nLab
Chern-Simons form

Context

\infty-Chern-Weil theory

Differential cohomology

Contents

Idea

A Chern-Simons form CS(A)CS(A) is a differential form naturally associated to a differential form AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) with values in a Lie algebra 𝔤\mathfrak{g}: it is the form trivializing (locally) a curvature characteristic form F AF A\langle F_A \wedge \cdots \wedge F_A \rangle of AA, for \langle \cdots \rangle an invariant polynomial:

d dRCS(A)=F AF A, d_{dR} CS(A) = \langle F_A \wedge \cdots \wedge F_A \rangle \,,

where F AΩ 2(X,𝔤)F_A \in \Omega^2(X,\mathfrak{g}) is the curvature 2-form of AA.

Therefore it is often also called a secondary characteristic form.

More generally, for A,AΩ 1(P,𝔤)A,A' \in \Omega^1(P, \mathfrak{g}) two 𝔤\mathfrak{g}-valued 1-forms and for A^Ω 1(P×[0,1],𝔤)\hat A \in \Omega^1(P \times [0,1],\mathfrak{g}) a “path of connections”, the Chern-Simons form relative to AA and AA' is a form that trivializes the difference between the two curvature characteristic forms

d dRCS(A,A)=(F A) k(F A) k. d_{dR}CS(A,A') = \langle (F_A)^k \rangle - \langle (F_{A'})^k \rangle \,.

Chern-Simons forms are of interest notably when the differential forms A,AA,A' are (local representatives of) connections on a GG-principal bundle PXP \to X, for instance if AΩ 1(P,𝔤)A \in \Omega^1(P,\mathfrak{g}) is an Ehresmann connection 1-form.

Often the term Chern-Simons form is taken to refer to the case where 𝔤\mathfrak{g} is a semisimple Lie algebra with binary invariant polynomial ,\langle -, -\rangle (e.g. the Killing form) in which case CS(A)CS(A) is the 3-form

Ad dRA+cA[AA]. \langle A \wedge d_{dR} A\rangle + c \langle A \wedge [A \wedge A] \rangle \,.

Even more specifically, often the term is understood to refer to the case where 𝔤𝔤𝔩(n)\mathfrak{g} \subset \mathfrak{gl}(n) is a matrix Lie algebra, for instance 𝔬(n)\mathfrak{o}(n) (for the orthogonal group) or notably 𝔲(n)\mathfrak{u}(n) (for the unitary group). In that case the invariant polynomials may be taken to be given by matrix traces: =tr()\langle \cdots \rangle = tr(\cdots ).

Details

It is sufficient to discuss properties of Chern-Simons forms for 𝔤\mathfrak{g}-valued 1-forms. The corresponding statements for connections on a GG-bundle follow straightforwardly.

Paths of connections

Let UU be a smooth manifold.

Definition

A smooth path of 𝔤\mathfrak{g}-valued 1-forms on UU is a smooth 1-form A^Ω 1(U×[0,1],𝔤)\hat A \in \Omega^1(U\times [0,1],\mathfrak{g})

Call this path pure shift if ι tA^=0\iota_{\partial_t} \hat A = 0, where t:U×[0,1][0,1]t : U \times [0,1] \to [0,1] \hookrightarrow \mathbb{R} is the canonical coordinate along the interval.

We say this path goes from A 0:=ψ 0 *A^A_0 := \psi_0^* \hat A to A 1:=ψ 1 *A^A_1 := \psi_1^* \hat A, where

ψ t:UU×*Id×tU×[0,1] \psi_t : U \simeq U \times * \stackrel{Id \times t}{\to} U \times [0,1]

picks the copy of UU at parameter tt.

So a smooth path is a smooth 1-form on the cylinder U×[0,1]U \times [0,1] and it is pure shift if it has no “leg” along the [0,1][0,1]-direction. We will see that ι tA^\iota_{\partial_t} \hat A encodes infinitesimal gauge transformations, while tA^\partial_t \hat A is the change by infinitesimal shifts minus infinitesimal gauge transformations of the connection.

Definition/Observation

Let PP be an invariant polynomial on 𝔤\mathfrak{g} of arity nn.

Consider the fiber integration

CS P(A 0,A 1):= [0,1]P(F A^F A^). CS_P(A_0,A_1) := \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,.

This defines a (2n1)(2n-1)-form CS P(A 0,A 1)Ω 2n1(U)CS_P(A_0,A_1) \in \Omega^{2n-1}(U).

We have that the exterior differential of this form is the difference of the curvature characteristic forms of A 0A_0 and A 1A_1:

d dRCS P(A 0,A 1)=P(F A 1F A 1)P(F A 0F A 0) d_{dR} CS_P(A_0,A_1) = P(F_{A_1} \wedge \cdots \wedge F_{A_1}) - P(F_{A_0} \wedge \cdots \wedge F_{A_0})
Proof

Write the fiber integration more explicitly as an integral

CS P(A 0,A 1)= [0,1]ψ t *ι tP(F A^F A^)dt. CS_P(A_0,A_1) = \int_{[0,1]} \psi_t^* \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \,.

Then use that d dRd_{dR} is linear and commutes with pullback, use Cartan's magic formula d dRι t+ι d dR= td_{dR} \circ \iota_{\partial_t} + \iota_{\partial} \circ d_{dR} = \mathcal{L}_{\partial_t} in view of the fact that P(F A^F A^)P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) is a closed form and then finally apply the Stokes theorem:

d dR 0 1ψ t *ι tP(F A^F A^)dt = 0 1d dRψ t *ι tP(F A^F A^)dt = 0 1ψ t *d dRι tP(F A^F A^)dt = 0 1ψ t *ddtP(F A^F A^)dt = 0 1(ddtP(F A^F A^))(t)dt =ψ 1 *P(F A^F A^)ψ 0 *P(F A^F A^) =P(F A 1F A 1)P(F A 0F A 0). \begin{aligned} d_{dR} \int_0^1 \psi^*_t \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t & = \int_0^1 d_{dR} \psi^*_t \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \psi^*_t d_{dR} \iota_{\partial_t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \psi^*_t \frac{d}{d t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t \\ & = \int_0^1 \left(\frac{d}{d t} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})\right)(t) d t \\ & = \psi^*_1 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) - \psi^*_0 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \\ &= P(F_{A_1} \wedge \cdots \wedge F_{A_1}) - P(F_{A_0} \wedge \cdots \wedge F_{A_0}) \end{aligned} \,.

Explicit formulas

Above we saw that a general expression for the Chern-Simons CS P(A 0,A 1)CS_P(A_0,A_1) obtained from a path of connections A^\hat A between A 0A_0 and A 1A_1 is

CS P(A 0,A 1)= [0,1]P(F A^F A^). CS_P(A_0, A_1) = \int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) \,.

We now unwind this to get explicit formulas for the Chern-Simons form in terms of wedge products of connection forms and their curvatures.

For A^\hat A a pure shift path, A^:tA t\hat A : t \mapsto A_t notice that the curvature 2-form of A^\hat A is

F A^(t)=F A t+( tA t)dt. F_{\hat A}(t) = F_{A_t} + (\partial_t A_t) \wedge d t \,.

Inserting this into the above expression yields

CS P(A 0,A 1)= 0 1P( tAF A tF A t). CS_P(A_0,A_1) = \int_0^1 P(\partial_t A \wedge F_{A_t} \wedge \cdots \wedge F_{A_t}) \,.

Notably if A 0=0A_0 = 0 and A^\hat A is the constant path A^:ttA\hat A : t \mapsto t A to A 1:=AA_1 := A such that

F A^ =td dRA+t 2[AA] =tF A+(t 2t)[AA] \begin{aligned} F_{\hat A} &= t d_{dR} A + t^2 [A \wedge A] \\ &= t F_A + (t^2 - t) [A \wedge A] \end{aligned}

this yields

CS P(A):= 0 1P(A(tF A+(t 2t)[AA]))(tF A+(t 2t)[AA]))). CS_P(A) := \int_0^1 P(A \wedge (t F_{A} + (t^2 - t) [A \wedge A])) \wedge \cdots (t F_{A} + (t^2 - t) [A \wedge A]))) \,.

This is just an integral over a polynomial in tt with constant coefficients in forms. Peforming the integral yields a bunch of coefficients c ic_i and with these the Chern-Simons form achieves the form

CS(A)=c 1AF AF A+c 2A[AA]F AF A+. CS(A) = c_1 \langle A \wedge F_A \wedge \cdots \wedge F_A \rangle + c_2 \langle A \wedge [A \wedge A] \wedge F_A \wedge \cdots F_A \rangle + \cdots \,.

Particularly for n=2n = 2 and using the definition of the curvature 2-form F A=d dRA+[AwdgeA]F_A = d_{dR} A + [A \wdge A] we get

CS(A)=AdA+cA[AA]. CS(A) = \langle A \wedge d A\rangle + c \langle A \wedge [A \wedge A]\rangle \,.

Gauged paths of connections

Above we defined CS(A 0,A 1)CS(A_0,A_1) for every path of connections form A 0A_0 to A 1A_1 which is pure shift . This is a possibly convenient but unnecessary restriction:

Notice that a general (gauged) path is a general 1-form A^Ω 1(U×[0,1],𝔤)\hat A \in \Omega^1(U \times [0,1], \mathfrak{g}) which we can decompose in the form

A^:tA t+λdt, \hat A : t \mapsto A_t + \lambda d t \,,

where λ\lambda is a 𝔤\mathfrak{g}-valued function. The parallel transport of λdt\lambda d t along [0,1][0,1] defines an element in GG and shift ( tA) t(\partial_t A)_t of the connection along [0,1][0,1] is now relative to the gauge transformation on AA induced by this function: the curvature 2-form now is

F A^:tF A t+(( tA) t+d Uλ(t)+[λ,A t])dt F_{\hat A} : t \mapsto F_{A_t} + ((\partial_t A)_t + d_U \lambda(t) + [\lambda,A_t]) \wedge d t
Proposition

The Chern-Simons form [0,1]P(F A^F A^)\int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) defined with respect to any gauged lift of a pure shift path of connections differs from that of the pure shift path by an exact term.

In \infty-Chern-Weil theory

We discuss now a more encompassing perspective on Chern-Simons forms the way it occurs in ∞-Chern-Weil theory.

Prerequisites

We need to collect a few notions described elsewhere, on which the following discussion is based.

For 𝔤\mathfrak{g} a Lie algebra or more generally an ∞-Lie algebra we have the following dg-algebras naturally associated with it:

Given nn \in \mathbb{N}, the Lie integration of 𝔤\mathfrak{g} to degree nn is the ∞-Lie groupoid which is the nn-truncation of the simplicial presheaf

exp(𝔤):U,[n]dgAlg(CE(𝔤),C (U)Ω (Δ n)), \exp(\mathfrak{g}) : U,[n] \mapsto dgAlg( CE(\mathfrak{g}), C^\infty(U)\otimes \Omega^\bullet(\Delta^n) ) \,,

where here and in the following Ω (Δ n)\Omega^\bullet(\Delta^n) denotes the de Rham complex dg-algebra of those smooth differential forms ω\omega on the standard smooth nn-simplex that have sitting instants in that for each kk \in \mathbb{N} every kk-face of Δ n\Delta^n has an open neighbourhood such that restricted to that neighbourhood ω\omega is constant in the direction perpendicular to the face.

This is a one-object ∞-Lie groupoid which we may write

BG=cosk n+1exp(𝔤), \mathbf{B}G = \mathbf{cosk}_{n+1} \exp(\mathfrak{g}) \,,

thus defining the ∞-Lie group GG that integrates 𝔤\mathfrak{g} in degree nn.

At ∞-Chern-Weil theory is explained that a resolution of BG\mathbf{B}G that serves to compute curvature characteristic forms in that it encodes pseudo-connections on GG-principal ∞-bundles is given by the simplicial presheaf

BG diff:=cosk n+1(U,[n]{C (U)Ω (Δ n) CE(𝔤) Ω (U)Ω (Δ n) W(𝔤)}), \mathbf{B}G_{diff} := \mathbf{cosk}_{n+1} \left( U,[n] \mapsto \left\{ \array{ C^\infty(U) \otimes \Omega^\bullet(\Delta^n) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n) &\leftarrow& W(\mathfrak{g}) } \right\} \right) \,,

where the vertical morphisms are the canonical ones.

Much of the subtlety of the full theory of connections of \infty-bundles comes from the finite coskeleton-truncation here. For the following discussion of Chern-Simons forms it is helpful to first ignore this issue by taking n=n = \infty, hence ignoring the truncation for the moment. This is sufficient for understand everything about Chern-Simons forms locally.

A cocycle in ∞-Lie algebra cohomology in degree kk is a morphism

CE(𝔤)CE(b k1):μ. CE(\mathfrak{g}) \leftarrow CE(b^{k-1} \mathbb{R}) : \mu \,.

Simply by composition (since we ignore the truncation for the moment), this integrates to a cocycle of the corresponding \infty-Lie groupoids

exp(μ):exp(𝔤)exp(b k1), \exp(\mu) : \exp(\mathfrak{g}) \to \exp(b^{k-1}\mathbb{R}) \,,

At ∞-Chern-Weil theory it is discussed how the proper lift of this through the extension BG diff\mathbf{B}G_{diff} that computes the abstractly defined curvature characteristic classes is given by finding the invariant polynomial ,W(𝔤)\langle -,-\rangle \in W(\mathfrak{g}) that is in transgression with μ\mu in that we have a commuting diagram

CE(𝔤) μ CE(b k1) W(𝔤) (cs,,)) W(b k1) inv(𝔤) , CE(b k) \array{ CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\stackrel{(cs, \langle-,-\rangle))}{\leftarrow}& W(b^{k-1}\mathbb{R}) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\stackrel{\langle -,-\rangle}{\leftarrow}& CE(b^k \mathbb{R}) }

with a choice of interpolating Chern-Simons element csW(𝔤)cs \in W(\mathfrak{g}), which induces by precomposition with its upper part the morphism

exp((cs,,)):BG diffexp(b k1) diff. \exp((cs,\langle-,-\rangle)) : \mathbf{B}G_{diff} \to \exp(b^{k-1}\mathbb{R})_{diff} \,.

By further projection to its lower part we get furthermore a morphism

exp(b k1) diff dRB k simp:=(U,[n]{Ω (U)Ω (Δ n)W(b k)}). \exp(b^{k-1}\mathbb{R})_{diff} \to \mathbf{\flat}_{dR}\mathbf{B}^k \mathbb{R}_{simp} := (U,[n] \mapsto \{ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow W(b^k \mathbb{R}) \}) \,.

Finally – and this is crucial now for obtaining the incarnation of Chern-Simons forms at integrals of curvature forms as in the above discussion – at ∞-Lie groupoid in the section on simplicial differential forms (see also circle n-bundles with connection the section Models from ∞-Lie integration) it is discussed that the operation that takes the nn-cells on the right and integrates the corresponding forms over the nn-simplex yields an equivalence

Δ :(U,[n]{Ω (U)Ω (Δ n)W(b k)})Ξ(d dRΩ k1()d dRΩ closed k()) \int_{\Delta^\bullet} : (U,[n] \mapsto \{ \Omega^\bullet(U) \otimes \Omega^\bullet(\Delta^n) \leftarrow W(b^k \mathbb{R}) \}) \;\;\;\;\to \;\;\;\; \Xi( \stackrel{d_{dR}}{\to}\Omega^{k-1}(-)\stackrel{d_{dR}}{\to}\Omega^k_{closed}(-))

to the image of the \mathbb{R}-Deligne complex of sheaves under the Dold-Kan correspondence.

Higher order Chern-Simons form

With all of the above in hand, we can make now the following observations:

For XX a smooth manifold and 𝔤\mathfrak{g} an ∞-Lie algebra with coefficient for pseudo-connections being BG diff\mathbf{B}G_{diff} as above, a morphism

A:XBG diff A : X \to \mathbf{B}G_{diff}

of simplicial presheaves (no resolution on the left, since we are concentrating on globally defined forms for the present purpose) is effectively a 𝔤\mathfrak{g}-values differential form on XX

For μ\mu a cocycle on 𝔤\mathfrak{g} and ,\langle -,-\rangle a corresponding invariant polynomial the composite

XBG diff dRB k chn X \to \mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

discussed above produces the corresponding curvature characteristic form.

A homotopy

():XΔ[1]BG diff (\nabla \to \nabla') : X \cdot \Delta[1] \to \mathbf{B}G_{diff}

is a smooth path in the space of 𝔤\mathfrak{g}-valued forms on XX. Under the adjunction

[XΔ[1], dRB k chn][X, dRB k chn Δ[1]] [X \cdot \Delta[1], \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}] \simeq [X, \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}^{\Delta[1]}]

this corresponds to a (k1)(k-1)-form on XX this is the Chern-Simons form

CS():X dRB k chn Δ[1]. CS(\nabla \to \nabla') : X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}^{\Delta[1]} \,.

The higher homotopies are higher order Chern-Simons forms.

The following proposition says this in a more precise way for ordinary Chern-Simons forms.

Ordinary Chern-Simons forms revisited

We now show how the traditional definition of Chern-Simons forms is reproduced by the general abstract mechanism.

Proposition

(ordinary Chern-Simons form)

Let 𝔤\mathfrak{g} be a Lie algebra, and ,W(𝔤)\langle -,-\rangle \in W(\mathfrak{g}) an invariant polynomial.

Then morphisms (of simplicial presheaves)

A:XBG diff A : X \to \mathbf{B}G_{diff}

are in canonical bijection with Lie-algebra valued 1-forms AΩ 1(X,𝔤)A \in \Omega^1(X,\mathfrak{g}). Morphisms

X dRB k chn X \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

are in canonical bijection with closed kk-forms on XX and composition with the morphism

BG diff dRB k chn \mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

discussed above and under this canonical identification the composite

F A:XABG diff dRB k chn \langle F_A \rangle : X \stackrel{A}{\to} \mathbf{B}G_{diff} \stackrel{}{\to} \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn}

is the corresponding curvature characteristic form.

Homotopies

(A 1γA 2):X×Δ[1]BG diff (A_1 \stackrel{\gamma}{\to} A_2) : X\times \Delta[1] \to \mathbf{B}G_{diff}

are in canonical bijection with smooth paths in the space of 𝔤\mathfrak{g}-valued 1-forms on XX and under composition with BG diff dRB k\mathbf{B}G_{diff} \to \mathbf{\flat}_{dR}\mathbf{B}^k \mathbb{R} these identify with the corresponding Chern-Simons form

F ACS(AγA)F A:X( dRB k chn) Δ[1]. \langle F_A\rangle \stackrel{CS(A \stackrel{\gamma}{\to} A')}{\to} \langle F_{A'}\rangle : X \to (\mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn})^{\Delta[1]} \,.
Proof

This is a straightforward unwinding of the definitions. We spell it out in the following in order to highlight the way the mechanism works.

By the Yoneda lemma and the definition of BG diff\mathbf{B}G_{diff}, a morphism XBG diffX \to \mathbf{B}G_{diff} is equivalently a diagram

C (X)Ω (Δ 0) CE(𝔤) Ω (X)Ω (Δ 0) (A,F A) W(𝔤). \array{ C^\infty(X) \otimes \Omega^\bullet(\Delta^0) &\leftarrow& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) \otimes \Omega^\bullet(\Delta^0) &\stackrel{(A,F_A)}{\leftarrow}& W(\mathfrak{g}) } \,.

Since CE(𝔤)CE(\mathfrak{g}) is trivial in degree 0 and since C (X)otimessΩ (Δ 0)C^\infty(X)\otimess \Omega^\bullet(\Delta^0) is trivial above degree 0, the top morphism is necessarily 0 and the commutativity of the diagram is an empty condition.

The bottom morphism on the other hand enccodes precisely a 𝔤\mathfrak{g}-valued form, as discussed in some detail at Weil algebra.

Composition with the morphism BG diff dRB k chn\mathbf{B}G_{diff} \to \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn} is composition of the bottom morphism of the above digram with W(𝔤)CE(b k1):W(\mathfrak{g}) \leftarrow CE(b^{k-1}\mathbb{R}) : \langle - \rangle followed by fiber integration of the resulting kk-form

Ω (X)Ω (Δ 0)(A,F A)W(𝔤)CE(b k1):F A \Omega^\bullet(X)\otimes \Omega^\bullet(\Delta^0) \stackrel{(A,F_A)}{\leftarrow} W(\mathfrak{g}) \stackrel{\langle - \rangle}{\leftarrow} CE(b^{k-1}\mathbb{R}) : \langle F_A \rangle

over the point. This fiber integration is of course trivial, so that we find that indeed X(A,F A)BG difflanglw dRB k chnX \stackrel{(A,F_A)}{\to} \mathbf{B}G_{diff} \stackrel{\langlw - \rangle}{\to} \mathbf{\flat}_{dR} \mathbf{B}^k \mathbb{R}_{chn} is the curvature characteristic form defined on \langle -\rangle on AA.

Next, a homotopy (AγA):XΔ[1]BG diff(A \stackrel{\gamma}{\to} A') : X \cdot \Delta[1] \to \mathbf{B}G_{diff} is (again by the Yoneda lemma) a diagram

C (X)Ω (Δ 1) λ CE(𝔤) Ω (X)Ω (Δ 0) (A,F A)γ(A,F A) W(𝔤). \array{ C^\infty(X) \otimes \Omega^\bullet(\Delta^1) &\stackrel{\lambda}{\leftarrow}& CE(\mathfrak{g}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) \otimes \Omega^\bullet(\Delta^0) &\stackrel{(A,F_A) \stackrel{\gamma}{\to} (A',F_{A'})}{\leftarrow}& W(\mathfrak{g}) } \,.

The top morphism defines an XX-parameterized family of 𝔤\mathfrak{g}-valued 1-form on the interval [0,1][0,1], which is canonically identified with a smooth function g:X×[0,1]Gg : X \times [0,1] \to G into the simply connected Lie group integrating] 𝔤\mathfrak{g} based at the identity, g(x,0)=eg(x,0) = e, by the formula

λ=g *θ \lambda = g^* \theta

where θΩ 1(G,𝔤)\theta \in \Omega^1(G, \mathfrak{g}) is the Maurer-Cartan form on GG,

or conversely by parallel transport

f(x,s)=Pexp( [0,s]λ(x,s)ds) f(x,s) = P \exp(\int_{[0,s]} \lambda(x,s) d s)

We may think of this as a smooth path of gauge transformations .

The bottom morphism encodes a 𝔤\mathfrak{g}-valued form

A^+λΩ 1(X×[0,1],𝔤) \hat A + \lambda \in \Omega^1(X \times [0,1] , \mathfrak{g})

with A^Ω 1(X,𝔤)C ([0,1])\hat A \in \Omega^1(X,\mathfrak{g}) \otimes C^\infty([0,1]) and λ\lambda as before, such that A^(s=0)=A\hat A(s = 0) = A and A^(s=1)=A\hat A(s = 1) = A'.

This is a smooth path in the space of 1-forms . In the case that λ=0\lambda = 0 this is a pure shift path in the terminology above. we look at this case in the following, for ease of notation.

Under composition with W(𝔤)CE(b k1):W(\mathfrak{g}) \leftarrow CE(b^{k-1}\mathbb{R}) : \langle -\rangle this becomes a kk-form

F A^Ω k(X)C (Δ 1)Ω k1(X)Ω 1(Δ 1). \langle F_{\hat A } \rangle \in \Omega^{k}(X)\otimes C^\infty(\Delta^1)\oplus \Omega^{k-1}(X)\otimes \Omega^1(\Delta^1) \,.

The fiber integration of this over Δ 1\Delta^1 is manifestly the same operation as that in the definition of the Chern-Simons form above.

As secondary characteristic forms

If a curvature characteristic form vanishes (for instance if the connection is flat or the degree of the curvature characteristic form is simply greater than the dimension of XX) the corresponding Chern-Simons form is a closed form. So in this case the de Rham cohomology class of the curvature characteristic form becomes trivial, but the Chern-Simons form provides another de Rham class. This is therefore called a secondary characteristic class.

Chern-Simons theory

In particular on a 3-dimensional smooth manifold XX necessarily the Chern-Simons 3-form is closed. The functional

(AΩ 1(X,𝔤)) XCS(A) (A \in \Omega^1(X,\mathfrak{g})) \mapsto \int_X CS(A)

is the action functional of the quantum field theory called Chern-Simons theory.

More generally, for XX a (2n1)(2n-1)-dimensional smooth manifold and ,,\langle -,\cdots, -\rangle an invariant polynomial of arity nn, the analous formula defines the action functional of (2n+1)(2n+1)-dimensional Chern-Simons theory.

In terms of \infty-Lie algebroids

As discussed at invariant polynomial, Chern-Simons elements int the Weil algebra W(𝔤)W(\mathfrak{g}) of a Lie algebra 𝔤\mathfrak{g} induce the transgression between invariant polynomials and cocycles in Lie algebra cohomology.

For

T vertP A vert 𝔤 TP (A,F A) inn(𝔤) TX (P i(F A)) ib n i𝔲(1) \array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} \\ \downarrow && \downarrow \\ T P &\stackrel{(A,F_A)}{\to}& inn(\mathfrak{g}) \\ \downarrow && \downarrow \\ T X &\stackrel{(P_i(F_A))}{\to}& \prod_i b^{n_i} \mathfrak{u}(1) }

the data of an Ehresmann connection on a GG-principal bundle expressed as a diagram of ∞-Lie algebroids with the curvature characteristic forms on the bottom, a choice of transgression element cs Pcs_P for an invariant polynomial PP in transgression with a Lie algebra cocycle μ\mu induces a diagram

𝔤 μ b n𝔲(1) inn(𝔤) (cs P,P) eb n𝔲(1) ib n i𝔲(1) p i b n+1𝔲(1). \array{ \mathfrak{g} &\stackrel{\mu}{\to}& b^n \mathfrak{u}(1) \\ \downarrow && \downarrow \\ inn(\mathfrak{g}) &\stackrel{(cs_P,P)}{\to}& e b^{n} \mathfrak{u}(1) \\ \downarrow && \downarrow \\ \prod_i b^{n_i}\mathfrak{u}(1) &\stackrel{p_i}{\to}& b^{n+1} \mathfrak{u}(1) } \,.

The pasting of this to the above Ehresmann connection expresses in the middle horizontal morphism the Chern-Simons form cs P(A)cs_P(A) and its curvature characteristic form P(F A)P(F_A)

T vertP A vert 𝔤 μ b n𝔲(1) TP A inn(𝔤) (cs P,P) eb n𝔲(1) TX (P i) ib n i𝔲(1) p i b n+1𝔲(1). \array{ T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &\stackrel{\mu}{\to}& b^n \mathfrak{u}(1) \\ \downarrow && \downarrow && \downarrow \\ T P &\stackrel{A}{\to}& inn(\mathfrak{g}) &\stackrel{(cs_P,P)}{\to}& e b^n \mathfrak{u}(1) \\ \downarrow && \downarrow && \downarrow \\ T X &\stackrel{(P_i)}{\to}& \prod_i b^{n_i} \mathfrak{u}(1) &\stackrel{p_i}{\to}& b^{n+1} \mathfrak{u}(1) } \,.

References

The article introducing the concept is

As it says in the introduction of this article, it was motivated by an attempt to find a combinatorial formula for the Pontrjagin class of a Riemannian manifold (i.e. that associated to the O(n)-principal bundle to which the tangent bundle is associated) and the Chern-Simons form appeared as a boundary term that obstructed to original attempt to derive the Pontrjagin class by integrating curvature classes simplex-by-simplex. But A combinatorial formula of the kind these authors were looking for was however (nevertheless) given later in

  • Jean-Luc Brylinski, Dennis McLaughlin? Čech cocycles for characteristic classes , Comm. Math. Phys. 178 (1996) (pdf)

The statements about “pure shift” paths are reviewed on the first few pages of

which discusses the relevance of Chern-Simons forms in differential K-theory.

The L-∞-algebra-formulation is discussed in SSS08.

An abstract algebraic model of the algebra of Chern’s characteristic classes and Chern-Simons secondary characteristic classes and of the gauge group action on this algebra (which also enables some noncommutative generalizations) is pioneered in 2 articles

  • Israel M. Gelfand, Mikhail M. Smirnov, The algebra of Chern-Simons classes, the Poisson bracket on it, and the action of the gauge group, Lie theory and geometry, 261–288, Progr. Math. 123, Birkhäuser 1994; Chern-Simons classes and cocycles on the Lie algebra of the gauge group, The Gelfand Mathematical Seminars, 1993–1995, 101–122, Birkhäuser 1996.

Revised on January 13, 2013 20:29:45 by Urs Schreiber (203.116.137.162)