Schreiber
infinity-Chern-Simons theory -- Lagrangians

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

Contents

The Lagrangians

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

  1. is controlled by a Chern-Simons element csW(𝔞)cs \in W(\mathfrak{a}) that exhibits the transgression of \langle \rangle to an invariant polynomial μCE(𝔞)\mu \in CE(\mathfrak{a});

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

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

L L_\infty-algebroids

Abstractly, an ∞-Lie algebroid is a synthetic-differential ∞-groupoid all whose k-morphisms are infinitesimal, for all k1k \geq 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 \mathbb{R}-algebra RR. If LL is graded RR-module, then the suspension of LL is denoted as L[1]L[-1] where L[1] i=L i1L[-1]_{i}=L_{i-1}. We say LL has finite type if it is degree-wise finitely generated. Given a chain complex (L,d)(L,d) of finite type, its dual L *L^{\ast} is the cochain complex Hom R (L,R)Hom_{R}^{-\bullet}(L,R). Hence if LL is concentrated in non-negative degrees, so is L *.L^{\ast}.

Definition

Let (L,d)(L,d) be a chain complex of RR-modules of finite type concentrated in non-negative degrees. A Lie \infty-algebroid structure on LL is a degree 1 RR-derivation d CEd_{CE} on the graded commutative algebra CE(L):=S R (L *[1])CE(L):=S^{\bullet}_{R}(L^{\ast}[-1]) such that: * d CEd_{CE} is a differential: d CEd CE=0d_{CE} \circ d_{CE} =0, * d CEd_{CE} is compatible with dd: If p:S R (L *[1])L *p \colon S^{\bullet}_{R}(L^{\ast}[-1]) \to L^{\ast} is the canonical projection, then pd CE| L *(f)(x)=f(dx)p \circ d_{CE} \vert_{L^{\ast}}(f)(x)=f(d x) for all fL *f \in L^{\ast} and xLx \in L.

Such a chain complex endowed with a Lie \infty-algebroid structure is called Lie \infty-algebroid. If LL is a Lie \infty-algebroid, and the underlying complex is concentrated in degrees 0,n10,\dots n-1, then we say LL is a Lie nn-algebroid.

In analogy with the case in Lie theory, if LL is a Lie \infty-algebroid, we call the DGCA (CE(L),d CE)(CE(L),d_{CE}) the Chevalley-Eilenberg algebra of LL. Note that CE(L)CE(L) is in fact a quasi-free DGCA i.e. it is free as a graded commutative algebra. A morphism of Lie \infty-algebroids L 1L 2L_1 \to L_2 is a dgca morphism CE(L 2)CE(L 1)CE(L_2) \to 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 EME \to M equipped with a bundle morphism ρ:ETM\rho \colon E \to T M 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])S^{\bullet}_{R}(L^{\ast}[-1]), where R=C (M)R=C^{\infty}(M) and L=Γ(E)L= \Gamma(E). Hence a Lie algebroid is a Lie \infty-algebroid in the sense of def. 1. Similarly, Courant algebroids are also examples of Lie \infty-algebroids (Roytenberg99).

The geometry of L L_\infty-algebroids

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

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

\infty-Chern-Simons elements

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

Definition

The Weil algebra of a Lie \infty-algebroid LL is the dg-algebra

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

where the differential d Wd_W is defined to be

(2)d W=[d CE 0 1 d CE]. d_W = \begin{bmatrix} d_{CE} & 0 \\ 1 & -d_{CE} \end{bmatrix}.

We may write this as

d W=d CE+d, d_W = d_{CE} + \mathbf{d} \,,

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

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

Proposition

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

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

Definition

An nn-cocycle on a Lie \infty-algebroid LL is a degree nn element μCE(L) \mu \in CE(L) such that d CEμ=0d_{CE}\mu=0.

This next definition generalizes adad-invariant polynomials from Lie theory to ∞-Lie theory:

Definition

An invariant polynomial on a Lie \infty-algebroid LL is an element

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

such that

dp=0 \mathbf{d} p = 0

and

d CEp=0 d_{CE} p = 0

and hance

(4)d Wp=0. d_W p =0.

The space of invariant polynomials is denoted inv(L)inv(L).

Clearly inv(L)W(L)inv(L) \subseteq W(L) is a dg-sub algebra with the trivial differential. Note that if LL is a Lie algebra, then the d Wd_W-closed condition is equivalent to the usual adad-invariance.

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

Definition

Let μ\mu be a cocycle on a Lie \infty-algebroid LL and pinv(L)p \in inv(L) an invariant polynomial. A Chern-Simons element for μ\mu and pp is an element csW(L)cs \in W(L) such that

(5)π(cs)=μ, \pi(cs) = \mu,

and

(6)d Wcs=p. d_W cs = p.

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

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

(8) cs μ p d Wcs \array{ && cs & \mapsto \mu\\ && \downarrow && \\ p &\mapsto& d_W cs&&\\ }
Remark

A Chern-Simons element cscs witnessing the transgression from \langle - \rangle to μ\mu is equivalently a commuting diagram of the form

(9)CE(𝔞) μ CE(b n) cocycle W(𝔞) cs W(b n) ChernSimonselement inv(𝔞) inv(b n) invariantpolynomial \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 nn-algebroid 𝔞\mathfrak{a} is a cocycle

(10)Σ^ coskexp(𝔞) conn Σ \array{ \hat \Sigma &\stackrel{\nabla}{\to}& \mathbf{cosk}\exp(\mathfrak{a})_{conn} \\ \downarrow^{\mathrlap{\simeq}} \\ \Sigma }

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

Ω vert (U×Δ k) A vert CE(𝔞) transitionfunction Ω (U×Δ k) A W(𝔞) connectionforms Ω (U) F A inv(𝔞) curvaturecharacteristicforms, \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 AF_A of the ∞-Lie algebroid valued differential forms AA on U×Δ kU \times \Delta^k with values in 𝔞\mathfrak{a} in the middle are horizontal.

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

Ω vert (U×Δ k) A vert CE(𝔞) μ CE(b n) :μ(A vert) Ω (U×Δ k) A W(𝔞) cs W(b n) :cs(A) ChernSimonsform Ω (U) F A inv(𝔞) inv(b n) :F A curvature. \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 nn-bundle with connection is quotiented by the discrete group Γ\Gamma of periods of μ\mu, such that the \infty-Chern-Weil homomorphism is given by sending the \infty-connections presented by (10) to

Σ^ cosk nexp(𝔞) conn exp(cs) B n(/Γ) conn Σ. \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 nn-form is locally given by the Chern-Simons form cs(A)cs(A). This is the Lagrangian of the \infty-Chern-Simons theory defined by (𝔞,)(\mathfrak{a},\langle - \rangle) and evaluated on the given \infty-connection. If Σ\Sigma is a smooth manifold of dimension nn, then the higher holonomy of this circle nn-bundle over Σ\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 Σcs(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 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

Revised on September 10, 2011 01:00:51 by Urs Schreiber (82.113.99.43)