Chern-Dold character

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Rational homotopy theory
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[[!include differential graded objects - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Chern-Dold character_ is the natural generalization of the [[Chern character]] from [[topological K-theory]] to any [[generalized (Eilenberg-Steenrod) cohomology theory]]. It is given essentially by [[rationalization]] of [[Brown representability theorem|coefficient spectra]].

## Definition

For $E$ a [[spectrum]] and $E^\bullet$ the [[generalized (Eilenberg-Steenrod) cohomology theory|generalized cohomology theory]] it [[Brown representability theorem|represents]]

$$
  E^\bullet(X) \;\simeq\; \pi_{-\bullet} Maps(X,E)
$$

the _[[Chern-Dold character]] for $E$_ ([Buchstaber 70](#Buchstaber70)) is the map induced by [[rationalization]] over the [[real numbers]]

$$
  E \overset{L_{\mathbb{R}}}{\longrightarrow} E_{\mathbb{R}}
$$

i.e. is

\[
  \label{ChernDoldCharacter}
  chd
  \;\colon\;
  E^\bullet(X)
  \;\simeq\;
  \pi_{-\bullet}Maps(X,E)
  \overset{
    \pi_{-\bullet}Maps(X,L_{\mathbb{R}})
  }{\longrightarrow}
  \pi_{-\bullet}Maps(X,E_{\mathbb{R}})
  \;\simeq\;
  E^\bullet_{\mathbb{E}}(X)
  \;\simeq\;
  H^\bullet(X, \pi_{\bullet}(E)\otimes_{\mathbb{Z}}\mathbb{R})
  \,.
\]

The very last equivalence in (eq:ChernDoldCharacter) is due to [Dold 56, Cor. 4](#Dold56) (reviewed in detail in [Rudyak 98, II.3.17](#Rudyak98), see also  [Gross 19, Def. 2.5](#Gross19)).

One place where this neat state of affairs (eq:ChernDoldCharacter) is made fully explicit is [Lind-Sati-Westerland 16, Def. 2.1](#LindSatiWesterland16). Many other references leave this statement somewhat in between the lines (e.g. [Buchstaber 70](#Buchstaber70), [Upmeier 14](#Upmeier14)) and, in addition, often without reference to Dold (e.g. [Hopkins-Singer 02, Sec. 4.8](#HopkinsSinger02), [Bunke 12, Def. 4.45](#Bunke12), [Bunke-Gepner 13, Def. 2.1](#BunkeGepner13), [Bunke-Nikolaus 14, p. 17](#BunkeNikolaus14)).

Beware that some authors say _[[Chern-Dold character]]_ for the full map in (eq:ChernDoldCharacter) (e.g.
[Buchstaber 70](#Buchstaber70),
[Upmeier 14](#Upmeier14), [Lind-Sati-Westerland 16, Def. 2.1](#LindSatiWesterland16)), while other authors mean by this only that last equivalence in (eq:ChernDoldCharacter) (e.g. [Rudyak 98, II.3.17](#Rudyak98), [Gross 19, Def. 2.5](#Gross19)).

## Exanples

Examples of [[Chern-Dold characters]]:

* [[Chern character]] on [[KU]];

* [[Pontrjagin character]] on [[KO]].

Further examples listed in [FSS 20](#FSS20)

## References

The identification of rational [[generalized (Eilenberg-Steenrod) cohomology|generalized cohomology]] as [[ordinary cohomology]] with [[coefficients]] in the rationalized [[stable homotopy groups]] is due to

* {#Dold56} [[Albrecht Dold]], _Relations between ordinary and extraordinary homology_, Matematika, 9:2 (1965), 8–14; Colloq. algebr. Topology, Aarhus Universitet, 1962, 2–9 ([mathnet:mat350](http://mi.mathnet.ru/eng/mat350)), reprinted in: J. Adams & G. Shepherd (Authors), _Algebraic Topology: A Student's Guide_ (London Mathematical Society Lecture Note Series, pp. 166-177). Cambridge: Cambridge University Press 1972 ([doi:10.1017/CBO9780511662584.015](https://doi.org/10.1017/CBO9780511662584.015))

reviewed in

* [Hilton 71, Theorem 3.18](#Hilton71)

* {#Rudyak98} [[Yuli Rudyak]], II.7.13 in: _On Thom Spectra, Orientability, and Cobordism_, Springer 1998 ([doi:10.1007/978-3-540-77751-9](https://doi.org/10.1007/978-3-540-77751-9))

* {#Gross19} Jacob Gross, _The homology of moduli stacks of complexes_ ([arXiv:1907.03269](https://arxiv.org/abs/1907.03269))

The combination of [Dold 56](#Dold56) to the [[Chern-Dold character]] on [[generalized (Eilenberg-Steenrod) cohomology theory]] is due (for [[complex cobordism cohomology]]) to

* {#Buchstaber70} [[Victor Buchstaber]], _The Chern–Dold character in cobordisms. I_,

  Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 ([mathnet:3530](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3530&option_lang=eng))

  English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 ([doi:10.1070/SM1970v012n04ABEH000939](https://iopscience.iop.org/article/10.1070/SM1970v012n04ABEH000939))

Review in

* [Hilton 71, p. 50](#Hilton71)

That the Chern-Dold character reduces to the original Chern character on K-theory is

* [Hilton 71, Theorem 5.8](#Hilton71).

That the Chern-Dold character is given by rationalization of representing spectra is made fully explicit in

* {#LindSatiWesterland16} [[John Lind]], [[Hisham Sati]], [[Craig Westerland]], Section 2.1: _Twisted iterated algebraic K-theory and topological T-duality for sphere bundles_, Ann. K-Th. 5 (2020) 1-42 ([arXiv:1601.06285](https://arxiv.org/abs/1601.06285))

This rationalization construction appears also (without attribution to [#Hilton 71](#Hilton71) or [Buchstaber 70](#Buchstaber70) or [Dold 56](#Dold56)) in the following articles (all in the context of [[differential cohomology]]):

* {#HopkinsSinger02} [[Mike Hopkins]], [[Isadore Singer]], Section [4.8, page 47](http://arxiv.org/PS_cache/math/pdf/0211/0211216v2.pdf#page=47) of _[[Quadratic Functions in Geometry, Topology,and M-Theory]]_, ([math.AT/0211216](http://arxiv.org/abs/math.AT/0211216)).

* {#Bunke12} [[Ulrich Bunke]], _Differential cohomology_ ([arXiv:1208.3961](http://arxiv.org/abs/1208.3961))

* {#BunkeGepner13} [[Ulrich Bunke]], [[David Gepner]], around def. 2.1 of: _Differential function spectra, the differential Becker-Gottlieb transfer, and applications to differential algebraic K-theory_ ([arXiv:1306.0247](http://arxiv.org/abs/1306.0247))

* {#Upmeier14} Markus Upmeier, _Refinements of the Chern-Dold Character: Cocycle Additions in Differential Cohomology_,  J. Homotopy Relat. Struct. 11, 291–307 (2016). ([arXiv:1404.2027](https://arxiv.org/abs/1404.2027), [doi:10.1007/s40062-015-0106-y](https://doi.org/10.1007/s40062-015-0106-y))

* {#BunkeNikolaus14} [[Ulrich Bunke]], [[Thomas Nikolaus]], _Twisted differential cohomology_, Algebr. Geom. Topol. Volume 19, Number 4 (2019), 1631-1710. ([arXiv:1406.3231](http://arxiv.org/abs/1406.3231), [euclid:euclid.agt/1566439272](https://projecteuclid.org/euclid.agt/1566439272))

More on the [[Chern-Dold character]] on [[complex cobordism cohomology]]:

* [[Victor Buchstaber]], A. P. Veselov, _Chern-Dold character in complex cobordisms and abelian varieties_ ([arXiv:2007.05782](https://arxiv.org/abs/2007.05782))

The observation putting this into the general context of [[differential cohomology diagrams]] (see there) of [[stable homotopy types]] in [[cohesion]] is due to

* {#BunkeNikolausVoelkl13} [[Ulrich Bunke]], [[Thomas Nikolaus]], [[Michael Völkl]], section 4.4. of _Differential cohomology theories as sheaves of spectra_ ([arXiv:1311.3188](http://arxiv.org/abs/1311.3188))

based on [Bunke-Gepner 13](#BunkeGepner13).

Further generalization of the [[Chern-Dold character]] to [[non-abelian cohomology]]:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalized]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

The [[equivariant Chern-Dold character]] in [[equivariant cohomology]]:

* [[Wolfgang Lück]], _Chern characters for proper equivariant homology theories and applications to K- and L-theory_, Journal für die reine und angewandte Mathematik, Volume 2002: Issue 543 ([doi:10.1515/crll.2002.015](https://doi.org/10.1515/crll.2002.015), [pdf](https://www.him.uni-bonn.de/lueck/data/eh.pdf))

* [[Wolfgang Lück]], _Equivariant Cohomological Chern Characters_, International Journal of Algebra and Computation, Vol. 15, No. 05n06, pp. 1025-1052 (2005) ([arXiv:math/0401047](https://arxiv.org/abs/math/0401047), [doi:10.1142/S0218196705002773](https://doi.org/10.1142/S0218196705002773))

* [[Wolfgang Lück]], _Equivariant Chern characters_, 2006 ([pdf](https://www.him.uni-bonn.de/lueck/data/goettingen_equi_Chern061130trans.pdf), [[LuckEquivariantChernCharacters.pdf:file]])

[[!redirects Dold-Chern characters]]

[[!redirects Chern character map]]
[[!redirects Chern character maps]]

[[!redirects Chern-Dold character]]
[[!redirects Chern-Dold characters]]

[[!redirects Chern-Dold character map]]
[[!redirects Chern-Dold character maps]]

[[!redirects Dold-Chern character map]]
[[!redirects Dold-Chern character maps]]


Chern-Simons 2-gerbe > history

see *[[Chern-Simons circle 3-bundle]]*


Chern-Simons element

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#### $\infty$-Chern-Simons theory
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#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _Chern-Simons element_ on an [[L-∞ algebroid]] (named after [[Shiing-shen Chern]] and [[James Simons]] who considered this for [[semisimple Lie algebras]]) is an element of its [[Weil algebra]] that exhibits a [[transgression]] between an [[∞-Lie algebroid cocycle]] and an [[invariant polynomial]].

It is construct that arises in the presentation of the [[∞-Chern-Weil homomorphism]] by an [[∞-anafunctor]] of [[simplicial presheaves]].

## Definition

We discuss [[∞-Lie algebra]]s and [[∞-Lie algebroid]]s $\mathfrak{a}$ of [[finite type]] in terms of their [[Chevalley-Eilenberg algebra]]s $CE(\mathfrak{a})$. For $\infty$-Lie algebras these are objects in the [[category]] [[dgAlg]] of [[dg-algebra]]s (over a given ground [[field]]). For $\infty$-Lie algebroids these are dg-algebras equipped with a lift of the degree-0 algebra to an algebra over a given [[Fermat theory]] $T$ and such that the [[differential]] is a $T$-[[derivation]] in this degree. (See [[∞-Lie algebroid]] for details). We shall write in the following $dgAlg$ also for the category of dg-algebras with this extra structure and leave the [[Fermat theory]] $T$ implicit.

+-- {: .num_defn}
###### Definition

For $\mathfrak{g}$ an [[∞-Lie algebra]] or more generally [[∞-Lie algebroid]], $\mu \in CE(\mathfrak{g})$ a [[∞-Lie algebra cocycle]] (a closed element of the [[Chevalley-Eilenberg algebra]]) and $\langle - \rangle \in W(\mathfrak{g})$ an [[invariant polynomial]], a **Chern-Simons element** exhibiting the _[[transgression]]_ between the two is an element

$$
  cs \in W(\mathfrak{g})
$$

such that

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

1. and $cs|_{CE(\mathfrak{g})} = \mu$

where the restriction is along the canonical morphism $W(\mathfrak{g}) \to CE(\mathfrak{g})$.

=--

+-- {: .num_remark}
###### Remark

Notice that a degree-$n$ [[∞-Lie algebroid cocycle]] $\mu$ is equivalently a morphism

$$
  CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathbb{R}) : \mu
$$

and an [[invariant polynomial]] of degree $n+1$ is equivalently a morphism

$$
  inv(\mathfrak{g}) \leftarrow inv(b^{n-1}\mathbb{R}) =
  CE(b^n \mathbb{R}) : \langle - \rangle
$$

in [[dgAlg]].

=--

+-- {: .num_prop}
###### Observation

A **Chern-Simons element** $cs$ an $\mathfrak{a}$ witnessing the [[transgression]] of $\langle - \rangle$ to $\mu$ is equivalently a morphism

$$
  W(\mathfrak{g}) \leftarrow W(b^{n-1} \mathbb{R})
  : cs
$$

such that we have a [[commuting diagram]] in [[dgAlg]]

\[
  \array{
    CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}& CE(b^{n-1} \mathbb{R})
    &&&
    cocycle
    \\
    \uparrow && \uparrow^{\mathrlap{i^*}}
    \\
    W(\mathfrak{g}) &\stackrel{cs}{\leftarrow}& W(b^{n-1} \mathbb{R})
    &&&
    Chern-Simons\;element
    \\
    \uparrow && \uparrow
    \\
    inv(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}&
    inv(b^{n-1} \mathbb{R})
    &&&
    invariant\;polynomial
  }
  \,,
  \label{TheDiagram}
\]

where the vertical morphisms are the canonical ones.

=--

+-- {: .num_remark}
###### Remark

If we think of

* $W(\mathfrak{g})$ as differential forms on the total space of the universal $G$-bundles;

* $CE(\mathfrak{g})$ as differential forms on the fiber

* $inv(\mathfrak{g})$ as differential forms on the base space

then the abov expresses the classical notion of transgression of forms from the fiber to the base of a fibe bundle (for instance [Borel, section 9](#Borel)).

=--

## Properties

### Uniqueness

+-- {: .num_pop}
###### Observation

For a given transgressive cocycle $\mu$ and transgressing invariant polynomial $\langle - \rangle$ the set of Chern-Simons elements witnessing the [[transgression]] is a [[torsor]] (based over the point and) over the additive [[group]]

$$
  \{\omega \in W(\mathfrak{g}) | d_{W(\mathfrak{g})} \omega = 0, i^* \omega = 0\}
$$

of Chern-Simons elements for vanishing cocycle and vanishing invariant polynomial.

=--

### Canonical $\infty$-Chern-Simons elements
  {#CanonicalChernSimonsElement}

Since the [[Weil algebra]] of an [[L-∞ algebra]] has trivial cohomolgy in positive degree, every [[invariant polynomial]] $\langle -,\cdots, -\rangle$ has a Chern-Simons element and there is a standard formula for it.

+-- {: .num_pop}
###### Proposition

Let $\mathfrak{g}$ be an [[L-∞ algebra]] with $k$-ary brackets $[-,\cdots, -]_k : \mathfrak{g}^{\otimes k} \to \mathfrak{g}$ and equipped with a quadratic [[invariant polynomial]] $\langle -,-\rangle$.

A Chern-Simons element for $\langle-,-\rangle$ is given by the formula

$$
  cs(A) =
  \langle A, d_{dR} A\rangle
  +
  \sum_{k = 1}^\infty
  \frac{2}{(k+1)!}
  \langle A, [A,\cdots,A]_k\rangle
  \,,
$$

where $A : W(\mathfrak{g}) \to \Omega^\bullet(\Sigma)$ is any $\mathfrak{g}$-[[infinity-Lie algebroid-valued differential form|valued form]]

=--

+-- {: .proof}
###### Proof

There is a canonical contracting [[homotopy]]

$$
  \tau : W(\mathfrak{g}) \to W(\mathfrak{g})
$$

satisfying $[d_W, \tau] = Id$

and the above element is

$$
  cs = \tau \langle -,-\rangle
  \,.
$$

To see this, let $\{t_a\}$ be a [[basis]] and $\{t^a\}$ the dual basis. Then the differential of the [[Chevalley-Eilenberg algebra]] can be written

$$
  d_{CE(\mathfrak{g})} t^a
  =
  -
  \sum_{k = 1}^\infty
   \frac{1}{k!}
   [t_{a_1}, t_{a_2}, \cdots, t_{a_k}]^a
   \,\,
   t^{a_1} \wedge t^{a_2}\wedge \cdots t^{a_k}
  \,,
$$

where

$$
  [-,-, \cdots, -] : \mathfrak{g}^{\otimes_k} \to \mathfrak{g}
$$

is the corresponding $k$-ary bracket.

Write

$$
  P_{a b} := \langle t_a , t_b\rangle
  \,,
$$

for the components of the [[invariant polynomial]] in this basis.

Then the claim is that

$$
  cs
  =
  2 P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b
  +
  \sum_{k = 1}^\infty
  C_{a b_1, \cdots, b_k}
   \,\,
  t^a \wedge t^{b_1} \wedge \cdots \wedge t^{b_k}
  \,,
$$

where the coefficients are

$$
  C_{a b_1, \cdots, b_k}
  :=
  \frac{1}{(k+1)!}
  (P_{a b} [t_{b_1}, \cdots, t_{b_k}]^b)
$$

Write $F(\mathfrak{g})$ for the [[free construction|free]] [[dg-algebra]] on the [[graded vector space]] $\mathfrak{g}^*$. In terms of the above basis this is generated from $\{t^a, \mathbf{d}t^a\}$. As discussed at [[Weil algebra]], there is a dg-algebra [[isomorphism]]

$$
  F(\mathfrak{g}) \stackrel{\simeq}{\to} W(\mathfrak{g})
$$

given by sending $t^a \mapsto t^a$ and $\mathbf{d}t^a \mapsto d_{CE} t^a + r^a$.

Let $h : F(\mathfrak{g}) \to F(\mathfrak{g})$ be the [[derivation]] which on generators is defined by

$$
  h : t^a \mapsto 0
$$
$$
  h : \mathbf{d}t^a \mapsto t^a
  \,.
$$

Notice that this is _not_ the homotopy that exhibits the triviality of $Id_{F(\mathfrak{g}^*)}$, rather that homotopy  is $\frac{1}{L} h$, where $L$ is the word length operator for element in $F(\mathfrak{g}^*)$ in terms of the generators $\{t^a , \mathbf{d}t^a\}$.

Therefore the homotopy $\tau$ is the composite top morphism in the diagram

$$
  \array{
    W(\mathfrak{g}) &\stackrel{\tau}{\to}& W(\mathfrak{g})
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \uparrow^{\mathrlap{\simeq}}
    \\
    F(\mathfrak{g}) &\stackrel{\frac{1}{L} h}{\to}& F(\mathfrak{g})
  }
  \,.
$$

Unwinding this, we find

$$
  \begin{aligned}
    cs & := \tau \left( P_{a b} r^{a} \wedge r^b \right)
    \\
    & =
     P_{a b} \tau
    \left(
      d_{W(\mathfrak{g})} t^a
      +
      \sum_{k = 1}^\infty
      [t_{a_1}, \cdots, t_{a_k}]^a
      t^{a_1} \wedge \dots t^{a_k}
    \right)
    \wedge
    \left(
      d_{W(\mathfrak{g})} t^b
      +
      \sum_{k = 1}^\infty
      [t_{b_1}, \cdots, t_{b_k}]^b
      t^{b_1} \wedge \dots t^{b_k}
    \right)
    \\
    & =
    P_{a b} t^a \wedge d_{W(\mathfrak{g})} t^b
    +
    \sum_{k = 1}^\infty
    \frac{2}{k! (k+1) }
    P_{a b} [t_{b_1}, \cdots t_{b_k}]^b t^{b_1} \wedge
    \cdots \wedge t^{b_k}
  \end{aligned}
  \,.
$$

=--

+-- {: .num_example}
###### Example

We consider the ordinary Chern-Simons element as an example of this formula: let $\mathfrak{g}$ be a [[semisimple Lie algebra]] and $\langle -,-\rangle$ the [[Killing form]] [[invariant polynomial]]. Then the above computation gives

$$
  \begin{aligned}
    cs & = \tau \left( P_{a b} r^a \wedge r^b \right)
    \\
    & =
    P_{a b}\tau
    \left(d_W t^a + \frac{1}{2}C^a{}_{a_1 a_2}t^{a_1} \wedge t^{a_2} \right)
    \wedge
    \left(d_W t^b + \frac{1}{2}C^b{}_{b_1 b_2}t^{b_1} \wedge t^{b_2} \right)
    \\
    & =
    P_{a b} t^a \wedge d_W t^b
    +
    \frac{2}{2! 3} P_{a b} t^a \wedge C^b_{b_1 b_2} t^{b_1} \wedge t^{b_2}
    \\
    & =
    P_{a b} t^a \wedge d_W t^b
    +
    \frac{1}{3} C_{a b c} t^a \wedge t^{b} \wedge t^{c}
  \end{aligned}
  \,.
$$

=--

## Origin and relation to other concepts
  {#OriginAndRelatedConcepts}

We discuss the general abstract structures of which Chern-Simons elements are presentations and how they are related to other structures.

The term _Chern-Simons element_ alludes to the term _[[Chern-Simons form]]_ and [[Chern-Simons theory]]. In the following we explain the relation.

### As presentations for the $\infty$-Chern-Weil homomorphism
  {#PresentationForChernWeil}

We explain here briefly how Chern-Simons elements provide a _presentation_ of a generalization of the [[Chern-Weil homomorphism]] -- the [[∞-Chern-Weil homomorphism]] in [[cohesive (∞,1)-topos]] theory -- in the sense in which [[(∞,1)-topos]]es have [[presentable (∞,1)-category|presentations]] by a [[model structure on simplicial presheaves]].

To warm up, we start with considering a traditional setup of [[Lie groupoid]] theory. Recall that for $G$ a [[Lie group]], we may form its [[delooping]] [[Lie groupoid]] dnoted $*//G$ or $\mathbf{B}G$. Then with $X$ any [[smooth manifold]], we have that the [[groupoid]] of [[morphism]]s of [[Lie groupoid]]s $X \to \mathbf{B}G$ is equivalent to that of $G$-[[principal bundle]]s on $X$:

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

Here we are thinking of [[Lie groupoid]]s as [[differentiable stack]]s, hence as [[object]]s in the [[(2,1)-topos]]

$$
  SmoothGrpd := Sh_{(2,1)}(SmoothMfd)
$$

of [[stack]]s/[[(2,1)-sheaves]] on the [[site]] [[SmoothMfd]] (equivalently on its [[small category|small]] [[dense subsite]] [[CartSp]] of [[Cartesian space]]s). (This is discussed in detail at _[[principal bundle]]_ ).

There is a _differential refinement_ of the [[Lie groupoid]] $\mathbf{B}G$, to the smooth groupoid

$$
  \mathbf{B}G_{conn}
   :=
  SmoothGrpd(\mathbf{P}_1(-), \mathbf{B}G)
  \,,
$$

where $\mathbf{P}_1(X)$ is the [[path groupoid]] of $X$. This is the [[(2,1)-sheaf]] given by the
[[(∞,1)-sheafification|(2,1)-sheafification]] of the assignment that sends a [[smooth manifold]] $U$ to the [[groupoid of Lie algebra-valued 1-forms]] on $U$.

There is a corresponding [[natural equivalence]]

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

of morphisms into $\mathbf{B}G_{conn}$ with the groupoid of $G$-[[principal bundle]]s with [[connection on a bundle|with connection]] on $X$. (This is described in detail at _[[connection on a bundle]]_ ).

In particular if $G = U(1)$ is the [[circle group]], a morphism $X\to \mathbf{B}U(1)_{conn}$ is a [[circle n-bundle with connection|circle bundle with connection]]. This

This allows already to consider a simple case of a [[characteristic class]] and its refinement to a [[differential characteristic class]]:
Let $U$ be the [[unitary group]]. There is a canonical morphism of [[Lie groupoid]]s $\mathbf{c}_1 : \mathbf{B}U \to \mathbf{B}U(1)$ given by the [[determinant]]. This -- or rather its image in [[cohomology]]

$$
  \mathbf{c}_1 : SmoothGrpd(- ,\mathbf{B}U) \to SmoothGrpd(-, \mathbf{B} U(1))
$$

is a smooth representative of the [[characteristic class]] called the _first [[Chern class]]_ . Its differential refinement is the evident morphism

$$
  \hat \mathbf{c}_1 : \mathbf{B}U_{conn} \to \mathbf{B}U(1)_{conn}
$$

that sends a $\mathfrak{u}$-[[Lie algebra valued 1-form|valued differential form]] to the [[trace]] of its Lie algebra value. Postcomposition with this is the refined [[Chern-Weil homomorphism]]

$$
  \array{
    SmoothGrpd(X, \mathbf{B}U)_{conn}
      &\stackrel{\hat \mathbf{c}_1}{\to}&
    SmoothGrpd(X, \mathbf{B}U(1)_{conn})
    \\
    \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}}
    \\
    U Bund_\nabla(X) &\to& U(1) Bund_\nabla(X)
  }
$$

with values in circle bundles with connection, hence in degree-2 [[ordinary differential cohomology]].

It is this kind of construction on [[Lie groupoid]]s that we now want to generalize to a notion of [[smooth ∞-groupoid]]s, to see that Chern-Simons elements are a means to constructi morphisms akind to the differential first Chern-class $\hat \mathbf{c}_1$.

A general abstract context for [[higher geometry]] equipped with [[differential cohomology]] is a [[cohesive (∞,1)-topos]] $\mathbf{H}$ of [[∞-groupoid]]s equipped with _cohesive structure_ , such as [[Smooth∞Grpd|smooth cohesive structure]].

An example for such is the [[∞-stack]]-analog of the [[stack]]-[[(2,1)-topos]] over [[SmoothMfd]]: the [[(∞,1)-sheaf (∞,1)-topos|∞-stack (∞,1)-topos]] [[Smooth∞Grpd]] $:= \hat Sh_{(\infty,1)}(SmoothMfd)$.

In that context we have for instance all the higher [[delooping]]s of $U(1)$: the circle Lie (n+1)-groups

$$
  \mathbf{B}^n U(1) \in Smooth\infty Grpd
  \,.
$$

This is such that the evident generalizations of the above classification statements hold: we have that morphisms $X \to \mathbf{B}^n U(1)$ form an [[n-groupoid]]

$$
  Smooth\infty Grpd(X, \mathbf{B}^n U(1))
   \simeq
  U(1) (n-1)Bund(X)
$$

equivalent to that of [[circle n-bundle with connection|circle n-bundles]]/$(n-1)$-[[bundle gerbe]]s on $X$.

If here $X = \mathbf{B}G$ is again the [[delooping]] of a [[Lie group]], this means that now also the higher [[characteristic class]]es are represented by morphisms

$$
  \mathbf{B}G \to \mathbf{B}^n U(1)
  \,.
$$

For instance for $G = Spin$ the [[spin group]], the first fractional [[Pontryagin class]] has a smooth incarnation given by a morphism of the form

$$
  \frac{1}{2}\mathbf{p}_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)
$$

corresponding under the above equivalence to the ordinary [[Chern-Simons circle 3-bundle]] on $\mathbf{B}G$.

Every [[cohesive (∞,1)-topos]] comes canonically and essentially uniquely equipped with all the intrinsic structure that we need for the discussion of a refinement of this to [[differential characteristic class]]es:

There is an  [[adjoint (∞,1)-functor|endo-(∞,1)-adjunction]]

$$
  (\mathbf{\Pi} \dashv \mathbf{\flat}) : Smooth\infty Grpd \to
  Smooth \infty Grpd
$$

where

* $\mathbf{\Pi}(X)$ is the path ∞-groupoid of a [[smooth ∞-groupoid]] $X$;

* $\mathbf{\flat}\mathbf{B}G$ is the coefficient object for flat ∞-connections on $G$-[[principal ∞-bundle]]s.

A [[morphism]] $\mathbf{\Pi}(X) \to \mathbf{B}^n U(1)$ encodes the flat [[higher parallel transport]] of a flat [[circle n-bundle with connection]], and we have that the [[n-groupoid]] of morphisms

$$
  Smooth \infty Grpd(\mathbf{\Pi}(X), \mathbf{B}^n U(1))
  \simeq
  U(1) n Bund_{\nabla_{flat}}(X)
$$

is that of flat [[circle n-bundles with connection]]/ (n-1)-[[connection on a bundle gerbe|bundle gerbes with connection]].

We observe that a _trivial_ circle $n$-bundle with connection is equivalently just a globally defined [[differential form|differential n-form]]. Therefore if we define the modified [[adjoint (∞,1)-functor|(∞,1)-adjunction]]

$$
  (\mathbf{\Pi}_{dR} \dashv \mathbf{\flat}_{dR})
  :
   */Smooth\infty Grpd
    \stackrel{\leftarrow}{\to}
   Smooth\infty Grpd
$$

by forming the [[(∞,1)-pullback]]

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

which is the coefficient object for _trivial_ principal $\infty$-bundles equipped with flat $\infty$-connection, one finds (discussed in detail here) that morphisms $X \to \mathbf{\flat}_{dR} \mathbf{B}^n U(1)$ correspond to trivial circle bundle with connection, hence to cocycles in [[de Rham cohomology]] of $X$;

$$
  \pi_0 Smooth \infty Grpd(X, \mathbf{\flat}_{dR} \mathbf{B}^n U(1))
  =
  \left\{
    \array{
      H_{dR}^n(X) & n \geq 2
      \\
      \Omega^1_{cl}(X) & n = 1
    }
  \right.
  \,.
$$

This now allows us to construct differential refinements:

one can show (detailed discussion is here) that there are canonical cocycles

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

in the degree $(n+1)$-[[de Rham cohomology]] of $\mathbf{B}^n U(1)$: these are the universal [[curvature characteristic form]]s on $\mathbf{B}^n U(1)$.

Then for $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ any smooth [[characteristic class]], the corresponding (unrefined) [[differential characteristic class]] is simply the composite

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

The (unrefined) [[∞-Chern-Weil homomorphism]] is postcomposition with this morphism:

$$
  (\mathbf{c}_{dR})_* : H^1(X,G) \to H_{dR}^{n+1}(X)
  \,.
$$

This is finally where the Chern-Simons elements come in:

+-- {: .standout}

Chern-Simons elements are a means to _present_ the composite morphism $\mathbf{c}_{dR}$ of [[smooth ∞-groupoid]]s by an [[∞-anafunctor]] between smooth [[Kan complex]]es.

=--

This presentation we describe in the next section.

(In fact a bit more is true: the serve to present the refinement of $\mathbf{c}_{dR}$ to a morphism $\hat \mathbf{c}$ with values in [[ordinary differential cohomology]]. This we come to further below.)

### Chern-Simons forms
  {#ChernSimonsForms}

We explain now how Chern-Simons elements arise as a presentation of a [[differential characteristic class]] $\mathbf{c}_{dR}$ by a [[span]] of [[simplicial presheaves]].

At the heart of the presentation of differenial characteristic classes by morphisms of simplicial presheaves is a differential refinement of the [[Lie integration]] of [[L-∞ algebra]]s and [[∞-Lie algebroid]]: for $\mathfrak{g}$ an ordinary [[Lie algebra]], one finds that the 3-[[coskeleton]] of the simplicial presheaf that assigns flat [[vertical differential form|vertical]] $\mathfrak{g}$-[[Lie algebra valued 1-form]]s

$$
  \exp(\mathfrak{g}) :
  (U,[k])
  \mapsto
  \left\{
     \array{
       \Omega^\bullet_{si, vert}(U \times \Delta^k)
        &\stackrel{A_{vert}}{\leftarrow}&
      CE(\mathfrak{g})
     }
  \right\}
$$

is the [[delooping]] of the [[simply connected]] [[Lie group]] $G$
integrating $\mathfrak{g}$

$$
 \mathbf{cosk}_3 \exp(\mathfrak{g}) \stackrel{\simeq}{\to}
  \mathbf{B}G
  \,.
$$

Similarly the [[Lie integration]] of the [[line Lie n-algebra]] $b^{n-1}\mathbb{R}$

$$
  \exp(b^{n-1}\mathbb{R})
  :
  (U,[k])
  \mapsto
  \left\{
     \array{
       \Omega^\bullet_{si,vert}(U \times \Delta^k)
       &\leftarrow&
       CE(b^{n-1}\mathbb{R})
     }
  \right\}
$$

is the $n$-fold [[delooping]] of $\mathbb{R}$:

$$
  \exp(b^{n-1}\mathbb{R})  \stackrel{\simeq}{\to}
  \mathbf{B}^n \mathbb{R}
  \,.
$$

Moreover, for $\mu : \mathfrak{g} \to b^{n-1}\mathbb{R}$ a degree-$n$ [[cocycle]] in [[Lie algebra cohomology]], simple postcomoposition gives its image under [[Lie integration]]

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

Under [[coskeleton|coskeletization]] on the left this carves out the [[period]]s of $\mu$ as a lattice in $\mathbb{R}$, which typically is the [[integer]]s, so that this descends to degree $n$-cocycle in [[Lie group cohomology]] with coefficients in $U(1) \simeq \mathbb{R}/\mathbb{Z}$

$$
  \exp(\mu) : \mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z}
  \,.
$$

(See [[Lie group cohomology]] and [[smooth ∞-groupoid]] for discussion of the refined notion of Lie group cohomology arising here.)

The differential refinement of these construction is based on the following fact (discussed in detail here)

1. the object $\mathbf{B}^n U(1) \in $ [[Smooth∞Grpd]] is equivalently presented by a quotient of the presheaf of [[Kan complex]]es given by

   $$
     \mathbf{B}^n U(1)_{diff} :
     (U \in SmoothMfd, [k] \in \Delta)
      \mapsto
     \left\{
       \array{
         \Omega^\bullet_{si,vert}(U \times \Delta^k)
         &\leftarrow& CE(b^{n-1} \mathbb{R})
         \\
         \uparrow && \uparrow
         \\
         \Omega^\bullet_{si}(U \times \Delta^k)
         &\stackrel{\omega}{\leftarrow}&
         W(b^{n-1} \mathbb{R})
       }
     \right\}
     \,,
   $$

   where on the right we have the set of horizontal morphisms in [[dgAlg]] that make a [[commuting diagram]] with the canonical vertical morphisms as indicated.

   $$
     \array{
       \mathbf{B}^n U(1)_{diff}
       \\
       \downarrow^{\mathrlap{\simeq}}
       \\
       \mathbf{B}^n U(1)
     }
     \,.
   $$

   (We may think of a morphism of simplicial presheaves $X \to \mathbf{B}^n U(1)_{diff}$ as a [[circle n-bundle with connection|circle n-bundle]]/$(n-1)$-bundle gerbe equipped with a _[[pseudo-connection]]_ . )

   Notice that the bottom morphism here encodes precisely a degree-$n$ [[differential form]] $\omega$ on $U \times \Delta^k$,

1. The morphism $curv : \mathbf{B}^n U(1) \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} U(1)$ is presented on this by the [[∞-anafunctor]]

   $$
     \array{
       \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim}
       \\
       \downarrow^{\mathrlap{\simeq}}
       \\
       \mathbf{B}^n U(1)
     }
   $$

   by the map that sends such a form $\omega$ to its [[curvature]] $d \omega$. If the [[pseudo-connection]]s that we are dealing with are genuine connections the [[curvature]] is a basic form down on $U$ and this means diagrammatically that it forms the [[pasting]] composite

   $$
     \array{
       \Omega^\bullet_{si,vert}(U \times \Delta^k)
       &\leftarrow& CE(b^{n-1} \mathbb{R})
       \\
       \uparrow && \uparrow
       \\
       \Omega^\bullet_{si}(U \times \Delta^k)
       &\stackrel{\omega}{\leftarrow}&
       W(b^{n-1} \mathbb{R})
       \\
       \uparrow && \uparrow
       \\
       \Omega^\bullet(U)
       &\stackrel{d \omega}{\leftarrow}&
       inv(b^{n-1} \mathbb{R})
     }
   $$

   and then picks out the bottom horizontal morphism.

Therefore our task of presenting $\mathbf{c}_{dR}$ amounts to computing the composition of [[∞-anafunctor]]s

$$
  \array{
    && \mathbf{B}^n U(1)_{diff} &\to& \mathbf{\flat}_{dR} \mathbf{B}^{n+1}U(1)_{sim}
    \\
    && \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}G &\stackrel{\exp(\mu)}{\to}& \mathbf{B}^n U(1)
  }
$$

To do se we need to complete componentwise to [[commuting diagram]]s. To this end we first complete the assignment of $\exp(\mathfrak{g})$ to a diagram

\[
  (U,[k])
  \mapsto
  \left\{
  \array{
    \Omega^\bullet(U \times \Delta^k)_{vert}
    &\stackrel{A_{vert}}{\leftarrow}&
    CE(\mathfrak{g})
    &&& transition\;function
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U \times \Delta^k)
    &\stackrel{A}{\leftarrow}&
    W(\mathfrak{g})
    &&&
    connection
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U)
    &\stackrel{\langle F_A \rangle}{\leftarrow}&
    inv(\mathfrak{g})
    &&&
    curvature\;characteristic\;form
  }
  \label{ConnectionDiagram}
  \right\}
  \,.
\]

Here in the middle row an unrestricted $\mathfrak{g}$-[[∞-Lie algebra valued differential form]] appears, which is the local $\mathfrak{g}$-[[connection on an ∞-bundle|∞-connection]]. And in the lower row all its [[curvature characteristic forms]] appear, obtained by evaluating the [[curvature]] $F_A$ in the [[invariant polynomial]]s on $\mathfrak{g}$.

This is such that a choice of Chern-Simons element witnessing the [[transgression]] of an [[invariant polynomial]] to $\mu$ allows to refine $\exp(\mu)$ to

\[
  \cdots
  \stackrel{\exp(\mu)_{conn}}{\mapsto}
  \left\{
  \array{
    \Omega^\bullet(U \times \Delta^k)_{vert}
    &\stackrel{A_{vert}}{\leftarrow}&
    CE(\mathfrak{g})
    &\stackrel{\mu}{\leftarrow}&
    CE(b^{n-1}\mathbb{R})
    &&& characteristic\;class
    \\
    \uparrow && \uparrow  && \uparrow
    \\
    \Omega^\bullet(U \times \Delta^k)
    &\stackrel{A}{\leftarrow}&
    W(\mathfrak{g})
    &\stackrel{cs}{\leftarrow}&
    W(b^{n-1}\mathbb{R})
    &&&
    Chern-Simons\;form
    \\
    \uparrow && \uparrow && \uparrow
    \\
    \Omega^\bullet(U)
    &\stackrel{\langle -\rangle}{\leftarrow}&
    inv(\mathfrak{g})
    &\stackrel{\langle -\rangle}{\leftarrow}&
    inv(b^{n-1}\mathbb{R})
    &&&
    curvature\;characteristic\;form
  }
  \label{ChernWeilDiagram}
  \right\}
  \,.
\]

Here now the middle row is the evaluationn of the connection form inside the Chern-Simons element. This is the corresponding [[Chern-Simons form]]

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

of the $\mathfrak{g}$-connection evaluated in the given Chern-Simons element. Its [[curvature]] is the [[curvature characteristic form]] $\langle F_A \rangle$ appearing in the bottom line of the diagram, which is obtained by evaluating the $\mathfrak{g}$-valued [[curvature]] in the given [[invariant polynomial]].

A more comprehensive account of this is at [[Chern-Weil homomorphism in Smooth∞Grpd]].

### Chern-Simons action functionals

By the above construction, every Chern-Simons element $cs \in W(\mathfrak{a})$ of degree $d$ on an [[∞-Lie algebroid]] $\mathfrak{a}$ induces an [[action functional]] on the space of [[∞-Lie algebroid valued forms]] on $\mathfrak{a}$ over a $d$-[[dimensional]] [[smooth manifold]] $\Sigma$

$$
  S_{cs} : \Omega(\Sigma, \mathfrak{a}) \to \mathbb{R}
$$

given by

$$
  (A,B,C, \cdots) \mapsto \int_\Sigma CS(A,B,C, \cdots)
  \,.
$$

This generalizes the action functional of ordinary [[Chern-Simons theory]] to general Chern-Simons elements. In the [examples](#Examples) below is a list of various [[quantum field theories]] that arise as generalized Chern-Simons theories this way.

For more details see [[schreiber:infinity-Chern-Simons theory]].

## Examples {#Examples}

### On semisimple Lie algebra-- Standard Chern-Simons action functional {#StandardCS}

Let $\mathfrak{g}$ be a [[semisimple Lie algebra]]. For the following computations, choose a [[basis]] $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{r^a\}$ denotes the corresponding degree-shifted basis of $\mathfrak{g}^*[1]$.

Notice that in terms of this the differential of the CE-algebra is

$$
  d_{CE(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c
$$

and that of the Weil algebra

$$
  d_{W(\mathfrak{g})} : t^a \mapsto -\frac{1}{2}C^a{}_{b c}t^b \wedge t^c
  + r^a
$$

and

$$
  d_{W(\mathfrak{g})} : r^a \mapsto  -C^a{}_{b c} t^b \wedge r^c
  \,.
$$

Let $P_{a b} r^a \wedge r^b \in W(\mathfrak{g})$ be the [[Killing form]] invariant polynomial. This being invariant

$$
  d_{W(\mathfrak{g})} P_{a b} r^a \wedge r^b
  =
  2 P_{a b} C^{a}{}_{d e} t^d \wedge r^e \wedge r^b = 0
$$

is equivalent to the fact that the coefficients

$$
  C_{a b c} := P_{a a'}C^{a'}{}_{b c}
$$

are skew-symmetric in $a$ and $b$, and therefore skew in all three indices.

+-- {: .un_prop}
###### Proposition

A Chern-Simons element for the [[Killing form]] invariant polynomial $ \langle -, - \rangle = P(-,-)$ is

$$
  \begin{aligned}
    cs &=
    P_{a b} t^a \wedge (d_{W(\mathfrak{g})} t^b)
    +
    \frac{1}{3} P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c
    \\
    & =
    P_{a b} t^a \wedge r^b
    -
    \frac{1}{6}
    P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c
  \end{aligned}
  \,.
$$

In particular the Killing form $\langle -,-\rangle$ is in transgression with the degree 3-cocycle

$$
  \mu = -\frac{1}{6}\langle -,[-,-]\rangle
  \,.
$$

=--

+-- {: .proof}
###### Proof

We compute

$$
  \begin{aligned}
    d_{W(\mathfrak{g})}
    \left(
       P_{a b} t^a \wedge r^b
       +
       \frac{1}{2}P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge t^c
    \right)
    & =
      P_{a b} r^a \wedge r^b
      \\
      & -\frac{1}{2} P_{a b} C^a{}_{d e} t^d \wedge t^e \wedge r^b
      \\
      & + P_{a b} C^b{}_{d e} t^a \wedge t^d \wedge r^e
      \\
      & - \frac{3}{6}
       P_{a a'}C^{a'}{}_{b c} t^a \wedge t^b \wedge r^c
      \\
      & =
      P_{a b} r^a \wedge r^b
      \\
      & + \frac{1}{2}C_{a b c} t^a \wedge t^b \wedge r^c
      \\
      & - \frac{1}{2}
       C_{a b c} t^a \wedge t^b \wedge r^c
      \\
      & = P_{a b } r^a \wedge r^b
  \end{aligned}
  \,.
$$

=--

Under a [[Lie algebra-valued form]]

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

this Chern-Simons element is sent to

$$
  cs(A) = P_{a b} A^a \wedge d A^b + \frac{1}{3} C_{a b c} A^a \wedge A^b \wedge A^c
  \,.
$$

If $\mathfrak{g}$ is a [[matrix Lie algebra]] then the Killing form is the [[trace]] and this is equivalently

$$
  cs(A) = tr(A \wedge d A) + \frac{2}{3} tr(A \wedge A \wedge A)
  \,.
$$

This is a familiar form of the standard [[Chern-Simons form]] in degree 3.

For $\Sigma$ a 3-[[dimensional]] [[smooth manifold]] the corresponding [[action functional]] $S_{CS} : \Omega^1(\Sigma, \mathfrak{g}) \to \mathbb{R}$

$$
  S_{CS} : A \mapsto \int_\Sigma cs(A)
$$

is the standard action functional of [[Chern-Simons theory]].

### Higher CS-forms on semisimple Lie algebras

For $\mu \in CE(\mathfrak{g})$ any higher order cocycle, $CS_\mu(A)$ is the corresponding higher order Chern-Simons form.

#### Action functional for Chern-Simons (super-)gravity

Higher Chern-Simons elements on the [[Poincare Lie algebra]] $\mathfrak{g} = \mathfrak{iso}(d,1)$ or the [[super Poincare Lie algebra]] $\mathfrak{g} = \mathfrak{siso}(d,1)$ yield [[action functional]]s for [[gravity]] and [[supergravity]].

(...)

See ([Zanelli](#Zanelli)).

#### Fractional secondary Pontryagin classes

For instance for $\mu_7$ the 7-cocycle on a [[semisimple Lie algebra]], $CS_{\mu_7}(A)$ is the corresponding Chern-Simons 7-form, corresponding to the second [[Pontryagin class]].

Notice that this we may also think of as a 7-cocycle on the corresponding [[string Lie 2-algebra]]. As such it is the one that classifies the extension to the [[fivebrane Lie 6-algebra]]. The corresponding Chern-Simons 7-form appears as the local conneciton data in the [[Chern-Simons circle 7-bundle with connection]] that obstructions the lift from a [[differential string structure]] to a [[differential fivebrane structure]].

### On strict Lie 2-algebras -- BF-theory action functional {#BF}

+-- {: .un_prop}
###### Proposition

Let $\mathfrak{g} = (\mathfrak{g}_2 \stackrel{\partial}{\to} \mathfrak{g})_1$ be a [[strict Lie 2-algebra]].

Then

* every [[invariant polynomial]] $\langle -\rangle_{\mathfrak{g}_1} \in inv(\mathfrak{g}_1)$ on $\mathfrak{g}_1$ is a Chern-Simons element on $\mathfrak{g}$, restricting to the trivial [[∞-Lie algebra cocycle]];

* for $\mathfrak{g}_1$ a [[semisimple Lie algebra]] and
  $\langle - \rangle_{\mathfrak{g}_1}$ the [[Killing form]], the corresponding Chern-Simons action functional on [[∞-Lie algebra valued forms]]

  $$
    \Omega^\bullet(X) \stackrel{(A,B)}{\leftarrow}
    W(\mathfrak{g}_2 \to \mathfrak{g}_1)
    \stackrel{(\langle - \rangle_{\mathfrak{g}_1}, d_W \langle - \rangle_{\mathfrak{g}_1} )}{\leftarrow}
    W(b^{n-1} \mathbb{R})
  $$

is the sum of the [[action functional]]s of [[topological Yang-Mills theory]] with [[BF-theory]] with [[cosmological constant]] (in the sense of [[gravity as a BF-theory]]):

$$
  CS_{\langle-\rangle_{\mathfrak{g}_1}}(A,B)
  =
  \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1}
  -
  2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1}
  +
  2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1}
  \,,
$$

where $F_A$ is the ordinary [[curvature]] 2-form of $A$.

=--

This is from ([SSSI](#SSSI-BFtheory)).

+-- {: .proof}
###### Proof

For $\{t_a\}$ a [[basis]] of $\mathfrak{g}_1$ and $\{b_i\}$ a basis of $\mathfrak{g}_2$ we have

$$
  d_{W(\mathfrak{g})} : \sigma t^a \mapsto
  d_{W(\mathfrak{g}_1)}
  +
  \partial^a{}_i \sigma b^i
  \,.
$$

Therefore with $\langle -\rangle_{\mathfrak{g}_1} = P_{a_1 \cdots a_n} \sigma r^{a_1} \wedge \cdots \sigma t^{a_n}$ we have

$$
  d_{W(\mathfrak{g})} \langle - \rangle_{\mathfrak{g}_1}
  =
  n P_{a_1 \cdots a_n}\partial^{a_1}{}_i  \sigma b^{i} \wedge \cdots \sigma t^{a_n}
  \,.
$$

The right hand is a polynomial in the shifted generators of $W(\mathfrak{g})$, and hence an [[invariant polynomial]] on $\mathfrak{g}$. Therefore $\langle - \rangle_{\mathfrak{g}_1}$ is a Chern-Simons element for it.

Now for $(A,B)$ an [[∞-Lie algebra-valued form]], we have that the 2-form curvature is

$$
  F_{(A,B)}^1 = F_A - \partial B
  \,.
$$

Therefore

$$
  \begin{aligned}
    CS_{\langle -\rangle_{\mathfrak{g}_1}}(A,B)
    & = \langle F_{(A,B)}^1\rangle_{\mathfrak{g}_1}
    \\
    & =
    \langle F_A \wedge F_A\rangle_{\mathfrak{g}_1}
    -
    2\langle F_A \wedge \partial B\rangle_{\mathfrak{g}_1}
    +
    2\langle \partial B \wedge \partial B\rangle_{\mathfrak{g}_1}
  \end{aligned}
  \,.
$$

=--

### On symplectic $\infty$-Lie algebroids -- The AKSZ Lagrangian
  {#Symplectic}

A [[symplectic Lie n-algebroid]] $(\mathfrak{a}, \omega)$ is a [[∞-Lie algebroid|Lie n-algebroid]] $\mathfrak{a}$ equipped with a binary non-degenerate [[invariant polynomial]] $\omega \in W(\mathfrak{a})$ of degree $n+2$.

The corresponding Chern-Simons elements of $\omega$ are the integrands for the [[action functional]]s of various [[TQFT]] [[sigma-model]]s.

With $\{-,-\} : CE(\mathfrak{a}) \otimes CE(\mathfrak{a}) \to CE(\mathfrak{a})$ the graded [[Poisson bracket]] induced by $\omega$ we have (see [Roytenberg](#Roytenberg)) that there exists a [[∞-Lie algebra cocycle]]  $\mu \in CE(\mathfrak{a})$ such that

$$
  d_{CE(\mathfrak{a})} = \{\mu, -\}
  \,.
$$

So in particular $\mu$ being a cocycle means that

$$
  d_{CE(\mathfrak{a})} \mu = \{\mu, \mu\} = 0
  \,.
$$

+-- {: .un_prop}
###### Proposition

The cocycle $\mu$ is in transgression with the invariant polynomial $\frac{n}{2}\omega$ via the Chern-Simons element

$$
  \begin{aligned}
    cs &= \frac{1}{2 }\iota_{\epsilon} \omega - \mu
  \end{aligned}
  \,,
$$

where $\epsilon$ is the Euler vector field ([Roytenberg](#Roytenberg)).

=--

Here $\mathbf{d}$ is the shift derivation in the [[Weil algebra]], in that $d_{W(\mathfrak{a})} = d_{CE(\mathfrak{a})} + \mathbf{d}$.

+-- {: .proof}
###### Proof

To safe typing signs, we write as if all functions were even graded. By standard reasoning the computation holds true then also for arbitrary grading.

Observe that

1. on unshifted generators we have

   $$
     (\mathbf{d}x^b) \omega_{a b} \{x^a  , -\} = \mathbf{d}
   $$

1. we have graded commutators

   * $[\mathbf{d}, \iota_v] = N$ (the degree operator)

   and

   * $[d_{CE(\mathfrak{a})}, \iota_v] = -d_{CE(\mathfrak{a})}\mathbf{d}^{-1}$.

   (as one checks on generators).

Therefore

$$
  \begin{aligned}
    d_{W(\mathfrak{a})} \frac{1}{2}\iota_{v} \omega
    &=
    [d_{W(\mathfrak{a})}, \iota_{v}] \frac{1}{2}\omega
    \\
    & =
    (n - d_{CE(\mathfrak{a})} \mathbf{d}^{-1} ) \frac{1}{2}\omega
    \\
    & =
    \frac{n}{2}\omega - \omega_{a b} \{\mu, x^a\} \mathbf{d}x^b
    \\
    &= \frac{1}{2} \omega + \mathbf{d}\mu
  \end{aligned}
  \,,
$$

where in the first line we used that by definition of [[invariant polynomial]] $d_{W(\mathfrak{a})} \omega = 0$. Similarly, using that by definition $d_{CE(\mathfrak{a})} \mu = 0$ we have

$$
  d_{W(\mathfrak{a})} \mu = \mathbf{d}\mu
  \,.
$$

So in total we have

$$
  d_{W(\mathfrak{a})} (\frac{1}{2} \iota_\epsilon \omega - \mu)
  =
  \frac{1}{2}\omega
  \,.
$$

=--

+-- {: .remark}
###### Remark

In local coordinates where $\omega = \omega_{a b} \mathbf{d}x^a \wedge \mathbf{d}x^b$ we have

$$
  cs = n \omega_{a b} x^a \wedge \mathbf{d}x^b + \mu
  \,.
$$

The Chern-Simons action functional corresponding to this Chern-Simons element
on $\mathfrak{a}$ is that considered in [[AKSZ theory]].

Below we spell out some low-dimensional cases explicitly.

=--

#### Higher phase space ---Hamiltonian and Lagrangian mechanics

The symplectic Lie $n$-algebroid $(\mathfrak{P}, \omega)$ may be thought of as an [[n-symplectic manifold]] that models the [[phase space]] of a physical system.

This means for $(\mathfrak{g},\langle-\rangle) = (\mathfrak{P}, \omega)$ a symplectic Lie $n$-algebroid, the general diagram \eqref{TheDiagram} exhibiting the transgression between cocycles and invariant polynomials via Chern-Simons elements may be labeled in terms of [[Hamiltonian mechanics]], [[Lagrangian mechanics]] and [[symplectic geometry]] as follows

\[
  \array{
    CE(\mathfrak{P})
    &\stackrel{H}{\leftarrow}&
    CE(b^{n}\mathbb{R})
    &&& Hamiltonian
    \\
    \uparrow && \uparrow
    \\
    W(\mathfrak{P})
    &\stackrel{L}{\leftarrow}&
    W(b^n \mathbb{R})
    &&&
    Lagrangian
    \\
    \uparrow && \uparrow
    \\
    inv(\mathfrak{P})
    &\stackrel{\omega}{\leftarrow}&
    inv(b^n \mathbb{R})
    &&&
    symplectic\;structure
  }
  \label{PhysicsDiagram}
\]

=--

See [[Hamiltonian]], [[Lagrangian]], [[symplectic structure]].

#### On a symplectic manifold -- The topological particle

For $X$ a [[smooth manifold]] we may regard its cotangent bundle
$\mathfrak{a} = T^* X$ as a Lie 0-algebroid and the canonical 2-form
$\omega \in W(\mathfrak{a}) = \Omega^\bullet(X)$ as a binary invariant
polynomial in degree 2.

The Chern-Simons element is the canonical 1-form $\alpha$ which in local coordinates is $\alpha = p_i d q^i$.

The corresponding action functional on the line

$$
  \int_{\mathbb{R}} \gamma^* (p_i\, d q^i)
$$

is the familiar term for the action functional of the particle (missing the kinetic term, which makes it "topological").

#### On a Poisson Lie algebroid -- The Poisson $\sigma$-model

+-- {: .un_prop}
###### Proposition

Let $\mathfrak{a} = \mathfrak{P}(X,\pi)$ by a [[Poisson Lie algebroid]]. This comes with the canonical [[invariant polynomial]] $\omega = \mathbf{d} \partial_i \wedge \mathbf{d} x^i$.

The corresponding [[∞-Lie algebroid cocycle]] is

$$
  \mu_{\omega} = \pi
$$

and a Chern-Simons element for this is

$$
  cs_\omega = \pi^{i j} \partial_i  \wedge \partial_j
  + \partial_i \wedge \mathbf{d}x^i
  \,.
$$

For $\Sigma$ a 2-[[dimensional]] [[smooth manifold]] the corresponding [[action functional]] on [[∞-Lie algebroid-valued forms]] $S : \Omega^\bullet(X, \mathfrak{P}(X,\pi)) \to \mathbb{R}$ is the actional functional of the [[Poisson sigma-model]]

$$
  S : (X, \eta) \mapsto \int_\Sigma (\eta \wedge d X + \pi(\eta \wedge \eta)
  \,.
$$

=--

+-- {: .proof}
###### Proof

We compute in a local coordinte patch:

$$
  \begin{aligned}
    d_{W(\mathfrak{P}(X,\pi))} (\pi^{i j} \partial_i \wedge \partial_j
   + \partial_i \wedge \mathbf{d} x^i)
    = &
   \mathbf{d}x^k (\partial_k \pi^{i j}) \partial_i \wedge \partial_j
   \\
   & + 2 \pi^{i j} (\mathbf{d}\partial_i) \wedge \partial_j
   \\
   & - (\partial_i \pi^{j k}) \partial_j \wedge \partial_k \wedge
       \mathbf{d}x^i
   \\
   & +
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
   \\
   & + (-)(-) 2\pi^{i j} \partial_i \wedge \mathbf{d}\partial_j
   \\
   = &
   (\mathbf{d}\partial_i)\wedge (\mathbf{d} x^i)
  \end{aligned}
  \,.
$$

=--

### On higher extensions of the super Poincare Lie algebra -- supergravity

See [[D'Auria-Fre formulation of supergravity]] for the moment.

## Related concepts

* [[∞-Lie algebroid cocycle]]

* **Chern-Simons element**

* [[invariant polynomial]]

## References

A classical reference on transgression of differential forms from the fiber to the base of a [[fiber bundle]] is section 9 of.

* [[Armand Borel]], _Topology of Lie groups and characteristic classes_  Bull. Amer. Math. Soc. Volume 61, Number 5 (1955), 397-432. ([EUCLID](http://projecteuclid.org/euclid.bams/1183520007))
{#Borel}

The general definition of Chern-Simons element on $\infty$-Lie algebras and $\infty$-Lie algebroids is definition 21 in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_\infty$-connections_ (web)
{#SSSI}

The examples of the [[BF-theory]] invariant polynomials and Chern-Simons elements are in prop. 18 and def. 26 and the BF-action functional itself is extracted below proposition 28.
{#SSSI-BFtheory}

Dedicated discussion of $\infty$-Chern-Simons theory is at

* [[Domenico Fiorenza]], [[Chris Rogers]] [[Urs Schreiber]], _[[schreiber:∞-Chern-Simons functionals]]_

A comprehensive account is in

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_ .

A survey of higher Chern-Simons elements and their action functionals as applied to [[gravity]] and [[supergravity]] is in

* Jorge Zanelli, _Lecture notes on Chern-Simons (super-)gravities_ [arXiv:0502193](http://arxiv.org/abs/hep-th/0502193)
{#Zanelli}

Symplectic Lie $n$-algebroids are discussed in

* [[Dmitry Roytenberg]], _Courant algebroids, derived brackets and even symplectic supermanifolds_ PhD thesis ([arXiv](http://arxiv.org/abs/math/9910078))

  _On the structure of graded symplectic supermanifolds and Courant algebroids_ ([arXiv](http://arxiv.org/abs/math/0203110))
{#Roytenberg}

A talk about the historical origins of the standard Chern-Simons forms see

* [[Jim Simons]], _Origin of Chern-Simons_ talk at Simons Center for Geometry and Physics (2011) ([video](http://media.scgp.stonybrook.edu/video/video.php?f=20110728_1_qtp.mp4))

[[!redirects Chern-Simons elements]]

[[!redirects generalized Chern-Simons theory]]
[[!redirects generalized Chern-Simons theories]]
[[!redirects ∞-Chern-Simons theory]]


Chern-Simons form

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

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

$$
  d_{dR} CS(A) = \langle F_A \wedge \cdots \wedge F_A \rangle
  \,,
$$

where $F_A \in \Omega^2(X,\mathfrak{g})$ is the [[curvature]] 2-form of $A$.

Therefore it is often also called a [[secondary characteristic form]].

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

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

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

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

$$
  \langle A \wedge d_{dR} A\rangle
  +
  c
  \langle
    A \wedge [A \wedge A]
  \rangle
  \,.
$$

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

## Details {#Details}

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

### Paths of connections {#PathsOfConnections}

Let $U$ be a [[smooth manifold]].

+-- {: .num_prop}
###### Definition

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

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

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

$$
  \psi_t :  U \simeq U \times * \stackrel{Id \times t}{\to} U \times [0,1]
$$

picks the copy of $U$ at parameter $t$.

=--

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

+-- {: .num_defn}
###### Definition/Observation

Let $P$ be an [[invariant polynomial]] on $\mathfrak{g}$ of arity $n$.

Consider the [[fiber integration]]

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

This defines a $(2n-1)$-form $CS_P(A_0,A_1) \in \Omega^{2n-1}(U)$.

We have that the exterior differential of this form is the difference of the [[curvature characteristic form]]s of $A_0$ and $A_1$:

$$
  d_{dR} CS_P(A_0,A_1)
  =
  P(F_{A_1} \wedge \cdots \wedge F_{A_1})
  -
  P(F_{A_0} \wedge \cdots \wedge F_{A_0})
$$

=--

+-- {: .proof}
###### Proof

Write the fiber integration more explicitly as an [[integral]]

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

Then use that $d_{dR}$ is linear and commutes with pullback, use [[Cartan's magic formula]] $d_{dR} \circ \iota_{\partial_t} + \iota_{\partial} \circ d_{dR} = \mathcal{L}_{\partial_t}$ in view of the fact that $P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})$ is a [[closed form]] and then finally apply the [[Stokes theorem]]:

$$
  \begin{aligned}
    d_{dR} \int_0^1 \psi^*_t  \iota_{\partial_t}
    P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t
   & =
    \int_0^1 d_{dR}  \psi^*_t  \iota_{\partial_t}
     P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t
    \\
   & =
    \int_0^1 \psi^*_t  d_{dR}   \iota_{\partial_t}
      P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t
    \\
   & =
    \int_0^1 \psi^*_t  \frac{d}{d t}
      P(F_{\hat A} \wedge \cdots \wedge F_{\hat A}) d t
    \\
   & =
    \int_0^1 \left(\frac{d}{d t}
      P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})\right)(t) d t
    \\
   & =
   \psi^*_1 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})
   -
   \psi^*_0 P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})
   \\
    &=
   P(F_{A_1} \wedge \cdots \wedge F_{A_1})
   -
   P(F_{A_0} \wedge \cdots \wedge F_{A_0})
  \end{aligned}
  \,.
$$

=--

### Explicit formulas

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

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

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  $\hat A$ a pure shift path, $\hat A : t \mapsto A_t$
notice that the [[curvature]] 2-form of $\hat A$ is

$$
  F_{\hat A}(t) = F_{A_t} + (\partial_t A_t) \wedge d t
  \,.
$$

Inserting this into the above expression yields

$$
  CS_P(A_0,A_1)
  =
  \int_0^1 P(\partial_t A \wedge F_{A_t} \wedge \cdots \wedge F_{A_t})
  \,.
$$

Notably if $A_0 = 0$ and $\hat A$ is the _constant_ path $\hat A : t \mapsto t A$ to $A_1  := A$ such that

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

this yields

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

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

$$
  CS(A)
  =
  c_1 \langle A \wedge F_A \wedge \cdots \wedge F_A \rangle
  +
  c_2 \langle A \wedge [A \wedge A] \wedge F_A \wedge \cdots F_A \rangle
  +
  \cdots
  \,.
$$

Particularly for $n = 2$ and using the definition of the
[[curvature]] 2-form $F_A = d_{dR} A + [A \wedge A]$ we get

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

### Gauged paths of connections

Above we defined $CS(A_0,A_1)$ for every path of connections form $A_0$ to $A_1$ which is _pure shift_ . This is a possibly convenient but unnecessary restriction:

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

$$
  \hat A : t \mapsto  A_t + \lambda d t
  \,,
$$

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

$$
  F_{\hat A}
  : t \mapsto
  F_{A_t}
  +
  ((\partial_t A)_t + d_U \lambda(t) + [\lambda,A_t]) \wedge d t
$$

+-- {: .num_prop}
###### Proposition

The Chern-Simons form $\int_{[0,1]} P(F_{\hat A} \wedge \cdots \wedge F_{\hat A})$ 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 {#InInfCSTheory}

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

### Prerequisites {#PrerequisitesHigher}

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

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

* the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$;

* the [[Weil algebra]] $W(\mathfrak{g})$;

* the algebra of [[invariant polynomial]]s $inv(\mathfrak{g})$.

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

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

where here and in the following $\Omega^\bullet(\Delta^n)$ denotes the [[de Rham complex]] dg-algebra of those smooth [[differential form]]s $\omega$ on the standard smooth $n$-simplex that have _sitting instants_ in that for each $k \in \mathbb{N}$ every $k$-face of $\Delta^n$ has an open neighbourhood such that restricted to that neighbourhood $\omega$ is constant in the direction perpendicular to the face.

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

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

thus defining the [[∞-Lie group]] $G$ that integrates $\mathfrak{g}$ in degree $n$.

At [[∞-Chern-Weil theory]] is explained that a [[resolution]] of $\mathbf{B}G$ that serves to compute [[curvature characteristic form]]s in that it encodes [[pseudo-connection]]s on $G$-[[principal ∞-bundle]]s is given by the simplicial presheaf

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

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 $n = \infty$, 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 $k$ is a morphism

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

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

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

At [[∞-Chern-Weil theory]] it is discussed how the proper lift of this through the extension $\mathbf{B}G_{diff}$ that computes the [[schreiber:differential cohomology in an (∞,1)-topos|abstractly defined]] curvature characteristic classes is given by finding the [[invariant polynomial]] $\langle -,-\rangle \in W(\mathfrak{g})$ that is in transgression with $\mu$ in that we have a commuting diagram

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

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

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

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

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

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 $n$-cells on the right and integrates the corresponding forms
over the $n$-simplex yields an equivalence

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

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

### Higher order Chern-Simons form {#HigherOrderChernSimonsForms}

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

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

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

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

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

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

discussed above produces the corresponding [[curvature characteristic form]].

A [[homotopy]]

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

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

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

this corresponds to a $(k-1)$-form on $X$ this is the Chern-Simons form

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

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 {#OrdinaryCSRevisited}

We now show how the [traditional definition](#PathsOfConnections) of Chern-Simons forms is reproduced by the [general abstract](#HigherOrderChernSimonsForms) mechanism.

+-- {: .num_prop}
###### Proposition
**(ordinary Chern-Simons form)**

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

Then morphisms (of simplicial presheaves)

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

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

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

are in canonical bijection with closed $k$-forms on $X$ and composition with the morphism

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

discussed [above](#PrerequisitesHigher) and under this canonical identification the composite

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

is the corresponding [[curvature characteristic form]].

Homotopies

$$
  (A_1 \stackrel{\gamma}{\to} A_2)  : X\times \Delta[1] \to \mathbf{B}G_{diff}
$$

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

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

=--

+-- {: .proof}
###### 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 $\mathbf{B}G_{diff}$, a morphism $X \to \mathbf{B}G_{diff}$ is equivalently a diagram

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

Since $CE(\mathfrak{g})$ is trivial in degree 0 and since $C^\infty(X)\otimess \Omega^\bullet(\Delta^0)$ 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 $\mathfrak{g}$-valued form, as discussed in some detail at [[Weil algebra]].

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

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

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

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

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

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

$$
  \lambda = g^* \theta
$$

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

or conversely by [[parallel transport]]

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

We may think of this as a smooth _path of gauge transformations_ .

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

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

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

This is a _smooth path in the space of 1-forms_ . In the case that $\lambda = 0$ this is a _pure shift path_ in the [terminology above](#PathsOfConnections). we look at this case in the following, for ease of notation.

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

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

The [[fiber integration]] of this over $\Delta^1$ is manifestly the same operation as that in the definition of the Chern-Simons form [above](#PathsOfConnections).

=--

## As secondary characteristic forms

If a [[curvature characteristic form]] _vanishes_ (for instance if the [[connection on a bundle|connection]] is flat or the degree of the curvature characteristic form is simply greater than the dimension of $X$) 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]] $X$ necessarily the Chern-Simons 3-form is closed. The [[functional]]

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

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

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

## In terms of $\infty$-Lie algebroids

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

For

$$
  \array{
    T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g}
    \\
    \downarrow && \downarrow
    \\
    T P &\stackrel{(A,F_A)}{\to}& inn(\mathfrak{g})
    \\
    \downarrow && \downarrow
    \\
    T X &\stackrel{(P_i(F_A))}{\to}& \prod_i b^{n_i} \mathfrak{u}(1)
  }
$$

the data of an [[Ehresmann connection]] on a $G$-[[principal bundle]] expressed as a diagram of [[∞-Lie algebroid]]s with the [[curvature characteristic form]]s on the bottom, a choice of transgression element $cs_P$ for an [[invariant polynomial]] $P$ in transgression with a Lie algebra cocycle $\mu$ induces a diagram

$$
  \array{
    \mathfrak{g} &\stackrel{\mu}{\to}& b^n \mathfrak{u}(1)
    \\
    \downarrow && \downarrow
    \\
    inn(\mathfrak{g}) &\stackrel{(cs_P,P)}{\to}& e b^{n} \mathfrak{u}(1)
    \\
    \downarrow && \downarrow
    \\
    \prod_i b^{n_i}\mathfrak{u}(1) &\stackrel{p_i}{\to}& b^{n+1} \mathfrak{u}(1)
  }
  \,.
$$

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

$$
  \array{
    T_{vert} P &\stackrel{A_{vert}}{\to}& \mathfrak{g} &\stackrel{\mu}{\to}&
    b^n \mathfrak{u}(1)
    \\
    \downarrow && \downarrow && \downarrow
    \\
    T P &\stackrel{A}{\to}& inn(\mathfrak{g})
    &\stackrel{(cs_P,P)}{\to}&
    e b^n \mathfrak{u}(1)
    \\
    \downarrow && \downarrow && \downarrow
    \\
    T X &\stackrel{(P_i)}{\to}& \prod_i b^{n_i} \mathfrak{u}(1)
    &\stackrel{p_i}{\to}&
    b^{n+1} \mathfrak{u}(1)
  }
  \,.
$$

## Related concepts

* [[∞-Lie algebroid valued differential forms]]

* [[connection on an ∞-bundle]]

* [[curvature]]

* [[Bianchi identity]]

* [[curvature characteristic form]]

* **Chern-Simons form**

## References

The article introducing  the concept is

* [[Shiing-Shen Chern]], [[James Simons]], _[[Characteristic forms and geometric invariants]]_  The Annals of Mathematics, Second Series, Vol. 99, No. 1 (Jan., 1974), pp. 48-69  ([jstor:1971013](http://www.jstor.org/pss/1971013)) ([pdf](http://www.dpmms.cam.ac.uk/~hk244/chern-simons.cp.pdf))

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 [[orthogonal group|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

* [[James Simons]], [[Dennis Sullivan]], _Structured vector bundles define differential K-theory_ ([arXiv](http://arxiv.org/abs/0810.4935?context=math.DG))

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

The [[L-∞-algebra]]-formulation is discussed in [SSS08](http://arxiv.org/abs/0801.3480).

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.

[[!redirects Chern-Simons forms]]

[[!redirects Chern-Simons 3-form]]
[[!redirects Chern-Simons 3-forms]]


Chern-Simons gerbe > history

see *[[Chern-Simons 2-gerbe]]*


Chern-Simons gravity

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#### Gravity
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#### $\infty$-Chern-Simons theory
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#Contents#
* table of contents
{:toc}

## Idea

### General

In the [[first order formulation of gravity]] ([[Cartan connection]]) the [[Einstein-Hilbert action]] in [[dimension]] 3 happens to be equivalent to the [[action functional]] of [[Chern-Simons theory]] for [[connection on a bundle|connections]] with values in the [[Poincare Lie algebra]] ([[spin connection]] and [[vielbein]], for a suitable quadratic [[invariant polynomial]]. Similarly for 3-dimensional [[supergravity]]. See [[3d quantum gravity]].

Generally, for every higher degree ($n$-ary) [[invariant polynomial]] on the Poincaré Lie algebra there is the corresponding [[higher dimensional Chern-Simons theory]] in dimension $2n-1$. These Chern-Simons type [[action functional]]s all include the [[Einstein-Hilbert action]] linear in the [[curvature]] as a summand, but generally contain higher degree curvature invariants. A theory of _Chern-Simons gravity_ is a field theory governed by such an [[action functional]].

While there is no experimental evidence for such higher curvature terms in the theory of [[gravity]], there exist parameter regions in which their predicted effects are smaller than could have been observed. Moreover, there are various suggestions that under [[Inönü-Wigner contraction]] of the [[AdS Lie algebra]] to the [[Poincare Lie algebra]], an $so(n,2)$-Chern-Simons gravity ([[anti de Sitter gravity]]) theory could be close to an ordinary Einstein-Hilbert theory.

In view of this, the fact that Chern-Simons action functionals are singled out by their nice formal properties has led to various speculations that possibly the fundamental theory of gravity is secretly a theory of Chern-Simons gravity after all (see the [References](References)). Whether that is true or not, certainly in the general mathematical context of [[schreiber:infinity-Chern-Simons theory]] the study of Chern-Simons gravity is natural and interesting.

**Warning** In parts of the literature the term _Chern-Simons gravity_  is used for the modification of the [[Einstein-Hilbert action]] in  4-dimensions obtained by adding to the standard action the [[Pontryagin class]] term $\int_X \langle R \wedge R\rangle$ for $R$ the [[Riemann curvature]]. This is however not a Chern-Simons action functional in the strict sense of the term.

### Problems in the non-perturbative regime
 {#ProblemsInTheNonPerturbativeRegime}

The above discussion of the relation between 3-dimensional [[Einstein-Hilbert action|Einstein-Hilbert gravity]] and [[Chern-Simons theory]] certainly makes sense in [[perturbation theory]]. But [[non-perturbative quantum field theory|non-perturbatively]] the relation between the two theories is at best much more subtle and apparently in fact not existent.

Citing from ([Witten 07, pages 4,5,6](#Witten07)):

> [[non-perturbative field theory|nonperturbatively]], the relation between [[3d quantum gravity|three-dimensional gravity]] and [[Chern-Simons theory|Chern-Simons gauge theory]] is unclear. For one thing, in Chern-Simons theory, nonperturbatively the [[vierbein]] may cease to be invertible. For example, there is a [[equations of motion|classical solution]] with $A = \omega = e = 0$. The viewpoint in ([Witten 88](#Witten88)) was that such non-geometrical configurations must be included to make sense of three-dimensional quantum gravity nonperturbatively. But it has has been pointed out (notably by [[Nathan Seiberg|N. Seiberg]]) that when we do know how to make sense of [[quantum gravity]], we take the invertibility of the vierbein seriously. For example, in [[perturbative string theory]], understood as a model of [[2d quantum gravity|quantum gravity in two spacetime dimensions]], the [[integration]] over [[moduli space]] of [[Riemann surfaces]] that leads to a sensible theory is derived assuming that the [[Riemannian metric|metric]] should be non-degenerate. There are other possible problems in the nonperturbative relation between threedimensional gravity and Chern-Simons theory. The equivalence between [[diffeomorphisms]] and [[gauge transformations]] is limited to diffeomorphisms that are continuously connected to the identity. However, in gravity, we believe that more general diffeomorphisms (such as [[modular transformations]] in [[perturbative string theory]]) play an important role. These are not naturally incorporated in the Chern-Simons description. One can by hand supplement the gauge theory description by imposing invariance under disconnected diffeomorphisms, but it is not clear how natural this is. Similarly, in quantum gravity, one expects that it is necessary to sum over the different topologies of spacetime. Nothing in the Chern-Simons description requires us to make such a sum. We can supplement the Chern-Simons action with an instruction to sum over threemanifolds, but it is not clear why we should do this. From the point of view of the Chern-Simons description, it seems natural to fix a particular [[Riemann surface]] $\Sigma$, say of [[genus]] $g$, and construct a [[space of quantum states|quantum Hilbert space]] by quantizing the Chern-Simons gauge fields on $\Sigma$. (Indeed, there has been remarkable progress in learning how to do this and to relate the results to [[Liouville theory]] [8-11].) In [[quantum gravity]], we expect topology-changing processes, such that it might not be possible to associate a Hilbert space with a particular spatial manifold. Regardless of one's opinion of questions such as these, there is a more serious problem with the idea that gravity and gauge theory are equivalent nonperturbatively in three dimensions. Some years after the [[AdS/CFT|gauge/gravity relation]] was suggested, it was discovered by Bañados, Teitelboim, and Zanelli [12] that in three-dimensional gravity with negative cosmological constant, there are black hole solutions. The existence of these objects, generally called [[BTZ black holes]], is surprising given that the classical theory is "trivial." Subsequent work [13,14] has made it clear that three-dimensional black holes should be taken seriously, particularly in the context of the AdS/CFT correspondence [15]. The BTZ black hole has a horizon of positive length and a corresponding [[Bekenstein-Hawking entropy]]. If, therefore, three dimensional gravity does correspond to a quantum theory, this theory should have a huge degeneracy of black hole states. It seems unlikely that this degeneracy can be understood in Chern-Simons gauge theory, because this essentially topological theory has too few degrees of freedom.

See also at _[[AdS3-CFT2 and CS-WZW correspondence]]_ for (pointers to) discussion of how some variant of Chern-Simons theory appears as one "sector" of [[AdS/CFT]] in 3 dimensions/2-dimensions (and indeed not as the quantum gravity sector).

## Related concepts

* [[gravity]], [[supergravity]]

  [[first-order formulation of gravity]]

* [[higher dimensional Chern-Simons theory]]

* [[quantum gravity]]

  * [[3d quantum gravity]]

## References
 {#References}

### In 3 dimensions

The original articles that considered 3-dimensional gravity as a [[Chern-Simons theory]] ([[anti de Sitter gravity]]) are

* A. Achúcarro and [[Paul Townsend]], _A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories_ , Phys. Lett. B180 (1986) 89.

* {#Witten88} [[Edward Witten]], _(2+1)-Dimensional Gravity as an Exactly Soluble System_ Nucl. Phys. B311 (1988) 46. ([web](http://adsabs.harvard.edu/abs/1988NuPhB.311...46W))

Further developments are in

* {#Witten07} [[Edward Witten]], _Three-Dimensional Gravity Revisited_ ([arXiv:0706.3359](http://arxiv.org/abs/0706.3359))

A review of some aspects is in

* Bastian Wemmenhove, _Quantisation of 2+1 dimensional Gravity as a Chern-Simons theory_ thesis (2002) ([pdf](http://staff.science.uva.nl/~bais/scripties/wemmenhoven.pdf))

Boundaries in 3d Chern-Simons gravity and relation to the [[WZW model]] are discussed for instance in

* Giovanni Arcioni, [[Matthias Blau]], Martin O'Loughlin, _On the boundary dynamics of Chern-Simons gravity_ ([arXiv:0210089](http://arxiv.org/abs/hep-th/0210089))

See also [[3d quantum gravity]].

### General and higher dimensions

An introduction and survey is in

* {#Zanelli05} [[Jorge Zanelli]], _Lecture notes on Chern-Simons (super-)gravities_ ([arXiv:hep-th/0502193](http://arxiv.org/abs/hep-th/0502193))

More along these lines is in

* [[Jorge Zanelli]], _Chern-Simons Forms in Gravitation Theories_, Class. & Quantum Grav. vol.29, 133001 (2012) ([arXiv:1208.3353](http://arxiv.org/abs/1208.3353))

Original articles include

* [[Ali Chamseddine]], _Topological gravity and supergravity in various dimensions_ Nuclear Physics B346 (1990) 213—234 ([web](http://adsabs.harvard.edu/abs/1990NuPhB.346..213C))

* Máximo Bañados, _Higher dimensional Chern-Simons theories and black holes_ ([pdf](http://worldscibooks.com/etextbook/4388/4388_chap01.pdf))

* Máximo Bañados, Ricardo Troncoso, Jorge Zanelli, _Higher dimensional Chern-Simons supergravity_ Phys. Rev. D 54, 2605–2611 (1996)

* [[Jorge Zanelli]], _Chern–Simons forms and transgression actions or the universe as a subsystem_ Journal of Physics: Conference Series Volume 68 Volume 68

* Hitoshi Nishino and Subhash Rajpoot, _Supersymmetric Lorentz Chern-Simons terms coupled to supergravity_ Phys. Rev. D 81, 085029 (2010)

### In 11d

A speculation that [[11-dimensional supergravity]] is naturally to be understood as a contraction limit of a Chern-Simons supergravity theory was put forward in

* [[Petr Ho?ava]], _M-Theory as a Holographic Field Theory_ ([arXiv:hep-th/9712130](http://arxiv.org/abs/hep-th/9712130))

with further developments in

* [[Horatiu Nastase]], _Towards a Chern-Simons M theory of $OSp(1|32)\times OSp(1|32)$_ ([arXiv:hep-th/0306269](http://arxiv.org/abs/hep-th/0306269))

Other approaches to 11d Chern-Simons supergravity include

* Fernando Izaurieta, Eduardo Rodríguez, _On eleven-dimensional Supergravity and Chern-Simons theory_ ([arXiv:1103.2182](http://arxiv.org/abs/1103.2182))

There is an 6-ary [[invariant polynomial]] of degree $12 = 2 \cdot 6$ on the [[M-theory super Lie algebra]]. Using its [[Chern-Simons element]] as the [[Lagrangian]] for an [[schreiber:infinity-Chern-Simons theory]] yields an 11-dimensional supersymmetric field theory different from but maybe related to [[11-dimensional supergravity]]. This is discussed in

* Mokhtar Hassaine, Ricardo Troncoso, [[Jorge Zanelli]], _Poincaré invariant gravity with local supersymmetry as a gauge theory for the M-algebra_ , Phys. Lett. B596 (2004) 132-137 ([arXiv:hep-th/0306258](http://arxiv.org/abs/hep-th/0306258))

* Mokhtar Hassaine, Ricardo Troncoso, [[Jorge Zanelli]], _11D Supergravity as a gauge theory for the M-algebra_ ([arXiv:hep-th/0503220](http://arxiv.org/abs/hep-th/0503220))

A discussion of [[Chern-Simons elements]] in higher [[supergravity]]  and their relation not quite to Chern-Simons forms but to their curvature-first-order-analog -- the _cosmo-cocycle condition_  -- is at

* _[[D'Auria-Fre formulation of supergravity]]_ .

### Boundary theories

Boundary [[higher dimensional WZW models]] for nonabelian [[higher dimensional Chern-Simons theory]] are discussed in

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional Chern-Simons Gravity_ ([arXiv:hep-th/0010020](http://arxiv.org/abs/hep-th/0010020))

* J. Gegenberg , G. Kunstatter, _Boundary Dynamics of Higher Dimensional AdS Spacetime_ ([arXiv:http://arxiv.org/abs/hep-th/9905228](http://arxiv.org/abs/hep-th/9905228))

[[!redirects Chern-Simons supergravity]]

[[!redirects higher dimensional Chern-Simons gravity]]
[[!redirects higher dimensional Chern-Simons supergravity]]


Chern-Simons invariant

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#### Physics
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[[!include physicscontents]]
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## Idea

The value of the [[action functional]] of [[Chern-Simons theory]] (see there for more) is sometimes referred to as the _Chern-Simons invariant_.

A related term is _[[secondary characteristic class]]_.

## Related concepts

* [[invariant polynomial]], [[Chern-Simons form]]

[[!redirects Chern-Simons invariants]]


Chern-Simons propagator

+-- {: .rightHandSide}
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[[!include physicscontents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Chern-Simons propagator_ is the [[Feynman propagator]] for [[Chern-Simons theory]] regarded as a [[Euclidean quantum field theory]]. Since the [[Feynman propagator]] only depends on the [[free field theory]]-[[equations of motion]] (the staring point of [[perturbative QFT]]), which here is $d A = 0$, its form is independent of the [[gauge group]] and in fact applies also to all [[higher Chern-Simons theories]] such as notably the [[AKSZ sigma-models]].

In the perturbative [[quantization of 3d Chern-Simons theory]] the CS propagator was first studied in [Axelrod-Singer 91](#AxelrodSinger91).

### As a smooth differential form on compactified configuration spaces

The Chern-Simons propagator was re-formulated as a [[Feynman amplitudes on compactified configuration spaces of points|smooth differential form on compactified configuration space]] in [Axelrod-Singer 93, p. 5-6](#AxelrodSinger93). In this specific form its mathematical nature was amplified in [Bott-Cattaneo 97, remark 3.6](#BottCattaneo97) and [Cattaneo-Mnev 10, Remark 11](#CattaneoMnev10). It was suggested that this serves to exhibit its [[Feynman amplitudes]] as exhibiting a [[graph complex]]-model for the [[de Rham cohomology]] of [[configuration spaces of points]] in [Kontsevich 93](#Kontsevich93), [Kontsevich 94](#Kontsevich94), which was proven in [Lambrechts-Volic 14](#LambrechtsVolic14). This role of the Chern-Simons propagator is advertized also in [Campos-Idrissi-Lambrechts-Willwacher 18, p. 66](#CamposIdrissiLambrechtsWillwacher18).

## References

Detailed review is in

* Konstantin Wernli, section 3 of _Perturbative Quantization of Split Chern-Simons Theory on Handlebodies and Lens Spaces by the BV-BFV formalism_, 2018

Original articles include the following

* {#AxelrodSinger91} [[Scott Axelrod]], [[Isadore Singer]], _Chern-Simons Perturbation Theory_, in S. Catto, A. Rocha (eds.) Proc. XXthe DGM Conf. World Scientific Singapore, 1992, 3-45; ([arXiv:hep-th/9110056](http://arxiv.org/abs/hep-th/9110056))

* {#AxelrodSinger93} [[Scott Axelrod]], [[Isadore Singer]],  _Chern--Simons Perturbation Theory II_, J. Diff. Geom. 39 (1994) 173-213 ([arXiv:hep-th/9304087](http://arxiv.org/abs/hep-th/9304087))

* {#BottCattaneo97} [[Raoul Bott]], [[Alberto Cattaneo]], Remark 3.6 in _Integral invariants of 3-manifolds_, J. Diff. Geom., 48 (1998) 91-133 ([arXiv:dg-ga/9710001](https://arxiv.org/abs/dg-ga/9710001))

* {#CattaneoMnev10} [[Alberto Cattaneo]], [[Pavel Mnev]], Remark 11 in _Remarks on Chern-Simons invariants_, Commun.Math.Phys.293:803-836,2010 ([arXiv:0811.2045](https://arxiv.org/abs/0811.2045))

* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))

* {#Kontsevich94} [[Maxim Kontsevich]], pages 11-12 of _Feynman diagrams and low-dimensional topology_, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics __120__, Birkhäuser (1994), 97–121 ([pdf](http://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], sections 6 and 7 of _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society ; no. 1079, 2014  ([arXiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

* [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], appendix B of _Perturbative quantum gauge theories on manifolds with boundary_, Communications in Mathematical Physics, January 2018, Volume 357, Issue 2, pp 631–730  ([arXiv:1507.01221](https://arxiv.org/abs/1507.01221), [doi:10.1007/s00220-017-3031-6](https://doi.org/10.1007/s00220-017-3031-6))

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], Najib Idrissi, [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

[[!redirects Chern-Simons propagators]]