Schreiber
infinity-Chern-Simons theory -- Lagrangians

This is a sub-entry of ∞-Chern-Simons theory . See there for background and details.

The Lagrangians

We describe some details of the construction of the ∞-Chern-Weil homomorphism induced by an invariant polynomial $⟨-,-⟩\in W(𝔞)$ on an ∞-Lie algebroid $𝔞$. As discussed at Chern-Weil homomorphism and ∞-connections, this model for the $\mathrm{\infty }$-Chern-Weil homomorphism;

1. is controlled by a Chern-Simons element $\mathrm{cs}\in W(𝔞)$ that exhibits the transgression of $⟨⟩$ to an invariant polynomial $\mu \in \mathrm{CE}(𝔞)$;

2. sends an $𝔞$-connection on an ∞-bundle given locally by an ∞-Lie algebroid valued differential form $A\in {\Omega }^1(U,𝔞)$ to the circle n-bundle with connection whose local connection $n$-form is Chern-Simons form $\mathrm{cs}(A)$.

The Lagrangian of $\mathrm{\infty }$-Chern-Simons theory is then of the form $A\mapsto {\int }_{\Sigma }\mathrm{cs}(A)$. Therefore the identification of $\mathrm{\infty }$-Chern-Simons theories essentially reduces to the identificaton of higher Chern-Simons elements $\mathrm{cs}$ on ∞-Lie algebroids. In the following we survey the relevant theory for this construction.

$L_{\mathrm{\infty }}$-algebroids

Abstractly, an ∞-Lie algebroid is a synthetic-differential ∞-groupoid all whose k-morphisms are infinitesimal, for all $k\ge 1$. Concretely, these objects may be defined as formal duals of certain cosimplicial algebras and equivalrently but more conveniently, using the monoidal Dold-Kan correspondence, of graded-commutative semifree dg-algebras. This perspective we describe now.

We fix a commutative $ℝ$-algebra $R$. If $L$ is graded $R$-module, then the suspension of $L$ is denoted as $L[-1]$ where $L[-1{]}_i=L_{i-1}$. We say $L$ has finite type if it is degree-wise finitely generated. Given a chain complex $(L,d)$ of finite type, its dual $L^{*}$ is the cochain complex ${\mathrm{Hom}}_R^{-•}(L,R)$. Hence if $L$ is concentrated in non-negative degrees, so is $L^{*}.$

Definition

Let $(L,d)$ be a chain complex of $R$-modules of finite type concentrated in non-negative degrees. A Lie $\mathrm{\infty }$-algebroid structure on $L$ is a degree 1 $R$-derivation $d_{\mathrm{CE}}$ on the graded commutative algebra $\mathrm{CE}(L):=S_R^{•}(L^{*}[-1])$ such that:

• $d_{\mathrm{CE}}$ is a differential: $d_{\mathrm{CE}}\circ d_{\mathrm{CE}}=0$,
• $d_{\mathrm{CE}}$ is compatible with $d$: If $p:S_R^{•}(L^{*}[-1])\to L^{*}$ is the canonical projection, then $p\circ d_{\mathrm{CE}}{\mid }_{L^{*}}(f)(x)=f(dx)$ for all $f\in L^{*}$ and $x\in L$.

Such a chain complex endowed with a Lie $\mathrm{\infty }$-algebroid structure is called Lie $\mathrm{\infty }$-algebroid. If $L$ is a Lie $\mathrm{\infty }$-algebroid, and the underlying complex is concentrated in degrees $0,\dots n-1$, then we say $L$ is a Lie $n$-algebroid.

In analogy with the case in Lie theory, if $L$ is a Lie $\mathrm{\infty }$-algebroid, we call the DGCA $(\mathrm{CE}(L),d_{\mathrm{CE}})$ the Chevalley-Eilenberg algebra of $L$. Note that $\mathrm{CE}(L)$ is in fact a quasi-free DGCA i.e. it is free as a graded commutative algebra. A morphism of Lie $\mathrm{\infty }$-algebroids $L_1\to L_2$ is a dgca morphism $\mathrm{CE}(L_2)\to \mathrm{CE}(L_1)$.

(want to expand this example out explicitly later)

In differential geometry, a Lie algebroid is traditionally thought of as a vector bundle $E\to M$ equipped with a bundle morphism $\rho :E\to TM$ to the tangent bundle and a Lie bracket on the space of sections satisfying some compatibility axioms. It is well-known that a Lie algebroid structure gives a degree 1 differential on the dgca $S_R^{•}(L^{*}[-1])$, where $R=C^{\mathrm{\infty }}(M)$ and $L=\Gamma (E)$. Hence a Lie algebroid is a Lie $\mathrm{\infty }$-algebroid in the sense of def. 1. Similarly, Courant algebroids are also examples of Lie $\mathrm{\infty }$-algebroids (Roytenberg99).

The geometry of $L_{\mathrm{\infty }}$-algebroids

In general, there is a simple geometric interpretation of our Lie $\mathrm{\infty }$-algebroids. Given a Lie $\mathrm{\infty }$-algebroid $L$, the degree 0 part of $\mathrm{CE}(L)$ is a commutative $ℝ$-algebra, which we interpret as the formal dual to the space over which the algebroid is defined. If this commutative algebra is $ℝ$ itself, then we have a Lie $\mathrm{\infty }$-algebroid over the point. It is easy to see this implies that $L$ is an $L_{\mathrm{\infty }}$-algebra.

The geometry behind our Lie $\mathrm{\infty }$-algebroids can be made more explicit by introducing the notion of a cosimplicial smooth algebra…..

$\mathrm{\infty }$-Chern-Simons elements

In this section, we introduce the $\mathrm{\infty }$-analogues of the machinery found in classical Chern-Weil theory which we will then use to construct Chern-Simons elements for Lie $\mathrm{\infty }$-algebroids.

Definition

The Weil algebra of a Lie $\mathrm{\infty }$-algebroid $L$ is the dg-algebra

(1)$$W(L)=(S^{•}(L^{*}[-1]\oplus L^{*}[-2]),d_W),$$W(L)= (S^{\bullet}(L^{\ast}[-1] \oplus L^{\ast}[-2]), d_{W}),

where the differential $d_W$ is defined to be

(2)$$d_W=\left[\begin{array}{cc}{d}_{\mathrm{CE}}& 0\\ 1& -d_{\mathrm{CE}}\end{array}\right].$$d_W = \begin{bmatrix} d_{CE} & 0 \\ 1 & -d_{CE} \end{bmatrix}.

We may write this as

$$d_W=d_{\mathrm{CE}}+d,$$d_W = d_{CE} + \mathbf{d} \,,

where $d$ is the shift operator and $[d_{\mathrm{CE}},d]=0$.

The Weil algebra $W(L)$ is, in fact, the (quasi-free DGCA) mapping cone of the identity $\mathrm{id}:\mathrm{CE}(L)\to \mathrm{CE}(L)$. Note that when $L$ is a Lie algebra, we recover the usual Weil algebra. It is straightforward to show that $W(L)$ is isomorphic to a free differential algebra and therefore is contractible (c.f. Prop. 6 in SSS09). Hence we have:

Proposition

If $L$ is a Lie $\mathrm{\infty }$-algebroid, then the cochain cohomology of $W(L)$ is trivial.

We also adopt the familiar notion of a cocycle in the Chevalley-Eilenberg algebra:

Definition

An $n$-cocycle on a Lie $\mathrm{\infty }$-algebroid $L$ is a degree $n$ element $\mu \in \mathrm{CE}(L)$ such that $d_{\mathrm{CE}}\mu =0$.

This next definition generalizes $\mathrm{ad}$-invariant polynomials from Lie theory to ∞-Lie theory:

Definition

An invariant polynomial on a Lie $\mathrm{\infty }$-algebroid $L$ is an element

(3)$$p\in S^{•}(L^{*}[-2])\subseteq W(L),$$p \in S^{\bullet}(L^{\ast}[-2]) \subseteq W(L),

such that

$$dp=0$$\mathbf{d} p = 0

and

$$d_{\mathrm{CE}}p=0$$d_{CE} p = 0

and hance

(4)$$d_{W}p=0.$$d_W p =0.

The space of invariant polynomials is denoted $\mathrm{inv}(L)$.

Clearly $\mathrm{inv}(L)\subseteq W(L)$ is a dg-sub algebra with the trivial differential. Note that if $L$ is a Lie algebra, then the $d_W$-closed condition is equivalent to the usual $\mathrm{ad}$-invariance.

Now we are ready to define $\mathrm{\infty }$-Chern-Simons elements. Recall that there is a canonical surjection $\pi :W(L)\twoheadrightarrow \mathrm{CE}(L)$ which restricts to the identity on $S^{•}(L^{*}[-1])$.

Definition

Let $\mu$ be a cocycle on a Lie $\mathrm{\infty }$-algebroid $L$ and $p\in \mathrm{inv}(L)$ an invariant polynomial. A Chern-Simons element for $\mu$ and $p$ is an element $\mathrm{cs}\in W(L)$ such that

(5)$$\pi (\mathrm{cs})=\mu ,$$\pi(cs) = \mu,

and

(6)$$d_W\mathrm{cs}=p.$$d_W cs = p.

If a Chern-Simons element exists for $\mu$ and $p$, then, in analogy with the classical case for fiber bundles, we say $p$ is in transgression with $\mu$. Note that we have a short exact sequence of complexes:

(7)$$\begin{array}{ccccc}⋮& & ⋮& & ⋮\\ \downarrow & & \downarrow & & \downarrow \\ {\ker }^n\pi & \to & W^n(L)& \stackrel{\pi }{\to }& {\mathrm{CE}}^n(L)\\ \downarrow & & \downarrow & & \downarrow \\ {\ker }^{n+1}\pi & \to & W^{n+1}(L)& \stackrel{\pi }{\to }& {\mathrm{CE}}^{n+1}(L)\\ \downarrow & & \downarrow & & \downarrow \\ ⋮& & ⋮& & ⋮\end{array}$$\array{ \vdots && \vdots && \vdots \\ \downarrow && \downarrow && \downarrow\\ ker^{n} \pi &\to& W^{n}(L)& \stackrel{\pi}{\to} &CE^{n}(L)\\ \downarrow && \downarrow && \downarrow \\ ker^{n+1} \pi &\to& W^{n+1}(L)& \stackrel{\pi}{\to} &CE^{n+1}(L)\\ \downarrow && \downarrow && \downarrow\\ \vdots && \vdots && \vdots }

Since $\mathrm{inv}(L)\subseteq \ker \pi$, the properties that a Chern-Simons element must satisfy can be understood in terms of the connecting homomorphism $H^n(\mathrm{CE}(L))\to H^{n+1}(\ker \pi ))$ of the corresponding long exact sequence in cohomology. In particular, if $\mu \in {\mathrm{CE}}^n(L)$ and $p\in {\mathrm{inv}}^{n+1}(L)$, then $\mathrm{cs}\in W^n(L)$ is a Chern-Simons element iff we have

(8)$$\begin{array}{cccc}& & \mathrm{cs}& \mapsto \mu \\ & & \downarrow & & \\ p& \mapsto & d_W\mathrm{cs}& & \\ \end{array}$$\array{ && cs & \mapsto \mu\\ && \downarrow && \\ p &\mapsto& d_W cs&&\\ }

Remark

A Chern-Simons element $\mathrm{cs}$ witnessing the transgression from $⟨-⟩$ to $\mu$ is equivalently a commuting diagram of the form

(9)$$\begin{array}{cccccc}\mathrm{CE}(𝔞)& \stackrel{\mu }{\leftarrow }& \mathrm{CE}(b^nℝ)& & & \mathrm{cocycle}\\ \uparrow & & \uparrow \\ W(𝔞)& \stackrel{\mathrm{cs}}{\leftarrow }& W(b^nℝ)& & & \mathrm{Chern}-\mathrm{Simons}\mathrm{element}\\ \uparrow & & \uparrow \\ \mathrm{inv}(𝔞)& \stackrel{⟨-⟩}{\leftarrow }& \mathrm{inv}(b^nℝ)& & & \mathrm{invariant}\mathrm{polynomial}\end{array}$$\array{ CE(\mathfrak{a}) &\stackrel{\mu}{\leftarrow}& CE(b^{n}\mathbb{R}) &&& cocycle \\ \uparrow && \uparrow \\ W(\mathfrak{a}) &\stackrel{cs}{\leftarrow}& W(b^n \mathbb{R}) &&& Chern-Simons element \\ \uparrow && \uparrow \\ inv(\mathfrak{a}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^n \mathbb{R}) &&& invariant polynomial }

in dgAlg. On the other hand, an n-connection with values in a Lie $n$-algebroid $𝔞$ is a cocycle

(10)$$\begin{array}{ccc}\hat{\Sigma }& \stackrel{\nabla }{\to }& \mathrm{cosk}\exp (𝔞{)}_{\mathrm{conn}}\\ {\downarrow }^{\simeq }\\ \Sigma \end{array}$$\array{ \hat \Sigma &\stackrel{\nabla}{\to}& \mathbf{cosk}\exp(\mathfrak{a})_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ \Sigma }

with coefficients in the simplicial presheaf ${\mathrm{cosk}}_{n+1}\exp (𝔞{)}_{\mathrm{conn}}$ that sends $U\in$ CartSp to the $(n+1)$-coskeleton of the simplicial set, which in degree $k$ is the set of commuting diagrams

$$\begin{array}{ccccc}{\Omega }_{\mathrm{vert}}^{•}(U\times {\Delta }^k)& \stackrel{A_{\mathrm{vert}}}{\leftarrow }& \mathrm{CE}(𝔞)& & \mathrm{transition}\mathrm{function}\\ \uparrow & & \uparrow \\ {\Omega }^{•}(U\times {\Delta }^k)& \stackrel{A}{\leftarrow }& W(𝔞)& & \mathrm{connection}\mathrm{forms}\\ \uparrow & & \uparrow \\ {\Omega }^{•}(U)& \stackrel{⟨F_A⟩}{\leftarrow }& \mathrm{inv}(𝔞)& & \mathrm{curvature}\mathrm{characteristic}\mathrm{forms}\end{array},$$\array{ \Omega^\bullet_{vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{a}) && transition function \\ \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{a}) && connection forms \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A \rangle}{\leftarrow}& inv(\mathfrak{a}) && curvature characteristic forms } \,,

such that the curvature forms $F_A$ of the ∞-Lie algebroid valued differential forms $A$ on $U\times {\Delta }^k$ with values in $𝔞$ in the middle are horizontal.

If $\mu$ is an ∞-Lie algebroid cocycle of degree $n$, then the ∞-Chern-Weil homomorphism operates by sending an $\mathrm{\infty }$-connection given by a Cech cocycle with values in simplicial sets of such commuting diagrams to the obvious pasting composite

$$\begin{array}{ccccccccc}{\Omega }_{\mathrm{vert}}^{•}(U\times {\Delta }^k)& \stackrel{A_{\mathrm{vert}}}{\leftarrow }& \mathrm{CE}(𝔞)& \stackrel{\mu }{\leftarrow }& \mathrm{CE}(b^nℝ)& :\mu (A_{\mathrm{vert}})& & & \\ \uparrow & & \uparrow & & \uparrow \\ {\Omega }^{•}(U\times {\Delta }^k)& \stackrel{A}{\leftarrow }& W(𝔞)& \stackrel{\mathrm{cs}}{\leftarrow }& W(b^nℝ)& :\mathrm{cs}(A)& & & \mathrm{Chern}-\mathrm{Simons}\mathrm{form}\\ \uparrow & & \uparrow \\ {\Omega }^{•}(U)& \stackrel{⟨F_A⟩}{\leftarrow }& \mathrm{inv}(𝔞)& \stackrel{⟨-⟩}{\leftarrow }& \mathrm{inv}(b^nℝ)& :⟨F_A⟩& & & \mathrm{curvature}\end{array}.$$\array{ \Omega^\bullet_{vert}(U \times \Delta^k) &\stackrel{A_{vert}}{\leftarrow}& CE(\mathfrak{a}) &\stackrel{\mu}{\leftarrow}& CE(b^{n}\mathbb{R}) & : \mu(A_{vert}) &&& \\ \uparrow && \uparrow && \uparrow \\ \Omega^\bullet(U \times \Delta^k) &\stackrel{A}{\leftarrow}& W(\mathfrak{a}) &\stackrel{cs}{\leftarrow}& W(b^n \mathbb{R}) & : cs(A) &&& Chern-Simons form \\ \uparrow && \uparrow \\ \Omega^\bullet(U) &\stackrel{\langle F_A \rangle}{\leftarrow}& inv(\mathfrak{a}) &\stackrel{\langle-\rangle}{\leftarrow}& inv(b^n \mathbb{R}) & : \langle F_A \rangle &&& curvature } \,.

Under the map to the coskeleton the group of such cocycles for line $n$-bundle with connection is quotiented by the discrete group $\Gamma$ of periods of $\mu$, such that the $\mathrm{\infty }$-Chern-Weil homomorphism is given by sending the $\mathrm{\infty }$-connections presented by (10) to

$$\begin{array}{ccccc}\hat{\Sigma }& \stackrel{\nabla }{\to }& {\mathrm{cosk}}_n\exp (𝔞{)}_{\mathrm{conn}}& \stackrel{\exp (\mathrm{cs})}{\to }& B^n(ℝ/\Gamma {)}_{\mathrm{conn}}\\ {\downarrow }^{\simeq }\\ \Sigma \end{array}.$$\array{ \hat \Sigma &\stackrel{\nabla}{\to}& \mathbf{cosk}_n\exp(\mathfrak{a})_{conn} &\stackrel{\exp(cs)}{\to}& \mathbf{B}^n (\mathbb{R}/\Gamma)_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ \Sigma } \,.

This is a cocycle for a circle n-bundle with connection, whose connection $n$-form is locally given by the Chern-Simons form $\mathrm{cs}(A)$. This is the Lagrangian of the $\mathrm{\infty }$-Chern-Simons theory defined by $(𝔞,⟨-⟩)$ and evaluated on the given $\mathrm{\infty }$-connection. If $\Sigma$ is a smooth manifold of dimension $n$, then the higher holonomy of this circle $n$-bundle over $\Sigma$ is the value of the Chern-Simons action. After a suitable gauge transformation this is given by the integral

$$\exp (iS(A))=\exp (i{\int }_{\Sigma }\mathrm{cs}(A)).$$\exp(i S(A)) = \exp(i \int_\Sigma cs(A)) \,.

This integration process we discuss in more detail in the following section.

References

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

• Hisham Sati, Urs Schreiber, Jim Stasheff, $L_{\mathrm{\infty }}$-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 $\mathrm{\infty }$-Chern-Simons theory is discussed in section 4.3 of

• Urs Schreiber, differential cohomology in a cohesive topos