencies of classical Chern-Weil theory

While very useful, the classical Chern-Weil homomorphism, even in its refined form where it takes values in ordinary differential cohomology, has two major deficiencies:

  1. It only differentially refines characteristic classes of classifying spaces BGB G for GG a Lie group.

    But there are applications where one needs differential refinements for instance of the higher connected covers in the Whitehead tower of Lie groups, which without further work are just topological groups. Central motivating examples are differential characteristic classes for the string 2-group and the fivebrane 6-group.

    Generally, therefore, one would want to have a generalization of the Chern-Weil homomorphism that applies to the case that GG is a more generally a smooth ∞-group : a smooth groupal A-∞ space.

  2. It only works on cohomology groups but does not keep track of gauge transformations/homotopies.

    A refinement of the Chern-Weil homomorphism from cohomology sets to cocycle ∞-groupoids gives access to its homotopy fibers: these are ∞-groupoids that encode the obstruction theory of the given differential characteristic class: they encode how trivializations of differential characteristic classes are related, how these relations are related, and so on.

Central motivating examples of such homotopy fibers of differential characteristic classes are differential string structures and differential fivebrane structures.

In the next paragraph we indicate how these deficiencies of the classical Chern-Weil homomorphism are removed by constructing it in the context of cohesive (∞,1)-toposes.

The general abstract formulation

We now indicate how traditional Chern-Weil theory briefly reviewed above may be formulated and generalized in a general abstract fashion in cohesive (∞,1)-topos theory.

Chern-Weil theory is traditionally discussed in terms of smooth universal connections on the universal principal bundles EGBGE G \to B G over the classifying space of GG, where the topological spaces EGE G and BGB G are both equipped in a clever way with smooth structure of sorts.

However, even when equipped with a smooth structure, the classifying space BGB G is far from being a faithful representative of the theory of smooth GG-principal bundles: all it remembers (in its topological homotopy type) is their equivalence classes, but it fails – with or without smooth structure – to carry any information about the morphisms of smooth GG-principal bundles: their gauge transformations.

This is not just a question of cosmetics: Chern-Weil theory produces (differential) characteristic classes and with every characteristic map representing a characteristic class, there is the associated extension theory that extracts the information encoded in the characteristic map: but these extensions are homotopy fibers and these homotopy fibers only come out right when the morphisms of bundles are faithfully taken into account.

The correct object to replace the smooth classifying space BGB G with is the corresponding moduli stack, which we shall write BG\mathbf{B}G (where here and in all of the following the boldface type is to indicate (good) smooth refinement of the topological structure that goes by the non-boldface symbols). This is nothing but the delooping Lie groupoid with a single object and GG worth of morphisms. For XX a smooth manifold, regarded as a Lie groupoid with only trivial morphisms, the groupoid of GG-principal bundles and smooth gauge transformations between them is

GBund(X)H(X,BG), G Bund(X) \simeq \mathbf{H}(X, \mathbf{B}G) \,,

where on the right the H(,)\mathbf{H}(-,-) denotes the groupoid of Morita morphisms of Lie groupoids.

Lie groupoids are certain concrete stacks on smooth manifolds. If we allow more generally also non-concrete such stacks – which we shall call just smooth groupoids – then there is also the differential refinement BG connBG\mathbf{B}G_{conn} \to \mathbf{B}G, such that

GBund conn(X)H(X,BG conn) G Bund_{conn}(X) \simeq \mathbf{H}(X, \mathbf{B}G_{conn})

is the groupoid of GG-principal bundles with connection.

If G=U(1)G = U(1) is the circle group, there is not just the Lie groupoid BU(1)\mathbf{B}U(1) but also the Lie 2-groupoid B 2U(1)\mathbf{B}^2 U(1) with a single object, a single morphism and U(1)U(1) worth of 2-morphisms. This is the beginning of a pattern: for each nn \in \mathbb{N} there is a smooth n-groupoid B nU(1)\mathbf{B}^n U(1) which is a smooth refinement of the topological Eilenberg-MacLane space K(,n+1)K(\mathbb{Z},n+1)

B nU(1)B nU(1)B nU(1)B n+1K(,n+1) \mathbf{B}^n U(1) \mapsto {\vert \mathbf{B}^n U(1)\vert} \simeq B^n U(1) \simeq B^{n+1} \mathbb{Z} \simeq K(\mathbb{Z}, n+1)

such that

U(1)nBund(X)H(X,B nU(1)) U(1) n Bund(X) \simeq \mathbf{H}(X, \mathbf{B}^n U(1))

is the n-groupoid whose objects are smooth circle n-bundles \simeq (n1)(n-1)-bundle gerbes on XX, whose morphisms are smooth gauge transformations, whose 2-morphisms are smooth gauge-of-gauge transformations, and so on.

Ordinary Lie groupoids and the Morita morphisms are to be regarded as living in a context called a (2,1)-topos . The more general context H\mathbf{H} needed for the above statement is an (∞,1)-topos of smooth ∞-groupoids. We write H:=\mathbf{H} := Smooth∞Grpd to indicate this.

This (∞,1)-topos is special in that it is a cohesive (∞,1)-topos. This implies that there is an intrinsic notion of ∞-Chern-Weil theory in such H\mathbf{H}. We name the main ingredients of this phenomenon:

  1. For all smooth \infty-groupoids XX, BG\mathbf{B}G, B nU(1)\mathbf{B}^n U(1), etc. in H\mathbf{H} there is a smooth path ∞-groupoid Π(X)\mathbf{\Pi}(X), Π(BG)\mathbf{\Pi}(\mathbf{B}G), Π(B nU(1))\mathbf{\Pi}(\mathbf{B}^n U(1)), etc.;

  2. such that for GG any smooth ∞-group, morphisms Π(X)BG\mathbf{\Pi}(X) \to \mathbf{B}G are (the higher parallel transport of) flat ∞-connections on GG-principal ∞-bundles:

    GBund flat(X)H(Π(X),BG). G Bund_{flat}(X) \simeq \mathbf{H}(\mathbf{\Pi}(X), \mathbf{B}G) \,.
  3. If the underlying GG-principal ∞-bundle of such a flat \infty-connection is trivial,

    X * Π(X) BG, \array{ X &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ \mathbf{\Pi}(X) &\to& \mathbf{B}G } \,,

    then it encodes a flat 𝔤\mathfrak{g}-valued differential form on XX, where 𝔤\mathfrak{g} is the L-∞ algebra of GG: its infinitesimal approximation.

  4. For BG=B nU(1)\mathbf{B}G = \mathbf{B}^n U(1) this is a closed smooth differential nn-form and the smooth \infty-groupoid

    dRB nU(1):=[Π(X),B nU(1)] B nU(1)* \mathbf{\flat}_{dR} \mathbf{B}^n U(1) := [\mathbf{\Pi}(X), \mathbf{B}^n U(1)] \prod_{\mathbf{B}^n U(1)} *

    is the coefficient object for de Rham cohomology on smooth ∞-groupoids in degree nn.

  5. There is a canonical curvature characteristic form

    curv:B nU(1) dRB n+1U(1) curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)

    on each B n+1U(1)\mathbf{B}^{n+1} U(1).

  6. For each smooth characteristic class [c][\mathbf{c}] for the smooth ∞-group GG

    c:BGB nU(1) \mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)

    the composite

    c dR:BGcB n(1)curv dRB n+1U(1) \mathbf{c}_{dR} : \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n (1) \stackrel{curv}{\to} \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)

    is the unrefined ∞-Chern-Weil homomorphism: it sends GG-principal ∞-bundles g:XBGg : X \to \mathbf{B}G to the closed differential form c dR(G):X dRB n+1U(1)\mathbf{c}_{dR}(G) : X \to \mathbf{\flat}_{dR}\mathbf{B}^{n+1}U(1).

  7. A circle n-bundle with connection is a circle nn-bundle twisted by the canonical curvature class:

    H conn(X,B nU(1)):=H(X,B nU(1)) H(X, dRB n+1U(1))H dR(X) \mathbf{H}_{conn}(X, \mathbf{B}^n U(1)) := \mathbf{H}(X, \mathbf{B}^n U(1)) \prod_{\mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)) } H_{dR}(X)
  8. A general connection on a principal ∞-bundle is an object that universally lifts the unrefined \infty-Chern-Weil homomorphism from de Rham cohomology to circle nn-bundles with connection.

    BG^ conn * ^ BG conn c diff B n1U(1) conn X g BG c B n1U(1) curv dRB nU(1) \array{ && \mathbf{B}\hat G_{conn} &\to& * \\ &{}^\mathllap{\hat \nabla}\nearrow& \downarrow && \downarrow \\ && \mathbf{B}G_{conn} &\stackrel{\mathbf{c}_{diff}}{\to}& \mathbf{B}^{n-1}U(1)_{conn} \\ &{}^{\mathllap{\nabla}}\nearrow & \downarrow && \downarrow && \\ X & \stackrel{g}{\to} & \mathbf{B}G & \stackrel{\mathbf{c}}{\to} & \mathbf{B}^{n-1} U(1) &\stackrel{curv}{\to}& \mathbf{\flat}_{dR} \mathbf{B}^{n} U(1) }
  9. such that the full \infty-Chern-Weil homomorphism is the operation that maps a GG-principal \infty-bundle with connection :XBG conn\nabla : X \to \mathbf{B}G_{conn} to the composite c diff():XB nU(1) conn\mathbf{c}_{diff}(\nabla) : X \to \mathbf{B}^n U(1)_{conn}.

    c diff:H(X,BG conn)H(X,B n1U(1) conn) \mathbf{c}_{diff} : \mathbf{H}(X, \mathbf{B}G_{conn}) \to \mathbf{H}(X, \mathbf{B}^{n-1} U(1)_{conn})

    is the \infty-Chern-Weil homomorphism.

  10. For an nn-dimensional smooth manifold Σ\Sigma the truncation morphism

    exp(iS Σ()):H(Σ,B nU(1) conn)H flat(Σ,B nU(1))τ 0H flat(Σ,B nU(1))U(1) \exp(i S_\Sigma(-)) : \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \stackrel{\simeq}{\to} \mathbf{H}_{flat}(\Sigma, \mathbf{B}^n U(1)) \stackrel{\tau_0}{\to} H_{flat}(\Sigma, \mathbf{B}^n U(1)) \simeq U(1)

    sends circle n-bundles with connection to the holonomy of their higher parallel transport over Σ\Sigma.

  11. Composed with the ∞-Chern-Weil homomorphism this is the action functional

    exp(iS Σ(c conn())):H(X,BG conn)U(1) \exp(i S_\Sigma(\mathbf{c}_{conn}(-))) : \mathbf{H}(X, \mathbf{B}G_{conn}) \to U(1)

    of the \infty-Chern-Simons theory associated with the given smooth cocycle c\mathbf{c}.

In this fashion the Chern-Weil homomorphism and its refinements and generalizations exists on general abstract grounds, and many of its general properties follow on these abstract grounds. In the next survey section we indicate how this general abstract construction translates into concrete Lie theoretic expressions.

Construction by Lie integration of L L_\infty-Chern-Simons elements

At least in good cases the general abstract ∞-Chern-Simons functional indicated above arises from Lie integration of infinitesimal data. We indicate the relevant notions of higher Lie theory and how Chern-Simons elements on L-∞ algebras serve to present Lagrangians for the general abstract \infty-Chern-Simons action functional.

To start with, notice that to every Lie algebra 𝔤\mathfrak{g} (here taken to be finite dimensional over \mathbb{R}) is associated its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}), a dg-algebra refinement of the Grassmann algebra 𝔤 *\wedge^\bullet \mathfrak{g}^*. This may usefully be thought of as the function algebra on the infinitesimal neighbourhood of the neutral element in the Lie group integrating 𝔤\mathfrak{g} (this is discussed in detail here).

One way to think of CE(𝔤)CE(\mathfrak{g}) is to notice that a class in Lie algebra cohomology of 𝔤\mathfrak{g} is a equivalently class in the ordinary cochain cohomology of CE(𝔤)CE(\mathfrak{g})

H LieAlg (𝔤,)H (CE(𝔤)). H^\bullet_{LieAlg}(\mathfrak{g}, \mathbb{R}) \simeq H^\bullet(CE(\mathfrak{g})) \,.

Traditionally this is taken as the definition of Lie algebra cohomology, but if one more thinks of deloopings of Lie algebras properly as being infinitesimal Lie groupoids b𝔤b \mathfrak{g}, then this becomes a theorem (discussed here):

H (b𝔤,). \cdots \simeq H^\bullet(b \mathfrak{g}, \mathbb{R}).

In fact CE(𝔤)=C (b𝔤)CE(\mathfrak{g}) = C^\infty(b \mathfrak{g}) captures the full information of a moduli stack: not only does CE()CE(-) encode the above cohomology groups and hence certain equivalence classes of Lie algebras, but it yields an equivalence of categories

LieAlgCEAlg opdgAlg op LieAlg \simeq CE Alg^{op} \hookrightarrow dgAlg^{op}

of the category of Lie algebras and the opposite category of Chevalley-Eilenberg algebras with morphisms the ordinary morphisms of dg-algebras.

This fact provides a quick but useful way to L-∞ algebras, the ∞-algebras over the Lie operad: the category

CE():L AlgdgAlg op CE(-) : L_\infty Alg \hookrightarrow dgAlg^{op}

of L-∞ algebras is the formal dual of the full subcategory of dgAlg on those dg-algebras whose underlying graded algebra is a Grassmann algebra on a \mathbb{N}-graded vector space 𝔤\mathfrak{g}.

(There is a further generalization to L-∞ algebroids, which however we shall not go into in the context of this introduction. More on this below.)

As before, a class in the cochain cohomology of the Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) is a class in the L-∞ algebra cohomology of 𝔤\mathfrak{g}. In fact, using L L_\infty-algebras Lie algebra cohomology and L-∞ algebra cohomology becomes representable:

we write b n1b^{n-1}\mathbb{R} for the line Lie n-algebra: the L L_\infty-algebra defined by the fact that CE(b n1)CE(b^{n-1}\mathbb{R}) has a single generator in degree nn and trivial differential. Using this, a cocycle in L-∞ algebra cohomology is a morphism of L-∞ algebras

μ:𝔤b n1. \mu : \mathfrak{g} \to b^{n-1}\mathbb{R} \,.

The shift in dimension on the right reflects that we are entitled to think of this equivalently as a morphism of infinitesimal ∞-groupoids of deloopings

bμ:b𝔤b n. b \mu : b \mathfrak{g} \to b^n \mathbb{R} \,.

Accordingly in a suitable context H\mathbf{H} of synthetic differential ∞-groupoids we have (again by this theorem)

H Lie n(𝔤,)π 0H(B𝔤,B n). H^n_{Lie}(\mathfrak{g}, \mathbb{R}) \simeq \pi_0 \mathbf{H}(\mathbf{B} \mathfrak{g}, \mathbf{B}^n \mathbb{R}) \,.

So there is an accurate infinitesimal analog of the general abstract cohomology of smooth ∞-groupoids, and this continues: there is a simple infinitesimal analog of groupal models for universal principal ∞-bundles: the Weil algebra W(𝔤)W(\mathfrak{g})

G EG𝔤 inn(𝔤)CE(𝔤) i * W(𝔤) \array{ G \\ \downarrow \\ \mathbf{E}G } \;\;\;\;\;\;\;\; \array{ \mathfrak{g} \\ \downarrow \\ inn(\mathfrak{g}) } \;\;\;\;\;\;\;\; \array{ CE(\mathfrak{g}) \\ \uparrow^{\mathrlap{i^*}} \\ W(\mathfrak{g}) }

is the unique free dg-algebra on the graded vector space 𝔤 *\mathfrak{g}^* such that the canonical projection of graded vector spaces 𝔤 *𝔤 *[1]𝔤 *\mathfrak{g}^* \oplus \mathfrak{g}^*[1] \to \mathfrak{g}^* extends to a dg-algebra homomorphism i *i^* as indicated.

Using the Weil algebra, we obtain the following notions:

For XX a smooth manifold, 𝔤\mathfrak{g}-valued differential forms are dg-algebra homomorphisms

Ω (X)W(𝔤):A. \Omega^\bullet(X) \leftarrow W(\mathfrak{g}) : A \,.

The curvature forms of such are the components

Ω (X)AW(𝔤) 1𝔤 *[1]:F A \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \leftarrow \wedge^1\mathfrak{g}^*[1] : F_A

of the shifted generators. Precisely if these all vanish does AA factor through i *i^* and we say it is flat .

An invariant polynomial on a Lie algebra or L-∞ algebra is an element in 𝔤 *[1]W(𝔤)\wedge^\bullet \mathfrak{g}^*[1] \hookrightarrow W(\mathfrak{g}) that is d Wd_{W}-closed.

An invariant polynomial \langle - \rangle is in transgression with an L L_\infty-algebra cocycle μCE(𝔤)\mu \in CE(\mathfrak{g}) if there is an element csW(𝔤)cs \in W(\mathfrak{g}) with

  1. i *cs=μi^* cs = \mu;

  2. d W(𝔤)cs=d_{W(\mathfrak{g})} cs = \langle - \rangle.

The element cscs we call the Chern-Simons element exhibiting the transgression.

For 𝔤\mathfrak{g}-valued form data AA , the composite

(1)Ω (X)AW(𝔤)csW(b n1):cs(A) \Omega^\bullet(X) \stackrel{A}{\leftarrow} W(\mathfrak{g}) \stackrel{cs}{\leftarrow} W(b^{n-1} \mathbb{R}) : cs(A)

is the Chern-Simons form of AA with respect to the given Chern-Simons element.

With all these L L_\infty-algebraic structures in place, the main statement is:

for μ:𝔤b n1\mu : \mathfrak{g} \to b^{n-1}\mathbb{R} a transgressive L-∞ algebra cocycle of degree (n1)(n-1) and

BG:=cosk nexp(𝔤) \mathbf{B}G := \mathbf{cosk}_{n}\exp(\mathfrak{g})

the (delooping) of the smooth (n1)(n-1)-group which is the universal Lie integration of 𝔤\mathfrak{g}, then

  1. μ\mu integrates to a smooth characteristic class

    c:=exp(μ):BGB n/Γ \mathbf{c} := \exp(\mu) : \mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\Gamma

    (where Γ\Gamma is some lattice inside \mathbb{R});

  2. the differential refinement c diff\mathbf{c}_{diff} of this c\mathbf{c} according to the above general abstract theory produces an \infty-Chern-Weil homomorphism that sends GG-principal ∞-bundles with ∞-connection \nabla to circle n-bundles with connection whose connection nn-form is locally gauge equivalent to the Chern-Simons form cs()cs(\nabla) (1).

It follows that the (exponentiated) action functional of the \infty-Chern-Simons theory induced by μ\mu is given by

Aexp(i Σcs(A)) A \mapsto \exp(i \int_\Sigma cs(A))

for the induced Chern-Simons element cscs.


Apart from producing general abstract statements about \infty-Chern-Simons theory, it is of interest to scan the space of all those higher gauge theory Lagrangians that are, maybe secretly, instances of \infty-Chern-Simons action functionals. Due to the additional generality of invariant polynomials and Chern-Simons element as we generalize from Lie algebras to L-∞ algebras and then further to L-∞ algebroids, this is a larger class than familiarity with ordinary Chern-Simons theory might suggest. We briefly list some examples and some of their properties. A detailed discussion is below in the section Examples.

This concludes our introduction and survey. We now turn to the detailed discussion.

1.) General abstract theory

General abstract \infty-Chern-Simons theory is defined in any cohesive (∞,1)-topos, as described in the section

Here we will work in the topos Smooth∞Grpd of smooth ∞-groupoids and in its infinitesimal cohesive neighbourhood, the (,1)(\infty,1)-topos SynthDiff∞Grpd of synthetic differential ∞-groupoids.

2.) The Lagrangians

The contents of this section is at

3.) The action functionals

The content of this section is at

4.) Covariant phase space

The contents of this section is at

5.) Examples

We discuss examples of \infty-Chern-Simons functionals.

The contents of this section is at

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Surveys, Expositions, lectures and talks

Expositions, surveys and lecture notes on \infty-Chern-Simons theory include the following

For more expositions, surveys and lectures see the References at differential cohomology in a cohesive topos.


The notion of Chern-Simons elements for L L_\infty-algebras and the associated \infty-Chern-Simons Lagrangians is due to

The induced construction of the ∞-Chern-Weil homomorphism with special attention to the Chern-Simons circle 3-bundle and the Chern-Simons circle 7-bundle is in

In the general context of cohesive (∞,1)-toposes \infty-Chern-Simons theory is discussed in section 4.3 of

The special case of the AKSZ sigma-model is discussed in

Discussion of symplectic Lie n-algebroids is in

  • Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv)

    On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv)

For more see differential cohomology in a cohesive topos -- references.

Revised on December 18, 2013 11:22:08 by Urs Schreiber (