Schreiber 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.

Abstract

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 $G$ 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 $G$-principal bundle.

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

In the cohesive (∞,1)-topos Smooth∞Grpd of smooth ∞-groupoids we give a natural general abstract procedure for integration of $L_{\mathbf{c}}$ over an $n$-dimensional parameter space $\Sigma \in \mathbf{H}$. The resulting smooth function $\exp(S_{\mathbf{c}}) : [\Sigma,\mathbf{B}G_{conn}] \to U(1)$ 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 $d \exp(S_{\mathbf{c}}) : [\Sigma, \mathbf{B}G_{conn}] \to \mathbf{\flat}_{dR} \mathbf{B}U(1)$ 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 $\mathbf{c}$ the Lie integration of the canonical Lie algebra 3-cocycle on $G$. 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:

For

• $B G \in$ Top the classifying space of (the topological group underlying) a compact Lie group $G$

• and $[c] \in H^n(B G, \mathbb{Z})$ a class in its integral cohomology – which we may call a characteristic class for $G$-principal bundles

we get for each smooth manifold $X$ an assignment

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

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

Let $[c]_\mathbb{R} \in H^{n+1}(B G, \mathbb{R})$ be the image of $[c]$ 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

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

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

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

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

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

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

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

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

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

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

In summary this yields the following picture:

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

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:

• the local ∞-connection $n$-form of $\mathbf{c}_{diff}$ is the Chern-Simons form $cs(\nabla)$ of a Chern-Simons element $cs$ of the invariant polynomial $\langle- \rangle$ evaluated on the given $G$-connection;

• the corresponding higher parallel transport as an assignment

  $$(\Sigma \stackrel{\phi}{\to} X) \mapsto \exp(\int_\Sigma \nabla_{\mathbf{c}_{diff}}) \in U(1)$$

  of $(n-1)$-dimensional manifolds in $X$ to the circle group is the action functional of the corresponding Chern-Simons theory.

Specifically

• for $\langle -,-\rangle$ the Killing form invariant polynomial on a semisimple Lie algebra, one calls $\mathbf{c}_{diff}$ the Chern-Simons circle 3-bundle; whose higher holonomy is the action functional of ordinary Chern-Simons theory;

• for $\langle -,-,-,-\rangle$ the next higher invariant polynomial on a semisimple Lie algebra, $\mathbf{c}_{diff}$ is a Chern-Simons circle 7-bundle, and so on.

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 $B G$ for $G$ 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 $G$ 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 $E G \to B G$ over the classifying space of $G$, where the topological spaces $E G$ and $B G$ are both equipped in a clever way with smooth structure of sorts.

However, even when equipped with a smooth structure, the classifying space $B G$ is far from being a faithful representative of the theory of smooth $G$-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 $G$-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 $B G$ with is the corresponding moduli stack, which we shall write $\mathbf{B}G$ (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 $G$ worth of morphisms. For $X$ a smooth manifold, regarded as a Lie groupoid with only trivial morphisms, the groupoid of $G$-principal bundles and smooth gauge transformations between them is

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

where on the right the $\mathbf{H}(-,-)$ 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 $\mathbf{B}G_{conn} \to \mathbf{B}G$, such that

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

is the groupoid of $G$-principal bundles with connection.

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

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

such that

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

is the n-groupoid whose objects are smooth circle n-bundles $\simeq$ $(n-1)$-bundle gerbes on $X$, 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 $\mathbf{H}$ needed for the above statement is an (∞,1)-topos of smooth ∞-groupoids. We write $\mathbf{H} :=$ 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 $\mathbf{H}$. We name the main ingredients of this phenomenon:

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

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

   $$G Bund_{flat}(X) \simeq \mathbf{H}(\mathbf{\Pi}(X), \mathbf{B}G) \,.$$

3. If the underlying $G$-principal ∞-bundle of such a flat $\infty$-connection is trivial,

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

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

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

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

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

5. There is a canonical curvature characteristic form

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

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

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

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

   the composite

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

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

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

   $$\mathbf{H}_{conn}(X, \mathbf{B}^n U(1)) := \mathbf{H}(X, \mathbf{B}^n U(1)) \prod_{\mathbf{H}(X, \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)) } H_{dR}(X)$$

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 $n$-bundles with connection.

   $$\array{ && \mathbf{B}\hat G_{conn} &\to& * \\ &{}^\mathllap{\hat \nabla}\nearrow& \downarrow && \downarrow \\ && \mathbf{B}G_{conn} &\stackrel{\mathbf{c}_{diff}}{\to}& \mathbf{B}^{n-1}U(1)_{conn} \\ &{}^{\mathllap{\nabla}}\nearrow & \downarrow && \downarrow && \\ X & \stackrel{g}{\to} & \mathbf{B}G & \stackrel{\mathbf{c}}{\to} & \mathbf{B}^{n-1} U(1) &\stackrel{curv}{\to}& \mathbf{\flat}_{dR} \mathbf{B}^{n} U(1) }$$

9. such that the full $\infty$-Chern-Weil homomorphism is the operation that maps a $G$-principal $\infty$-bundle with connection $\nabla : X \to \mathbf{B}G_{conn}$ to the composite $\mathbf{c}_{diff}(\nabla) : X \to \mathbf{B}^n U(1)_{conn}$.

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

   is the $\infty$-Chern-Weil homomorphism.

10. For an $n$-dimensional smooth manifold $\Sigma$ the truncation morphism

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

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

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

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

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

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

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

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

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 $b \mathfrak{g}$, then this becomes a theorem (discussed here):

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

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

$$LieAlg \simeq CE Alg^{op} \hookrightarrow dgAlg^{op}$$

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

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

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 $\mathfrak{g}$.

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

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

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

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

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

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

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

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 $W(\mathfrak{g})$

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

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

Using the Weil algebra, we obtain the following notions:

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

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

The curvature forms of such are the components

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

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

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

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

1. $i^* cs = \mu$;

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

The element $cs$ we call the Chern-Simons element exhibiting the transgression.

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

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

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

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

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

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

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

1. $\mu$ integrates to a smooth characteristic class

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

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

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

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

$$A \mapsto \exp(i \int_\Sigma cs(A))$$

for the induced Chern-Simons element $cs$.

Examples

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.

• ordinary Chern-Simons theory

  Take $\mathfrak{g}$ to be a semisimple Lie algebra and $\langle -\rangle$ the Killing form invariant polynomial. The corresponding $\infty$-Chern-Simons Lagrangian is the ordintay Lagrangian of ordinary Chern-Simons theory

  $$A \mapsto cs(A) = \langle A \wedge [A \wedge A]\rangle + c \langle A \wedge F_A\rangle \,.$$

  The circle n-bundle with connection given by ordinary Chern-Simons theory is known as the Chern-Simons circle 3-bundle . The non-triviality of its underlying class is the obstruction to lifting the $G$-principal bundle to a string structure. The non-triviality of its connection is the obstruction to having a differential string structure. In general it defines a twisted differential string structure.

• Dijkgraaf-Witten theory

  Let $G$ be a discrete group, and

  $$\alpha : \mathbf{B}G \to \mathbf{B}^3 U(1)$$

  a characteristic class. This is equivalently given by a 3-cocycle in group cohomology $H^3_{Grp}(G, U(1))$.

  Due to discreteness, there cannot be any way to obtain this by Lie integration of infinitesimal data. Nevertheless, the general abstract story from above still applies and produces an action functional: that of Dijkgraaf-Witten theory.

• Yetter model

  Let $G$ be a discrete 2-group and $\alpha : \mathbf{B}G \to \mathbf{B}^4 U(1)$ a characteristic class. The associated $\infty$-Chern-Simons theory is the higher analogy of Dijkgraaf-Witten theory: the Yetter model.

• magnetic dual Chern-Simons theory

  Take $\mathfrak{g}$ to be an semisimple Lie algebra, and $\mu$ the Lie algebra cocycle in transgression with the Killing form $\langle \rangle$. Write $\mathfrak{g}_\mu$ for the Lie 2-algebra extension classified by $\mu$: the string Lie 2-algebra.

  A $\mathfrak{g}_\mu$-valued field configuration is a pair $(A,B) \in \Omega^1(\Sigma, \mathfrak{g}) \times \Omega^2(\Sigma)$

  Then let $\langle -,-,-,-\rangle$ be the next nontrivial invariant polynomial. The corresponding $\infty$-Chern-Simons Lagrangian (in 7 dimension) is

  $$cs(A,B) = \langle A \wedge [A \wedge A] \wedge [A \wedge A] \wedge [A \wedge A]\rangle + (terms\;involving F_A). \,,$$

  The circle n-bundle with connection given by 7-dimensional Chern-Simons theory is the Chern-Simons circle 7-bundle . The non-triviality of its underlying class is the obstruction to lifting the string 2-group-principal 2-bundle to a fivebrane structure. The non-triviality of its 7-connection is the obstruction to having a differential fivebrane structure. In general it defines a twisted differential fivebrane structure.

• BF-theory and topological Yang-Mills

  (see below for the moment).

• AKSZ theories

  Let $\mathfrak{g} = \mathfrak{P}$ be a symplectic Lie n-algebroid and consider its defining invariant polynomial $\omega$.

  With $\pi \in CE(\mathfrak{P})$ the Hamiltonian whose Hamiltonian vector field under the graded symplectic structure $\omega$ is $d_{CE(\mathfrak{P})}$ we find that the corresponding Chern-Simons element is

  $$cs = \frac{1}{2} \iota_{\epsilon} \omega + \pi \,,$$

  where $\epsilon$ is the Euler vector field.

  This is the Lagrangian of the AKSZ theory with target $(\mathfrak{P}, \omega)$.

  For $n = 1$ this is the Poisson sigma-model.

  For $n = 2$ this is the Courant sigma-model.

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

• . cohesive (∞,1)-topos – Higher Holonomy and ∞-Chern-Simons functional

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

2.) The Lagrangians

The contents of this section is at

• infinity-Chern-Simons theory – Lagrangians

3.) The action functionals

The content of this section is at

• infinity-Chern-Simons theory – action functionals.

4.) Covariant phase space

The contents of this section is at

• infinity-Chern-Simons theory – covariant phase space

5.) Examples

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

The contents of this section is at

• infinity-Chern-Simons theory – examples

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

  • cohomology

    • Principal ∞-bundles – theory, presentations and applications

    • Fivebrane structures

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • A higher stacky perspective on Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • nonabelian 7d Chern-Simons theory and the 5-brane

    • Extended higher cup-product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

    • The brane bouquet

    • Obstruction theory for parameterized higher WZW terms

  • Higher geometric prequantum theory

    • Classical field theory via higher differential geometry

    • L-∞ algebras of local observables from higher prequantum bundles

    • Local prequantum field theory

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory

References

Surveys, Expositions, lectures and talks

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

• Urs Schreiber,

  Chern-Simons terms on higher moduli stacks, Talk at Hausdorff Institute, Bonn (2011) (pdf)

  $\infty$-Chern-Simons functionals, Talk at Higher Structures 2011 (pdf)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory

• Urs Schreiber, geometry of physics Lectures notes

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

General

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_\infty$-connections (web)

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

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes

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

• Urs Schreiber, differential cohomology in a cohesive topos

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory

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.