Chern-Weil theory

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#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#### Bundles
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[[!include bundles - contents]]
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#### Differential cohomology
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[[!include differential cohomology - 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

Chern-Weil theory studies the refinement of [[characteristic class]]es of [[principal bundle]]s in ordinary [[cohomology]] to [[de Rham cohomology]] and further to [[ordinary differential cohomology]].

The central operation that models this refinement is the construction of the [[Chern-Weil homomorphism]] from $G$-[[principal bundle]]s to [[de Rham cohomology]] by choosing a [[connection on a bundle|connection]] $\nabla$ and evaluating its [[curvature]] form $F_\nabla$ in the [[invariant polynomial]]s $\langle -\rangle$ of the [[Lie algebra]] $\mathfrak{g}$ to produce the [[curvature characteristic form]] $\langle F_\nabla \rangle$. Its de Rham cohomology class refines a corresponding [[characteristic class]] in [[integral cohomology]].

Concretely, the [[Chern-Weil homomorphism]] is presented by the following construction:

For

* $B G \in$ [[Top]] the [[classifying space]] of (the [[topological group]] underlying) a [[compact space|compact]] [[Lie group]] $G$

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

we get for each [[smooth manifold]] $X$  an assignment

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

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

Let $[c]_\mathbb{R} \in H^n(B G, \mathbb{R})$ be the image of $[c]$ in [[real cohomology]], induced by the evident inclusion of [[coefficients]] $\mathbb{Z} \hookrightarrow \mathbb{R}$.

The **first main statement** of Chern-Weil theory is that there is an [[invariant polynomial]]

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

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

$$
  \phi
  \;\colon\;
   inv(\mathfrak{g})
   \overset{\simeq}{\longrightarrow}
   H^\bullet(B G, \mathbb{R})
  \,.
$$

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

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

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

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

The **third main statement**, says that this construction may be refined by combining [[integral cohomology]] and [[de Rham cohomology]] to [[ordinary differential cohomology]]: the $n$-form $\langle F_\nabla \wedge \cdots F_\nabla\rangle$ may be realized itself as the [[curvature]] $n$-form of a [[circle n-bundle with connection]] $\hat \mathbf{c}$.

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

In summary this yields the following picture:

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

A central **implication** of the last step is that with the refinement from [[curvatures]] in [[de Rham cohomology]] to [[circle n-bundles with connection]] in [[differential cohomology]] is that these come with a notion of [[higher parallel transport]] and higher [[holonomy]]:

* the local connection form of $\hat \mathbf{c}$ is the [[Chern-Simons form]] $cs(\nabla)$ of a [[Chern-Simons element]] $cs$ of the [[invariant polynomial]] $\langle- \rangle$ evaluated on the given $G$-[[connection on a bundle|connection]];

* the corresponding [[higher parallel transport]] as an assignment

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

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

Specifically

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

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

So the refined Chern-Weil homomorphism provides a large family of [[gauge theory|gauge]] [[quantum field theories]] of Chern-Simons type in odd [[dimension]]s whose field configurations are always [[connection on a bundle|connections]] on [[principal bundle]]s and whose [[Lagrangians]] are [Chern-Simons element]]s on a [[Lie algebra]].

But the notion of [[invariant polynomial]]s and [[Chern-Simons element]]s naturally exists much more generally for [[L-∞ algebras]], and even more generally for [[L-∞ algebroid]]s. We claim here that in this fully general case there is still a natural analog of the Chern-Weil homomorphism -- which we call the _[[∞-Chern-Weil homomorphism]]_ . Accordingly this gives rise to a wide class of [[action functional]]s for [[gauge theory|gauge]] [[quantum field theories]], which may be called _[[schreiber:∞-Chern-Simons theories]]_.

## Generalizations

### To principal $\infty$-bundles

The notions of [[Lie group]], [[Lie algebra]], [[principal bundle]] and all the other ingredients of ordinary Chern-Weil theory generalize to notions in [[higher category theory]] such as  [[∞-Lie group]], [[∞-Lie algebra]], [[principal ∞-bundle]] etc. The generalization of Chern-Weil theory to this context is discussed at

* [[∞-Chern-Weil theory]].

### In noncommutative geometry

There is a [[noncommutative geometry|noncommutative]] analogue discussed in ([AlekseevMeinrenken2000](#AlekseevMeinrenken)).

## Examples and applications

* [[de Rham theorem]]

* [[Gauss-Bonnet theorem]]

* [[differential string structure]]

* [[differential fivebrane structure]]

## History {#History}

> (following notes provided by [[Jim Stasheff]])

The beginnings of the [[rational homotopy theory]] of [[Lie group]]s $G$ and hence their [[dg-algebra]]-description in terms of the [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ originate in the first half of the 20th century.

In his survey of what was known in 1936 on the homology of compact Lie groups

* [[Eli Cartan]], _La topologie des espaces représentatifs des groupes de Lie_ Act. Sci. Ind., No. 358, Hermann, Paris, (1936).

  reprinted in _Cartan's Complete Works_ vol $I_2$ pp. 1307-1330

E. Cartan conjectured that there should be a general result implying that the [[homology]] of the [[classical Lie group]]s is the same as the homology of a product of odd-dimensional [[sphere]]s. In particular, he lists the [[Poincare polynomial]]s for classical simple compact Lie groups.

In

* [[Heinz Hopf]],  _Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen_  (German)  Ann. of Math. (2)  42,  (1941). 22--52.

Hopf showed that such a characterization in terms of homology groups as [[intersection pairing]] algebras holds for any [[compact space|compact]] [[dimension|finite dimensional]] [[connected]] [[orientable]] [[manifold]] with a map $m:M\times M\to M$ such that left and right translation have non-zero degrees.

Later in

* [[Shiing-shen Chern]], _Differential geometry of fiber bundles_ Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pages 397-411,  Amer. Math. Soc., Providence, R. I. (1952)

with the development of [[cohomology]], especially [[de Rham cohomology]], this was stated as $H^\bullet(G)$ being [[isomorphic]] to  an [[exterior algebra]] on odd dimensional generators: the generating [[Lie algebra cohomology]] [[cocycle]]s $\mu \in CE(\mathfrak{g})$, $d_{CE(\mathfrak{g})} \mu = 0$.

Henri Cartan in

* [[Henri Cartan]], _Notions d'algébre différentielle; applications aux groupes de Lie et aux variétés où opère un groupe de Lie_ ,  Coll. Topologie Algébrique Bruxelles (1950)
15-28

  section 7, titled _Classes caracteristiques (reelles) d'un espace fibre principal_

at the end (1951) of an era of deRham cohomology dominence (prior to Serre's thesis) abstracted the [[differential geometry|differential geometric]] approach of Chern-Weil and the [[Weil algebra]] $W(\mathfrak{g})$ to the [[dg-algebra]] context with his notion of _$\mathfrak{g}$-algebras $A$_ . This involves what is known sometimes as the [[Cartan calculus]].  In addition to the differential $d$ of [[differential form]]s on a [[principal bundle]], Cartan abstracts the inner product aka contraction of differential forms with [[vector field]]s $X$ and the [[Lie derivative]] $\mathcal{L}_X$ with respect to vector fields.  that is, he posits 3 operators on a differential-graded-commutative alggebra (dgca):

$d$ of degree 1, $i_X$ of degree -1 and $L_X$ of degree 0 for X in $\mathfrak{g}$ subject to the relations:

* $[\iota_X,\iota_Y] = \iota_{[X,Y]}$

* $ [\mathcal{L}_X,\iota_Y]= \iota_{[X,Y]} $

and perhaps most useful

* $\mathcal{L}_X = d \iota_X + i\iota_X d$

This is what he terms a $\mathfrak{g}$-algebra.

For Cartan, an infinitesimal [[connection on a bundle|connection]] on a [[principal bundle]] $P \to X$ are  projectors (at each point $p$ of $P$) $\phi_p: T_p P\to T_p^{vert}$ equivariant with repect to the $G$-action.
This can be abstracted to a morphism

$$
  \Omega^\bullet(P) \leftarrow \mathfrak{g}^* : A
$$

of [[graded vector space]]s of degree 1 -- equivalently a [[Lie-algebra valued 1-form]] $A \in \Omega^1(P,\mathfrak{g})$ -- such that the two [[Ehresmann connection|Ehresmann condition]]s hold:

1. restricted to the fibers the 1-form $A$ is the [[Maurer-Cartan form]]

   $\iota_X A(h) = \iota_X h$

1. the form is equivariant in that

   $\mathcal{L}_X A (h) = A(\mathcal{L}_X h)$

  for all $X\in \mathfrak{g}$ and $h\in \mathfrak{g}^*$.  This data Cartan calls an _algebraic connection_ .

He then extends such an $A$ to a [[homomorphism]] of [[graded algebra]]s

$$
  \Omega^\bullet(P) \leftarrow CE(\mathfrak{g}) : A
$$

from the [[Chevalley-Eilenberg algebra]] $\wedge^\bullet \mathfrak{g}^*$.

In general, this will not respect the differentials, hence not be a morphism of [[dg-algebra]]s. In fact, the deviation gives the [[curvature]] of the connection: the curvature tensor is the map $h\mapsto d_{dR} A(h)-A(d_{CE} h)$.

> Jim: HAVE TO BREAK OFF NOW - WHAT WILL COME NEXT IS the [[Weil algebra]] $W(\mathfrak{g})$ as a Cartan $\mathfrak{g}$-algebra

* Weil, _Geometrie differentielle des espaces fibres_ (1949, unpublished)
  appears in Vol. 1, pp. 422-436, of his Collected Papers.

## Related concepts

* [[Sullivan model of classifying space]]

* [[Chern-Weil homomorphism]]

## References

[[!include Chern-Weil homomorphism -- references]]

More review:

* [[Fei Han]], _Chern-Weil theory and some results on classic genera_ ([pdf](http://math.berkeley.edu/~alanw/240papers03/han.pdf))

More monographs:

* [[Johan Louis Dupont]], _Fibre bundles and Chern-Weil theory_, Lecture Notes Series __69__, Dept. of Math., University of Aarhus, Aarhus, 2003, 115 pp. [pdf](http://www.johno.dk/mathematics/fiberbundlestryk.pdf)

* [[Johan Louis Dupont]], _Curvature and characteristic classes_, Lecture Notes in Math. __640__, Springer-Verlag, Berlin-Heidelberg-New York, 1978.

* [[Mikhail Postnikov]], _Лекции по геометрии. Семестр 4, Дифференциальная геометрия_ — М.: Наука, 1988

* [[Raoul Bott]], [[Loring Tu]], _Differential forms in algebraic topology_, Graduate Texts in Mathematics __82__, Springer 1982. xiv+331 pp.

* [[Loring Tu]], _Differential Geometry -- Connections, Curvature, and Characteristic Classes_, Springer (2017) [[ISBN:978-3-319-55082-4](https://www.springer.com/gp/book/9783319550824)]

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, 1999.

Lecture notes with an eye on [[Morse theory]] in terms of [[supersymmetric quantum mechanics]]:

* [[Weiping Zhang]], _Lectures on Chern-Weil theory and Witten deformations_ , Nankai Tracts in Mathematics - Vol. 4  ([web](http://www.worldscibooks.com/mathematics/4756.html))

Chern-Weil theory in the context of [[noncommutative geometry]]

* {#AlekseevMeinrenken} [[Anton Alekseev]], [[Eckhard Meinrenken]], _The non-commutative Weil algebra_, Invent. Math. **139** 1 (2000) 135-172 [[doi:10.1007/s002229900025](http://dx.doi.org/10.1007/s002229900025)]

[[!redirects Chern-Weil theorem]]
[[!redirects Chern-Weil theorems]]


Chern-Weil theory in Smooth∞Grpd

> much of the material below has been or is being reworked into the entries [[Smooth∞Grpd]] and [[connection on a smooth principal ∞-bundle]]

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#### $\infty$-Chern-Weil theory
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[[!include infinity-Chern-Weil theory - contents]]
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#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

In every [[cohesive (∞,1)-topos]] there is an intrinsic notion of Chern-Weil theory. We discuss the concrete realization of this in the cohesive $(\infty,1)$-topos [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s. This s the case that subsumes ordinary [[Chern-Weil theory]] of [[SmoothMfd|smooth]] [[principal bundle]]s with [[connection on a bundle|connection]] and generalizes it to [[connections on smooth principal ∞-bundles]].

## Motivation {#Motivation}

The central motivation for the study of a higher generalization of ordinary [[Chern-Weil theory]] is the interest in extending the [[Chern-Weil homomorphism]] for a given [[Lie group]] $G$ to the higher connected covers of $G$ through the whole [[Whitehead tower]] of $G$. Beyond the simply connected cover, these higher connected covers are still [[topological group]]s but fail to be (finite dimensional) [[Lie group]]s. They do however have natural realizations as [[smooth ∞-group]]s. Higher Chern-Weil theory is the extension of [[Chern-Weil theory]] from Lie groups to such smooth $\infty$-groups. It allows the refinement of differential characteristic classes to _fractional_ differential characteristic classes, that capture finer cohomological information.

### Fractional differential classes

We give some examples of such _fractional characteristic classes_
that occur in practice.

It is a familiar classical fact that the first [[Pontryagin class]]

$$
  p_1 : \mathcal{B}SO \to \mathcal{B}^4 \mathbb{Z}
  \,,
$$

which represents the generator of the fourth [[integral cohomology]] of the [[classifying space]] $\mathcal{B} SO$ of the [[special orthogonal group]] allows a division by 2 when pulled back one step along the [[Whitehead tower]] to the classifying space of the [[spin group]], in that there is a [[commuting diagram]]

$$
  \array{
     \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z}
     \\
     \downarrow && \downarrow^{\mathrlap{\cdot 2}}
     \\
     \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4 \mathbb{Z}
  }
  \,,
$$

in [[Top]], where the top horizontal morphism represents a generator of the 4th integral cohomology of the classifying space of the [[spin group]] and the right vertical morphism is induced by multiplication by 2 on the additive group of [[integer]]s.

This means that for $X$ [[manifold]] with [[spin structure]] exhibited by a classifying map $\hat g$

$$
  \array{
     && \mathcal{B} Spin &\stackrel{\frac{1}{2}p_1}{\to}& \mathcal{B}^4 \mathbb{Z}
     \\
     & {}^{\mathllap{\hat g}}\nearrow & \downarrow && \downarrow^{\mathrlap{\cdot 2}}
     \\
     X &\stackrel{g}{\to}& \mathcal{B} SO &\stackrel{p_1}{\to}& \mathcal{B}^4\mathbb{Z}
  }
$$

of its [[tangent bundle]] $T X$, the characteristic class $p_1(T X) : X \stackrel{g}{\to} \mathcal{B}SO \stackrel{p_1}{\to} \mathcal{B}^4 \mathbb{Z}$ of $T X$ regarded as an $SO$-[[associated bundle]] contains less information than the class $\frac{1}{2}p_1(T X) : X \stackrel{\hat g}{\to} \mathcal{B}Spin \stackrel{\frac{1}{2}p_1}{\to} \mathcal{B}^4 \mathbb{Z}$. For instance if the 4th cohomology of $X$ happens to be 2-[[torsion]], the former class entirely vanishes, while the latter need not.

This familiar situation poses no problem to classical [[Chern-Weil theory]], because both the [[special orthogonal group]] as well as the [[spin group]] of course have canonical structures of [[Lie group]]s, so that the [[Chern-Weil homomorphism]] may be applied to either.
We shall write $\mathbf{B} \mathrm{Spin}$ for the smooth refinement of  the classifying space $B \mathrm{Spin}$: the delooping Lie groupoid of $\mathrm{Spin}$
or equivalently the moduli stack for smooth $\mathrm{Spin}$-principal bundles. Here and in the
following the boldface indicates smooth (or otherwise cohesive) refinements.
Accordingly, there is a
smooth refinement $\frac{1}{2}\mathbf{p} : \mathbf{B} \mathrm{Spin} \to \mathbf{B}^3 U(1)$
of the first Pontryagin class, which takes smooth
$\mathrm{Spin}$-principal bundles to their first Pontryagin class.
This in turn has
has a further differential refinement $\frac{1}{2}{\hat {\mathbf{p}}} :
\mathbf{B}\mathrm{Spin}_{\mathrm{conn}} \to \mathbf{B}^3 U(1)_{\mathrm{conn}}$
that takes $\mathrm{Spin}$-principal bundles with connection to their Chern-Simons 2-gerbes
with connection.

All this is still captured by the traditional (refined) Chern-Weil homomorphism.
But this is no longer the case as we keep climbing up the [[Whitehead tower]] of the [[orthogonal group]]. In the next step the second [[Pontryagin class]] $p_2 : \mathcal{B}SO \to \mathcal{B}^8 \mathbb{Z}$ may be divided by 6 when pulled back to the classifying space of the [[string group]] (SSSII)

$$
  \array{
     \mathcal{B} String &\stackrel{\frac{1}{6}p_2}{\to}& \mathcal{B}^8 \mathbb{Z}
     \\
     \downarrow && \downarrow^{\mathrlap{\cdot 6}}
     \\
     \mathcal{B} SO &\stackrel{p_2}{\to}& \mathcal{B}^8 \mathbb{Z}
  }
  \,,
$$

As before, this means that if a space $X$ admits a [[string structure]], then the characteristic class $p_2(X)$ contains less information than the fractional refinement $\frac{1}{6}p_2(X)$ that it admits. In particular, the former may vanish if the degree 8 cohomology group of $X$ has 6-[[torsion]], while the latter need not vanish.

For purposes of ordinary cohomology this is no problem, but for the differential refinement by ordinary [[Chern-Weil theory]] it is: the [[string group]] does not admit a [[Lie group]] structure
that would make it a smooth version of the homotopy fiber of
$\frac{1}{2}p_1$ and hence standard [[Chern-Weil theory]] cannot produce the differential refinement of the fractional class $\frac{1}{6}p_2$.

But $\infty$-Chern-Weil theory can: there is a natural smooth refinement of the [[string group]] to a [[Lie 2-group]]: the [[string 2-group]]. We write $\mathbf{B}String$ for the corresponding [[delooping]] [[∞-Lie groupoid]]. The fractional second Pontryagin class does lift to this smooth refinement to produce a [[characteristic class]]

$$
  \frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \to \mathbf{B}^7 U(1)
$$

internal to $\mathbf{H} = $ [[?LieGrpd]]. Since this now lives in a smooth context, it does now have a differential Chern-Weil refinement

$$
  \frac{1}{6}\hat \mathbf{p}_2 : \mathbf{H}_{conn}(-, \mathbf{B}String)
  \to
  \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))
$$

that takes smooth $String$-[[principal 2-bundle]]s with 2-connection to degree 8-cocycles in [[ordinary differential cohomology]].

This kind of refinement we discuss in a bit more detail in the next section.

### Higher differential spin structures

These refined differential invariants of
fractional characteristic classes are relevant in the discussion of higher differential spin
structures.
(See the first part of
(SatiSchreiberStasheff II for a review).)
Ordinary [[spin structure]]s on a [[manifold]] may be understood
as trivializations of what are called
[[quantum anomaly]] [[Pfaffian line bundle]]s on the configuration
space of the spinning [[relativistic particle|quantum particle]] propagating on that manifold.
(This physical origin is after all the origin of the term
_spin structure_ .)
When these point-like super-particles are generalized to higher-dimensional $p$-[[brane]]s,
the trivialization of the corresponding Pfaffian line bundles correspond to
[[string structure]]s for $p = 1$
(this goes back to
(Killingback)
and
(Witten)
and has been made rigorous in
(Bunke)
then to [[fivebrane structure]]s for $p = 5$
(SatiSchreiberStasheff II)).

More precisely, the Pfaffian line bundles appearing here come equipped with a [[connection on a bundle|connection]],
and what matters is a trivialization of these bundles as bundles with connection.
This refinement translates to differential refinements of the string structures and the fivebrane structures on $X$.
The differential form data of a twisted
[[differential string structure]] constitutes
what in the physics literature is called the
[[Green-Schwarz mechanism]]. While this
still can and has been captured with tools of ordinary Chern-Weil theory and ordinary differential cohomology
(Freed,
Waldorf)
it has a natural formulation in higher Chern-Weil theory.
Going beyond that, the [[nLab:dual heterotic string theory|magnetic dual]] Green-Schwarz mechanism can be seen to encode a twisted differential fivebrane structure and this is not practical to be studied without some higher geometry.

The following restates this in a bit more technical detail.

For $G = Spin$ the [[spin group]], the first nontrivial [[characteristic class]] is the first fractional [[Pontryagin class]] given by a [[cocycle]] $\frac{1}{2}p_1 : \mathcal{B}G \to K(\mathbb{Z}, 4)$ in ordinary [[integral cohomology]] $H^4(\mathcal{B}Spin, \mathbb{Z})$. This induces a map

$$
  H^1(X, Spin) = H(X, \mathcal{B}Spin) \to H^4(X, \mathbb{Z})
$$

from isomorphism classes of topological $Spin$-[[principal bundle]]s to degree 4 integral cohomology.

If we assume that $X$ is a [[smooth manifold]] then we may consider the set

$$
Spin Bund(X)/ \sim = H(X,\mathbf{B}Spin)
$$

of [[isomorphism]]-classes of _smooth_ $Spin$-[[principal bundle]]s. Here and in all of the following, the boldface in "$\mathbf{B}G$" indicates a refinement, here of the bare classifying space $\mathcal{B}G$ to a smooth incarnation.

Then ordinary [[Chern-Weil theory]] provides a refinement of the fractional Pontryagin class $H(X, \mathbf{B}Spin) \to H^4(X,\mathbb{Z})$ to a map to [[ordinary differential cohomology]] $H_{diff}^4(X)$

$$
  \frac{1}{2} \hat p_1 : H(X, \mathbf{B}Spin) \to H_{diff}^4(X)
  \,.
$$

The first point of passing to a [[higher category theory]]-refinement of this situation is that it allows to refine, in turn, these morphisms of cohomology _sets_ to morphisms

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

and

$$
  \frac{1}{2} \hat \mathbf{p}_1 : \mathbf{H}_{conn}(X, \mathbf{B}Spin) \to \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))
$$

of [[cocycle]] [[∞-groupoid]]s: here $\mathbf{H}(X,\mathbf{B}G)$ is the [[groupoid]] whose objects are smooth $Spin$-[[principal bundle]]s, and whose morphisms are smooth homomorphisms between these. Similarly $\mathbf{H}(X,\mathbf{B}^3 U(1))$ denotes the [[3-groupoid] whose objects are smooth circle 2-group-[[principal ∞-bundles|principal 3-bundles]], while $\mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))$ is accordingly the [[3-groupoid]] whose objects are  [[circle n-bundle with connection|circle 3-bundles with connection]], whose morphisms are homomorphisms between these, whose 2-morphisms are higher homotopies between those. The original morphism of cohomology sets is the [[decategorification]] of this, the restriction to connected components:

$$
  \frac{1}{2}p_1 = \pi_0(\frac{1}{2}\mathbf{p}_1)
  \,.
$$

This refinement to cocylce $\infty$-groupoids notably has the consequence that it allows us to produce the [[homotopy fiber]]s of these morphisms.  To see the relevance of this, recall (from _[[string structure]]_ ) that the [[homotopy fiber]] of the
bare fractional Pontryagin class, which is the
[[(∞,1)-pullback]]/[[homotopy pullback]]

$$
  \array{
     \mathbf{H}(X,\mathbf{B}String) &\to& *
     \\
     \downarrow &\swArrow_\simeq& \downarrow
     \\
     \mathbf{H}(X,\mathbf{B}G)
     &\stackrel{\frac{1}{2} \mathbf{p}_1}{\to}&
     \mathbf{H}(X, \mathbf{B}^3 U(1))
  }
  \,,
$$

defines the $\infty$-groupoid $\mathbf{H}(X, \mathbf{B}String)$ of [[string structure]]s on $X$ ( _smooth_ , but not _differential_ ).

We can now replace the class $\frac{1}{2}\mathbf{p}_1$ by its differential refinement $\frac{1}{2}\hat \mathbf{p}_1$ and obtain an [[∞-groupoid]] $String_{diff}(X)$ that differentially refines the 2-groupoid $\mathbf{H}(X,\mathbf{B}String)$ of String-structures as the [[(∞,1)-pullback]]

$$
  \array{
     String_{diff}(X) &\to& *
     \\
     \downarrow &\swArrow_\simeq& \downarrow
     \\
     \mathbf{H}_{conn}(X,\mathbf{B}G)
     &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}&
     \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))
  }
  \,.
$$

This $String_{diff}(X)$ we may call the $\infty$-groupoid of _[[differential string-structures]]_ . A cocycle in there is naturally identified with a tuple consisting of

* a smooth $Spin$-[[principal bundle]] $P \to X$ with [[connection on a bundle|connection]] $\nabla$;

* the [[Chern-Simons 2-gerbe]] with connection $CS(\nabla)$ induced by this;

* a choice of trivialization of this Chern-Simons 2-gerbe -- this is the [[homotopy]] [[2-morphism]] in the middle of the above pullback diagram.

We may think of this as a refinement of [[secondary characteristic class]]es: the first Pontryagin [[curvature characteristic form]] $\langle F_\nabla \wedge F_\nabla \rangle$ itself is constrained to vanish, and so the [[Chern-Simons form]] 3-connection itself constitutes cohomological data.

So far this uses mostly just a little bit of [[(∞,1)-category theory]] or at least some [[homotopy theory]]. The first glimpse of something beyond ordinary [[Chern-Weil theory]] appearing is the $\infty$-groupoid $\mathbf{H}(X,\mathbf{B}String)$ which may be thought of as the [[2-groupoid]] of _smooth_ [[string 2-group]]-[[principal 2-bundle]]s.

But suppose we fix an $X$ such that $H(X, \mathbf{B}String)$ is nontrivial. Then we can continue the proceed to higher degrees:

the next topological characteristic class is the second fractional [[Pontryagin class]] $\frac{1}{6}p_2 : \mathcal{B}String \to \mathcal{B}^7 U(1)$. Since the [[string group]] does not have the structure of a [[Lie group]], this cannot be refined to [[differential cohomology]] using ordinary [[Chern-Weil theory]]. However, in terms of $\infty$-Chern-Weil theory it can:

we may obtain a differential refinement

$$
  \frac{1}{6}\hat \mathbf{p}_2 :
  \mathbf{B}String_{conn}
  \to
  \mathbf{B}^7 U(1)_{conn}
$$

that maps smooth [[string 2-group]]-[[principal 2-bundle]]s with 2-connectins to their [Chern-Simons circle 7-bundle with connection](#FivebraneStructure). This is an example of the higher version of the [[Chern-Weil homomorphism]].

And naturally we are then entitled to form its [[homotopy fiber]]s and produce the [[n-groupoid|7-groupoid]] of _[differential fivebrane structures](#DiffFivebraneStrucs)_ -- $Fivebrane_{diff}(X)$. For that notice (see [[fivebrane structure]]) that the homotopy fiber of the smooth but non-differential cocycles

$$
  \array{
    \mathbf{H}(X, \mathbf{B}Fivebrane) &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    \mathbf{H}(X, \mathbf{B}String)
    &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}&
    \mathbf{H}(X, \mathbf{B}^7 U(1))
  }
$$

is the [[n-groupoid|7-groupoid]] of smooth [[fivebrane structure]]s on $X$. Its differential refinement

$$
  \array{
    Fivebrane_{diff}(X) &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    \mathbf{H}_{conn}(X, \mathbf{B}String)
    &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}&
    \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))
  }
$$

we may therefore call the 7-groupoid $Fivebrane_{diff}(X)$ of _differential fivebrane structures_ . Cocycles in here are naturally identified with tuples of

* a $String$-[[principal 2-bundle]] $P \to X$, equipped with a [2-connection](#InfinityLieAlgebraConnection) $\nabla$;

* the Chern-Simons [[circle n-bundle with connection|circle 7-bundle]] $CS_7(\nabla)$ with connection induced by it;

* a choice of trivialization of $CS_7(\nabla)$.

These are the kind of structures that $\infty$-Chern-Weil theory studies.

## Idea

Ordinary [[Chern-Weil theory]] is about refinements of [[characteristic class]]es of $G$-[[principal bundle]]s for $G$ a [[Lie group]] (equivalently of the [[classifying space]] $\mathcal{B}G$ of that Lie group) from ordinary [[cohomology]] to [[differential cohomology]].

Under _$\infty$-Chern-Weil theory_ we want to understand the generalization of this to [[(∞,1)-category theory]]: where [[Lie group]]s are generalized to [[∞-Lie group]]s, [[Lie algebra]]s are generalized to [[∞-Lie algebra]]s and [[principal bundle]]s to [[principal ∞-bundle]]s.

So $\infty$-Chern-Weil theory produces [[differential cohomology]]-refinements of [[characteristic class]]es of $G$-[[principal ∞-bundle]]s for $G$ an [[∞-Lie group]], equivalently of the corresponding [[classifying space]]s $\mathcal{B}G$.

Ordinary Chern-Weil theory is formulated in the context of [[differential geometry]]. We need to widen this context somewhat in order that it can accomodate the relevant higher structures and so we place ourselves in the context of the [[(∞,1)-topos]] $\mathbf{H} = $ [[?LieGrpd]] of [[∞-Lie groupoids]].

In every $(\infty,1)$-topos that admits a notion of differential cohomology,  there is a general abstract notion of refinement of [[characteristic class]]es in [[cohomology]] to [[curvature characteristic forms|curvature characteristic classes]] in [[ordinary differential cohomology]].

The main construction in ∞-Chern-Weil theory is a concrete _model_ or _presentation_ of this abstract operation. This model is constructed in terms of [[Lie integration]] of objects in [[∞-Lie algebra cohomology]]. This construction is the higher analog of the [[Chern-Weil homomorphism]]. Its crucial intermediate step is the definition and construction of _[[connection on an ∞-bundle|∞-connections]]_ on [[principal ∞-bundle]]s.

This model itself is after all built on concrete familiar constructions in [[differential geometry]] and can be studied and appreciated in itself without recourse to the higher topos theory that we claim it provides a model for. The so inclined reader can ignore all the general abstract discussion in the following and concentrate on the concrete differential geometry.

ere is how this entry here proceeds.

A warmup for the full theory that connects to classical constructions is given at

* [Introduction](#PreparatoryConcepts)

Then in

* [∞-Chern-Weil theory](#ChernWeil)

we discuss the general definition of $\infty$-connections and of the Chern-Weil homomorphism and discuss some general properties. Then we turn to discussing

* [Examples](#Examples)

## Preparatory concepts {#PreparatoryConcepts}

General $\infty$-Chern-Weil theory, as described below, is naturally formulated in the context of [[(infinity,1)-topos]]-theory and some of its aspects can only be understood from that perspective.

However, unwinding the abstract higher topos theoretic concepts in terms of 1-categoriecal models yields concrete structures in familiar contexts of [[differential geometry]] that connect to various classical and familiar concepts. Since a full appreciation of the abstract formulation benefits from having a feeling for how these concrete models work out, the reader may at this point wish to look into some such basic aspects. These may be found behind the following link

* [[infinity-Chern-Weil theory -- preparatory concepts]].

## $\infty$-Chern-Weil theory {#ChernWeil}

For $G,A$ [[nLab:∞-group]]s in an [[∞-connected (∞,1)-topos]] $\mathbf{H}$ with [[nLab:delooping]]s $\mathbf{B}G$ and $\mathbf{B}A$, respectively, every [[nLab:characteristic class]] $c : \mathbf{B}G \to A$ serves to pull back the canonical intrinsic curvature form $curv_A : A \to \mathbf{\flat}_{dR} \mathbf{B}A$ to an intrinsic differential form $curv_A\circ c  : \mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}A$ on $\mathbf{B}G$.

For $G$ an ordinary [[nLab:Lie group]] regarded naturally as an object in $\mathbf{H} = $ [[nLab:?LieGrpd]], we show that the ordinary [[nLab:Chern-Weil homomorphism]] for $G$-[[nLab:principal bundle]]s may be understood as a concrete _model_ for this simple abstract situation, which applies to those characteristic classes $c$ that happen to be in the image of the Lie intgeration of Lie algebra cocycles.

More generally, this construction applies for $G$ an [[nLab:∞-Lie group]] with [[nLab:∞-Lie algebra]] $\mathfrak{g}$ and $c$ a characteristic class on $\mathbf{B}G$ that arises from Lie integration of a cocycle in the [[nLab:∞-Lie algebra cohomology]] of $\mathfrak{g}$.

The ordinary [[nLab:Chern-Weil homomorphism]] uses a [[nLab:connection on a bundle]] $\nabla$ as an intermediate tool for interpolating from a $G$-[[nLab:principal bundle]] to its curvature characteristic, represented by the [[nLab:curvature characteristic form]] $\langle F_\nabla \rangle$, where $F_\nabla$ is the [[nLab:curvature]] of $\nabla$ and $\langle - \rangle$ is an [[nLab:invariant polynomial]] on $\mathfrak{g}$. The choice of connection in this construction may be understood as providing a correspondence space in the following construction.

We know from the discussion of abelian differential cohomology [above](#AbGerbe) that the intrinsic morphism

$$
  \mathbf{B}^n \mathbb{R}/\mathbb{Z} \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}/\mathbb{Z}
$$

in $\mathbf{H} = \infty LieGrpd$ is modeled in $[CartSp^{op}, sSet]_{proj,cov}$ by the correspondence

$$
  \array{
     \mathbf{B}^n \mathbb{R}/\Gamma_{diff,simp} &\stackrel{curv}{\to}&
     \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}_{simp}
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{B}^n \mathbb{R}/\Gamma_{simp}
  }
  \,.
$$

If we write

$$
  \exp(b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R}))
  :
  \mathbf{cosk}_{n+1}(
  (U,[k])
  \mapsto
  \left\{
    \array{
      C^\infty(U)\otimes \Omega^\bullet(\Delta^k)
      &\leftarrow&
      CE(b^{n-1}\mathbb{R})
      \\
      \uparrow && \uparrow
      \\
      \Omega^\bullet(U)\otimes \Omega^\bullet(\Delta^n)
      &\leftarrow&
      W(b^{n-1}\mathbb{R})
    }
  \right\}
  )
$$

and so forth, then this correspondence is

$$
  \array{
     \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R}))
     &\to&
     \exp(* \to b^n \mathbb{R})
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     \exp(b^{n-1}\mathbb{R}\to *)
  }
  \,.
$$

If now $\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\mathbb{Z}$ is modeled by the Lie integration of a cocycle $\mu$ on a Lie $k$-algebra

$$
  \mathbf{cosk}_{k+1} \exp(\mathfrak{g})
  \stackrel{\exp(\mu)}{\to}
  \exp(b^{n-1}\mathbb{R})/\Gamma
  =
  \exp(b^{n-1}\mathbb{R} \to *)/\Gamma
$$

for $k \geq n-1$, then the total intrinsic differential form $\mathbf{B}G \to \mathbf{B}^n \mathbb{R}/\Gamma \to \mathbf{\flat}_{dR}\mathbf{B}^n \mathbb{R}$ is modeled by the zig-zag of morphisms

$$
  \array{
     && \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R}))
     &\to&
     \exp(* \to b^n \mathbb{R})
     \\
     && \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{cosk}_{k+1}\exp(\mathfrak{g})
     &\stackrel{\exp(\mu)}{\to}&
     \exp(b^{n-1}\mathbb{R}\to *)
  }
$$

in $[CartSp^{op}, sSet]$. In order to compute with such zig-zags of morphisms, in particular in order to compute [[nLab:homotopy fiber]]s, it is helpful to complete this to a single correspondence. There is a fairly evident choice for the tip of this total corresponence, namely

$$
  \mathbf{B}G_{diff}
  :=
  \mathbf{cosk}_{n+1} \exp(\mathfrak{g} \to inn(\mathfrak{g}))
  \,.
$$

It remains to complete the square and extend the [[nLab:∞-Lie algebra cohomology|∞-Lie algebra cocycle]] $\mu : \mathfrak{g} \to b^{n-1}\mathb{R}$ to a morphism $(\mathfrak{g} \to inn(\mathfrak{g})) \to (b^{n-1}\mathbb{R} \to inn(b^{n-1}\mathbb{R}))$. This is accomplished by an [[nLab:invariant polynomial]]

$$
  \langle -\rangle_\mu
  :
  inn(\mathfrak{g}) \stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to}
  inn(b^{n-1}\mathbb{R})
  \to
  b^n \mathbb{R}
$$

which is in transgression with $\mu$, witnessed by the [[nLab:Chern-Simons form|Chern-Simons element]] $cs_\mu$. Using this, we obtain the total diagram

$$
  \array{
     \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g}))
     &\stackrel{(\langle - \rangle_\mu, cs_\mu)}{\to}&
     \exp(b^{n-1}\mathbb{R}\to inn(b^{n-1}\mathbb{R}))
     &\to&
     \exp(* \to b^n \mathbb{R})
     \\
     \downarrow^{\mathrlap{\simeq}}
     && \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{cosk}_{k+1}\exp(\mathrak{g})
     &\stackrel{\exp(\mu)}{\to}&
     \exp(b^{n-1}\mathbb{R}\to *)
  }
  \,.
$$

By the fact that this commutes, we have that the correspondence

$$
  \left(
  \array{
    \mathbf{B}G_{diff,simp}
    &\to&
    \mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}_{simp}
    \\
    \downarrow
    \\
    \mathbf{B}G_{simp}
  }
  \right)
  \;\;
  :=
  \;\;
  \left(
  \array{
    \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to inn(\mathfrak{g}))
    &\to&
    \exp(* \to b^n \mathbb{R})
    \\
    \downarrow
    \\
    \mathbf{cosk}_{k+1}\exp(\mathfrak{g} \to *)
  }
  \right)
$$

in $[CartSp^{op}, sSet]_{proj,cov}$ models the intrinsic curvature characteristic form $\mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1} \mathbb{R}$.

We may identify cocycles with values in $\mathbf{B}G_{diff}$ as _(pseudo)-$\infty$-connections_ on the underlying $\mathbf{B}G$-cocycle. If their curvature is represented by a cocycle in $\mathbf{\flat}_{dR} \mathbf{B}^{n+1}\mathbb{R}$ which is given by a globally defined form, then these are _genuine_ $\infty$-connections. In either case, they serve as an intermediate step in computing the curvature characteristics.

### $\infty$-Lie algebra valued connections {#InfinityLieAlgebraConnection}

> The content of this section is at [[connection on an infinity-bundle]].

### Curvature characteristics {#InfChernWeil}

+-- {: .un_def }
###### Definition
**(Chern-Weil curvature characteristics)**

Let $\langle -\rangle : inn(\mathfrak{g}) \to b^{p} \mathbb{R}$ be an [[nLab:invariant polynomial]] on the [[nLab:∞-Lie algebra|Lie n-algebra]] $\mathfrak{g}$. Postcomposition with the corresponding diagram of dg-algebras

$$
  \array{
    CE(\mathfrak{g}) &\leftarrow& 0
    \\
    \uparrow && \uparrow
    \\
    W(\mathfrak{g}) &\stackrel{\langle -\rangle}{\leftarrow}&
    CE(b^p \mathbb{R})
  }
$$

induces a morphism of [[nLab:simplicial presheaves]]

$$
   \langle F_{(-)} \rangle
   :
  \mathbf{B}G_{diff} \to
  \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp}
$$

into the $(n+1)$-[[nLab:simplicial skeleton|coskeleton]] of the model for the de Rham coefficient object $\mathbf{\flat}_{dR}\mathbf{B}^{p+1}\mathbb{R}$ discussed [above](#U1FromLieIntegration).

For $\nabla : \hat X \to \mathbf{B}G_{diff}$ a connection, we call the
induced intrinsic de Rham cocycle

$$
  \langle F_\nabla \rangle :
  \hat X \stackrel{\nabla}{\to}
  \mathbf{B}G_{diff}
  \stackrel{\langle -\rangle}{\to}
  \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1}\mathbb{R}_{simp}
$$

the Chern-Weil [[nLab:curvature characteristic form]] of $\nabla$ with respect to $\langle -\rangle$.

=--

+-- {: .un_lemma }
###### Lemma

For $\nabla : \hat X \to \mathbf{B}G_{conn} \hookrightarrow \mathbf{B}G_{diff}$ a genuine connection, the induced curvature characteristic forms are globally defined closed forms, in that their cocycle factors through the sheaf $\Omega^{p+1}_{cl}(-)$ of closed $(p+1)$-forms:

$$
  \array{
     && \mathbf{B}G_{conn} &\stackrel{\langle F_{(-)}\rangle}{\to}& \Omega^{p+1}_{cl}(-)
     \\
    & \nearrow & \downarrow && \downarrow
    \\
    \hat X &\stackrel{}{\to}&
    \mathbf{B}G_{diff}
    &\stackrel{\langle F_{(-)}\rangle}{\to}&
    \mathbf{cosk}_{n+1} \mathbf{\flat}_{dR} \mathbf{B}^{p+1} \mathbb{R}_{simp}
  }
  \,.
$$

=--

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

for given $(U,[k])$ notice that $\langle F_{\nabla}\rangle(U,[k]) \in \Omega^\bullet(U\times \Delta^k)$ is closed and for $\nabla$ a genuine connection has no leg along $\Delta^k$: for $\partial_t$ a vector field along $\Delta^k$ we have $\iota_{\partial_t} \langle F_A\rangle = 0$. Therefore the [[nLab:Lie derivative]] along a vector $\partial_t$ along the simplex vanishes:

$$
  \mathcal{L}_t \langle F_A\rangle = d \iota_t \langle F_A\rangle +
  \iota_t d \langle F_A\rangle = 0
  \,.
$$

=--

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

As for the groupal case [above](#AbGerbesConnection), we hence find that the genuine $\infty$-connections are selected among all pseudo-connections as those whose curvature characteristic has a 0-[[nLab:truncated]] cocycle representative.

=--

So a genuine $\infty$-Lie algebra valued connection is a cocycle with values in the $(n+1)$-coskeleton of the simplicial presheaf of diagrams, which over $U,[k]$ assigns the set of diagrams

$$
  \array{
    C^\infty(U \times \Delta k)_{vert}
    &\leftarrow&
    CE(\mathfrak{g})
    &&&
    cocycle\;for\;underlying\;G-principal\;\infty-bundle
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U)\otimes \Omega^\bullet(X)
    &\leftarrow&
    CE(inn(\mathfrak{g})) = W(\mathfrak{g})
    &&&
    connection\;and\;curvature
    \\
    \uparrow && \uparrow
    \\
    \Omega^\bullet(U) &\leftarrow& inv(\mathfrak{g})
    &&&
    curvature\;characteristic\;forms
  }
  \,,
$$

(with $\Omega^\bullet(U \times \Delta^k)_{vert}$ the dg-algebra of [[vertical differential form]]s on the bundle $U \times \Delta^k \to U$), where the top morphism encodes the cocycle for the underlying $G = \tau_n\exp(\mathfrak{g})$-[[nLab:principal ∞-bundle]], where the middle morphism encodes the connection data and the bottom morphism the [[nLab:curvature characteristic form]]s.

Such $\infty$-Lie algebra valued connections were introduced in SSSI and further studied in SSSIII.

### Higher order Chern-Simons forms

See at [[Chern-Simons form]] the section In ∞-Chern-Weil theory.

### Chern character {#ChernCharacter}

Above we have considered [∞-Lie algebra valued connections](#InfinityLieAlgebraConnection) and their [curvature characteristic forms](#InfChernWeil). We now wish to show how these model the intrinsic [[schreiber:Chern character in an (∞,1)-topos]].

$$
  ch_{\mathbf{B}G} : \mathbf{B}G \to \mathbf{\Pi}(\mathbf{B}G) \to \mathbf{\Pi}(\mathbf{B}G)\otimes R
  \,.
$$

Since our ambient [[(∞,1)-topos]] is assumed to be [[locally ∞-connected (∞,1)-topos|locally ∞-connected]] we have in addition to the notion of [[Postnikov tower in an (∞,1)-category]] the notion of [[nLab:Whitehead tower in an (∞,1)-topos]]. Both notions are dual to each other: for $A \in \mathbf{H}$ any object and

$$
  \mathbf{\Pi}(A)
  \to
  \cdots
  \to
  \tau_{\leq 2}\mathbf{\Pi}(A)
  \to
  \tau_{\leq 1}\mathbf{\Pi}(A)
  \to \tau_{\leq 0}\mathbf{\Pi}(A)
  = *
$$

the [[nLab:Postnikov tower in an (∞,1)-category|intrinsic Postnikov tower]] of its [[path ∞-groupoid]], the [[nLab:pasting]] composite of [[nLab:(∞,1)-pullback]]s

$$
  \array{
    \vdots && && && \vdots
    \\
    \downarrow && && && \downarrow
    \\
    A_2 && &\to& \cdots &\to& \mathbf{B}\mathbf{\pi}_3(A) &\to& *
    \\
    \downarrow && && && && \downarrow
    \\
    A_1 && &\to& \cdots && && \mathbf{B}\mathbf{\pi}_2(A) &\to& *
    \\
    \downarrow
    && && && && \downarrow  && \downarrow
    \\
    A
    &\to&
    \mathbf{\Pi}A
    &\to&
    \cdots
    &\to&
    \tau_{\leq 3} \mathbf{\Pi}A
    &\to&
    \tau_{\leq 2} \mathbf{\Pi}A
    &\to&
    \tau_{\leq 1} \mathbf{\Pi}A
  }
  \,,
$$

defines the [[nLab:Whitehead tower in an (∞,1)-topos|Whitehead tower]]

$$
  * \to \cdots \to A_3 \to A_2 \to A_1 \to A_0 = A
$$

of $A$.

Since our $\mathbf{H}$ is assumed to be even [[nLab:∞-connected (∞,1)-topos|∞-connected]], the Postnikov tower of $\mathbf{\Pi}(A)$ is the image under $LConst : \infty Grpd \to \mathbf{H}$ of the ordinary [[nLab:Postnikov tower]] of $\Pi(A)$ in $\infty Grpd$. Accordingly, we have $\mathbf{B} \mathbf{\pi}_n(A) = LConst B^n \pi_n \Pi(A)$

The point now is that in $\mathbf{H} = $ [[nLab:?LieGrpd]] we may form smooh refinements of these discrete extensions: every discrete $(n+1)$-group $\mathbf{B}^{n+1}\mathbb{Z}$ we want to refine to a smooth $n$-group $\mathbf{B}^n U(1)$. By the discussion at geometric realization, both have equivalent underlying $\infty$-groupoids

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

For every direct summand abelian group $\mathbb{Z}$ in one of the $\mathbf{\pi}_n(A)$ we can ask for a refinement of the cocycle from coefficients $\mathbf{B}^n \mathbb{Z}$ to $\mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z}$. This does not change the geometric realization, up to equivalence, but does change the smooth structure. And it allows to refine to differential coefficients by postcomposing further with
$curv : \mathbf{B}^{n-1}\mathbb{R}/\mathbb{Z} \to
 \mathbf{\flat}_{dR}\mathbf{B}^{n}\mathbb{R}$.

For instance for $A = \mathbf{B}Spin \in \mathbf{H} = \infty Lie Grpd$ the [[nLab:delooping]] of the [[nLab:spin group]], we refine the internal Whitehead tower to

$$
  \array{
    \vdots
    \\
    \mathbf{B}Fivebrane &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& *
    \\
    \downarrow && \downarrow &&  \downarrow
    \\
    \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1)
  }
  \,,
$$

where the deloopings of the [[nLab:string 2-group]] and the [[nLab:fivebrane 6-group]] appear.

The result of such a smooth refinement is that we may apply the intrinsic curvature classes and the intrinsic de Rham theorem to obtain cocycles in realified cohomology, for instance

$$
  \mathbf{B}Spin
  \to
  \mathbf{B}^3 U(1)
  \stackrel{curv}{\to}
  \mathbf{\flat}_{dR} \mathbf{B}^4 U(1)
  =
  \mathbf{\flat}_{dR} \mathbf{B}^4 \mathbb{R}
  \,.
$$

If we have a $Spin$-principal bundle $X \to \mathbf{B}G$, we may form over it the covering circle $n$-group bundles on which these higher cocycles naturally live

$$
  \array{
    P_2 &\to& \mathbf{B}Fivebrane &\to& *
    \\
    \downarrow && \downarrow && \downarrow
    \\
    P_1 &\to& \mathbf{B}String &\to& \mathbf{B}^7 U(1) &\to& *
    \\
    \downarrow && \downarrow &&  && \downarrow
    \\
    X &\to& \mathbf{B}Spin &\to& \cdots &\to& \mathbf{B}^3 U(1)
  }
  \,.
$$

Here for $X$ an ordinary space, $X = \tau_0 X$, the higher circle $n$-group principal bundes $P_k$ have the property that also $\tau_0 P_k = X$. Therefore the 0-truncation of the entire composite

$$
  P_1 \to \mathbf{B}^7 U(1) \to \mathbf{\flat}_{dR}\mathbf{B}^8 \mathbb{R}
$$

defines a closed 8-form on $X$. This is the curvature characeristic form given by the Chern-Weil homomorphism in this degree. Its refinement to Deligne cohomology in this construction lives naturally not on $X$, but on the covering $P_1$ of $X$.

(...)

## Examples {#Examples}

### Principal 1-bundles with connection

We spell out here how the general theory of [∞-Lie algebra valued connection](#InfinityLieAlgebraConnection) reduces to the standard notion of [[nLab:connection on a bundle|connections]] on ordinary $G$-[[nLab:principal bundle]]s and how the [∞-Chern-Weil homomorphism](#InfChernWeil) reduces on these to the ordinary [[nLab:Chern-Weil homomorphism]].

Let $\mathfrak{g}$ be a (finite dimensional) [[nLab:Lie algebra]]. Then

$$
  \mathbf{cosk}_2 \exp(\mathfrak{g}) \simeq \mathbf{B}G
$$

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

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

The coefficient object $\mathbf{B}G_{conn}$ of [genuine ∞-Lie algebra connections](#InfinityLieAlgebraConnection) for $\mathfrak{g}$ an ordinary [[nLab:Lie algebra]] is weakly equivalent to the simplicial presheaf

$$
  \mathbf{B}G_{conn}
  \stackrel{\simeq}{\to}
  \Xi[G\times \Omega^1(-,\mathfrak{g}) \stackrel{
   \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1(-,\mathfrak{g})]
$$

that assigns objectwise the [[nLab:groupoid of Lie-algebra valued 1-forms]].

This is moreover isomorphic to the simplicial presheaf

$$
  \cdots =
  [CartSp^{op},sSet](\mathbf{P}_1(-),\mathbf{B}G])
$$

of morphisms out of the [[nLab:path groupoid]].

The flat coefficient object $\mathbf{\flat}\mathbf{B}G$ is modeled by the subobject

$$
  \Xi[G\times \Omega^1_{flat}(-,\mathfrak{g}) \stackrel{
   \overset{Ad_{p_1}(p_2)+ p_1 d p_1^{-1}}{\to}}{\underset{p_2}{\to}} \Omega^1_{flat}(-,\mathfrak{g})]
$$

of groupoids of Lie-algebra valued forms with vanishing [[nLab:curvature]] 2-form.

This is isomorphic to

$$
  \cdots =
  [CartSp^{op},sSet](\mathbf{\Pi}_1(-),\mathbf{B}G])
$$

of morphism out of the [[nLab:fundamental groupoid]].

=--

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

The statements about morphisms out of the path groupoid are discussed in detail in [SchrWalI](http://arxiv.org/abs/0705.0452).

=--

+-- {: .un_cor }
###### Corollary

For $X$ a [[nLab:paracompact space|paracompact]] [[nLab:smooth manifold]] and $\{U_i \to X\}$ a [[nLab:good open cover]] we have a natural equivalence of groupoids

$$
  [CartSp^{op}, sSet](C(\{U_i\}), \mathbf{B}G_{conn})
  \simeq
  G Bund_\nabla(X)
$$

with the groupoid of smooth $G$-principal bundles with connection on $G$.

For $\langle - \rangle : inn(\mathfrak{g}) \to b^p \mathbb{R}$ an [[nLab:invariant polynomial]] on $\mathfrak{g}$, the [induced morphism](#InfChernWeil)

$$
  [CartSp^{op}, sSet](C(\U_i\}), \mathbf{B}G_{diff})
  \stackrel{\langle F_{(-)}\rangle}{\to}
  \Omega^{p+1}_{cl}(X)
$$

is that of the ordinary [[nLab:Chern-Weil homomorphism]].

=--

We have seen that a refinement of the Chern-Weil homomorphism is available. The above morphism extends to a morphism

$$
  \langle F_{(-)}\rangle :  \mathbf{H}(-, \mathbf{B}G)
  \to
  \mathbf{H}(-,
    \tau_{1} \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R}
  )
$$

in $\mathbf{H} = \infty LieGrpd$ represented by

$$
 \array{
  [CartSp^{op},sSet](-, \mathbf{B}G_{diff})
  &\to&
  [CartSp^{op},sSet](-, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R})
  \\
  \downarrow^{\simeq}
  \\
  [CartSp^{op},sSet](-, \mathbf{B}G)
  }
  \,.
$$

For $X$ a smooth manifold with good cover $\{U_i \to X\}$ we have that
$ [CartSp^{op},sSet](C\{U_i\}, \mathbf{cosk}_2 \mathbf{\flat}_{dR}\mathbf{B}^{p+1} \mathbb{R})$ is the groupoid whose objects are closed $p+1$-forms on $X$ and whose morphisms are given by $p$-forms modulo exact forms.

Let $i \in I$ range over a set of generators for all [[nLab:invariant polynomial]]s. Then

$$
  \mathbf{H}(-,\mathbf{B}G)
  \to
  \prod_i \tau_1\mathbf{\flat}_{dR}\mathbf{B}^{n_i} \mathbb{R}
$$

is an approximation to the intrinsic Chern-character. We may consider its [[nLab:homotopy fiber]]s over a given set $Q_i$ of curvature characteristic forms.

Assume $\nabla, \nabla' : C(\{U_i\}) \to \mathbf{B}G_{diff}$ are two genuine connections with coinciding curvature characteristic classes $\{Q_i\}$. Then in the homotopy fiber they are coboundant cocycles precisely if all the [[nLab:Chern-Simons form]]s $CS_i(\nabla,\nabla')$ vanish modulo an exact form.

This equivalence relation is that which defines [[nLab:Simons-Sullivan structured bundle]]s. Their [[nLab:Grothendieck group]] completion yields [[nLab:differential K-theory]].

### Principal 2-bundles with connection

(...)

Let $\mathfrak{g}$ be a Lie [[nLab:strict 2-group]] coming from a [[nLab:differential crossed module]] $(\mathfrak{g}_2 \to \mathfrak{g}_1)$. Then we have two candidate Lie 2-groups integrating this: on the one hand the [[nLab:strict 2-group]] coming from the [[nLab:crossed module]] $(G_2 \to G_1)$ that integrates $(\mathfrak{g}_2 \to\mathfrak{g}_1)$ degreewise as ordinary Lie algebras, and on the other hand $cosk_k\exp(\mathfrak{g})$.

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

The morphism

$$
  \tau_2 \exp(\mathfrak{g}) \to
  \mathbf{B}(G_2 \to G_1)
$$

given by evaluating 2-dimensional [[nLab:parallel transport]] is a weak equivalence.
=--

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

Use the 3-dimensional nonabelian Stokes theorem from the appendix of [SchrWalII](http://arxiv.org/abs/0802.0663).

=--

+-- {: .un_cor }
###### Corollary

The object $\mathbf{B}(G_2 \to G_1)_{conn}$ assigns to $U \in CartSp$ the [[nLab:2-groupoid of Lie 2-algebra valued forms]] over $U$.

=--

This is described in detail in [SchrWalII](http://arxiv.org/abs/0802.0663), subject to the extra constraint that the 2-form curvature vanishes.

+-- {: .un_cor }
###### Corollary

A genuine connection on a $(G_2 \to G_1)$-[[nLab:principal 2-bundle]] with given cocycle $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)$ is a cocycle $X \stackrel{\simeq}{\leftarrow} C(\{U_i\}) \to \mathbf{B}(G_2 \to G_1)_{conn}$ given as follows:

1. on $U_i$ a pair of forms $A_i \in \Omega^1(U_i, \mathfrak{g}_1)$, $B_i \in \Omega^2(U_i, \mathfrak{g}_2)$;

1 on $U_i \cap U_j$ a function $g_{i j} \in C^\infty(U_{i}\cap U_j , G_1)$ and a 1-form $a_{i j} \in \Omega^1(U_i \cap U_j, \mathfrak{g}_2)$ such that ...

1. and so forth

=--

This is described in detail in [SchrWalIII](), subject to the extra constraint that the 2-form curvature vanishes.

(...)

### Twisted differential $String-$ and $Fivebrane$-structures {#DiffStringStruc}

We discuss now in detail refined Chern-Weil morphisms

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

that send [∞-connections](#InfinityLieAlgebraConnection) on $G$-[[principal ∞-bundle]]s to [[circle n-bundles with connection]] that represent a given [[characteristic class]]. $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ with coefficients in the [circle n-groupoid](#http://ncatlab.org/nlab/show/Lie+infinity-groupoid#BnU1).

Specifically, we consider the first two steps in the smooth refinement of the Whitehead tower of the [[orthogonal group]] $O$ that are controled by [[∞-Lie algebra cohomology]].

The smooth Whitehead tower of $O$ in [[?LieGrpd]] starts as

$$
  \cdots \to \mathbf{B}Spin \to \mathbf{B} SO \to \mathbf{B}O
  \,,
$$

where

* the [[delooping]] $\mathbf{B} SO$ of the [[special orthogonal group]] is the $\mathbb{Z}_2$-[[principal bundle]] over $\mathbf{B}O$ classified by the [[cocycle]] $\mathbf{B}O \to \mathbf{B} \mathbb{Z}_2$ that sends an elemen $k \in O$ to $+1$ if it is in the connected component of the identity and to $-1$ if it is not. This means we have an [[(∞,1)-pullback]] diagram

  $$
    \array{
      \mathbf{B} SO &\to& *
      \\
      \downarrow && \downarrow
      \\
      \mathbf{B} O &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B} \mathbb{Z}_2
    }
    \;
  $$

* the [[delooping]] $\mathbf{B} Spin$ of the [[spin group]] is the is the $\mathbf{B}\mathbb{Z}_2$-[[principal 2-bundle]] over $\mathbf{B} SO$ classified by the [[Stiefel-Whitney class]] $\mathbf{B} SO \to \mathbf{B}^2 \mathbb{Z}$

  $$
    \array{
      \mathbf{B} Spin &\to& *
      \\
      \downarrow && \downarrow
      \\
      \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2
    }
    \;
  $$

Since these two steps are controled by the [[torsion]]-group $\mathbb{Z}_2$ they have no nontrivial refinement to differential cohomology. The next step however is controled by what in the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]] is the first [[fractional Pontryagin class]] $\frac{1}{2}p_1 : \mathcal{B} Spin \to \mathcal{B}^4 \mathbb{Z}$ and which lifts through the [[schreiber:path ∞-groupoid]] functor $\Pi : \infty LieGrpd \to \infty Grpd$ to a characteristic class in $\mathbf{H} = $ [[?LieGrpd]] (as discussed there) $\frac{1}{2} p_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1)$ with coefficients in the smooth circle 3-groupoid. This cocycle does arise as the [[Lie integration]] $\exp(\mu)$ of the canonical [[Lie algebra cohomology|Lie algebra 3-cocycle]] $\mu = \langle -,[-,-]\rangle: \mathfrak{so} \to b^2 \mathbb{R}$.

The [[principal ∞-bundle|principal 3-bundle]] that this classifies is the [[delooping]] $\mathbf{B} String$ of the [[string 2-group]] $String$

$$
  \array{
    \mathbf{B} String &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B} Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)
  }
  \,.
$$

Notice that the fact that this is an [[(∞,1)-pullback]] implies that for any $X\in \mathbf{H} = \infty LieGrpd$ also

$$
  \array{
    \mathbf{H}(X,\mathbf{B} String) &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}(X,\mathbf{B} Spin) &\to& \mathbf{H}(X,\mathbf{B}^3 U(1))
  }
  \,,
$$

which exhibits the [[2-groupoid]] $\mathbf{H}(X,\mathbf{B}String)$ of  [[string structure]]s.

As we refine in this diagram the bottom morphism to differential cohomology, we obtain correspondingly differential string structures.

#### The string-lifting Chern--Simons $3$-bundle with connection {#StringStructure}

We describe the special case of the general [$\infty$-Chern--Weil homomorphism](#InfChernWeil) for [$\infty$-Lie algebra valued connections](#InfinityLieAlgebraConnection) corresponding to the [[characteristic class]] $\frac{1}{2}p_1\colon \mathbf{B}Spin \to \mathbf{B}^3 U(1)$: the first fractional [[Pontryagin class]] of the [[spin group]] $\mathbf{B}Spin$. The $\mathbf{B}^3 U(1)$-differential cocycle that it produces from a given $Spin$-[[principal bundle]]
is the [[Chern-Simons 2-gerbe|Chern?Simons 2-bundle]] with connection whose class is the obstruction for the existence of a [[string structure]].

The content of this subsection is at [[Chern-Simons 2-gerbe|Chern?Simons 2-gerbe]] in [the section on $\infty$-Chern--Weil theory](Chern-Simons+2-gerbe#InInfChernWeil).

#### Differential string structures

The content of this section is at [[differential string structure]].

#### The Fivebrane-lifting Chern-Simons 7-bundle with connection {#FivebraneStructure}

The content of this section is at [[Chern-Simons circle 7-bundle]].

#### Differential fivebrane structures {#DiffFivebraneStrucs}

Let

$$
  \frac{1}{6}\hat p_2 :
  \mathbf{H}_{conn}(-,\mathbf{B}String)
  \to
  \mathbf{H}_{diff}(-, \mathbf{B}^7 U(1))
$$

be the differential refinement of the second fractional Pontryagin class discussed [above](#FivebraneStructure).

**Definition**

For $X \in \mathbf{H} =$ [[nLab:?LieGrpd]], the $\infty$-groupoid  of **differential fivebrane-structures** $Fivebrane_{diff}(X)$ is the [[nLab:homotopy fiber]] of    $\frac{1}{6}p_2(X) : \mathbf{H}(X,\mathbf{B}String) \to \mathbf{H}_{diff}(X, \mathbf{B}^7 U(1))$.

More generally, the $\infty$-groupoid of **twisted differential fivebrane structures** is the [[nLab:(∞,1)-pullback]] $Fivebrane_{diff,tw}(X)$ in

$$
  \array{
    Fivebrane_{diff,tw}(X) &\to& H_{diff}(X,\mathbf{B}^7 U(1))
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}String)
    &\stackrel{\frac{1}{6}\hat p_2}{\to}&
    \mathbf{H}_{diff}(X,\mathbf{B}^7 U(1))
  }
  \,.
$$

In terms of the underlying $\infty$-Lie algebra valued local connection data, i.e. before Lie integration in the above sense , this has been considered in SSSIII

(...)

### $\infty$-Chern-Simons theory

The refined higher Chern-Weil homomorphism takes values in
[[circle n-bundles with connection]] in [[ordinary differential cohomology]]. Each of these comes with a notion of higher [[holonomy]] over $n$-dimensional curves $\Sigma_n \to X$. The map that takes a [[connection on an infinity-bundle]] to this holonomy is a generalization of the [[action functional]] of [[Chern-Simons theory]].

Therefore the higher Chern-Weil homomorphism defines a class of [[sigma-model]] [[quantum field theories]] that we call

[[schreiber:infinity-Chern-Simons theory]].

See there for more details.

Special noteworthy cases are

* the class of [[AKSZ sigma-models]] ([FRS11](#FRS))

* higher [[Chern-Simons supergravity]]

  (see [[D'Auria-Fre formulation of supergravity]])

## Related entries

* [[schreiber:differential cohomology in a cohesive topos]]

  * **∞-Chern-Weil theory**

    * [[schreiber:Principal ∞-bundles -- models and general theory]]

    * [[schreiber:Cech cocycles for differential characteristic classes]]

    * [[schreiber:Twisted differential structures]]

      * [[schreiber:Twisted Differential String and Fivebrane Structures]]

  * [[schreiber:∞-Chern-Simons theory]]

    * [[schreiber:Higher Chern-Weil Derivation of AKSZ Sigma-Models]]

## References

An explicit presentation of the $\infty$-Chern-Weil homomorphism in terms of [[simplicial presheaves]] and the application to [[differential string structure]]s and [[differential fivebrane structure]]s is considered in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech Cocycles for Differential characteristic Classes]]_

The special case that gives the [[AKSZ sigma-model]] is discussed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]_
 {#FRS}

A general abstract account is in

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

For a commented list of related literature see

differential cohomology in cohesive topos -- references

[[!redirects ∞-Chern-Weil theory]]

[[!redirects ∞-Chern-Weil homomorphism]]

[[!redirects Chern-Weil homomorphism in Smooth∞Grpd]]

[[!redirects infinity-Chern-Weil theory]]


Cheshire cat principle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[nuclear physics]] ([[quantum chromodynamics]]) the _Cheshire cat principle_ ([NNZ 85](#NNZ85)) is the observation that in the [[quark bag model]] for [[confinement|confined]] [[quarks]] inside [[hadrons]] the [[radius]] of the "bag" is not actually [[observable]]. Hence the "bag" of the the bag model disappears, leaving only its [[confinement|confined]] content behind, whence the colorful term.

From [Rho et al. 16](AdS-QCD#RhoEtAl16):

> One can make $[$[[chiral perturbation theory]]$]$ consistent with [[QCD]] by suitably matching the [[correlators]] of the [[effective field theory|effective theory]] to those of [[QCD]] at a [[scale]] near $\Lambda$. Clearly this procedure is not limited to only one set of [[vector mesons]]; in fact, one can readily generalize it to an infinite number of hidden [[gauge fields]] in an [[effective field theory|effective]] [[Lagrangian density|Lagrangian]]. In so doing, it turns out that a fifth dimension is "deconstructed" in a (4+1)-dimensional (or 5D) [[Yang-Mills theory|Yang–Mills]] type form. We will see in Part III that such a structure arises, [[top-down model building|top-down]], in [[string theory]].

> $[...]$

> $[$this [[holographic QCD]]$]$ model comes out to describe — unexpectedly well — low-energy properties of both [[mesons]] and [[baryons]], in particular those properties reliably described in quenched [[lattice QCD]] simulations.

> $[...]$

> One of the most noticeable results of this [[AdS/QCD|holographic model]] is the first derivation of vector dominance (VD) that holds both for mesons and for baryons. It has been
somewhat of an oddity and a puzzle that Sakurai’s [[vector meson dominance|vector dominance]] — with the lowest [[vector mesons]] [[rho meson|ρ]] and ω — which held very well for [[pion|pionic]] [[form factors]] at low momentum transfers famously failed for nucleon form factors. In this [[AdS/QCD|holographic model]], the [[vector meson dominance|VD]] comes out automatically for both the [[pion]] and the [[nucleon]] provided that the infinite $[$[[Kaluza-Klein mechanism|KK-]]$]$tower is included. While the [[vector meson dominance|VD]] for the [[pion]] with the infinite tower is not surprising given the successful Sakurai VD, that the [[vector meson dominance|VD]] holds also for the [[nucleons]] is highly nontrivial. $[...]$ It turns out to be a consequence of a [[AdS/QCD|holographic]] [[Cheshire cat principle|Cheshire Cat phenomenon]]

\linebreak

From [Nielsen-Zahed 09](Cheshire+cat+principle#NielsenZahed09):

> The [[Skyrmion]] is the ultimate [[soliton|topological]] [[quark bag model|bag model]] with zero size bag radius, lending further credence to the [[Cheshire cat principle]].

## Related concepts

* [[chiral perturbation theory]]

* [[quark bag model]]

* [[confinement]]

## References

Original article:

* {#NNZ85} S. Nadkarni, [[Holger Bech Nielsen]], [[Ismail Zahed]], _Bosonization Relations As Bag Boundary Conditions_, Nucl. Phys. B 253 (1985) 308 (doi:10.1016/0550-3213(85)90533-4)

Further developments:

* [[Mannque Rho]], _Cheshire Cat Hadrons_, Phys. Rept. 240 (1994) 1-142 ([arXiv:hep-ph/9310300](https://arxiv.org/abs/hep-ph/9310300), doi:10.1016/0370-1573(94)90002-7)

  updated as: _The Cheshire Cat Hadrons Revisited_ ([arXiv:hep-ph/0206003](https://arxiv.org/abs/hep-ph/0206003))

Review:

* [[Mannque Rho]], Lecture I in: _Lectures on Effective Field Theories for Nuclei, Nuclear Matter and Dense Matter_, 2002 ([arXiv:nucl-th/0202078](https://arxiv.org/abs/nucl-th/0202078), [cds:539674](https://cds.cern.ch/record/539674))

See also:

* [[Mannque Rho]], _Fractionalized Quasiparticles and the Hadron-Quark Duality in Dense Baryonic Matter_ ([arXiv:2004.09082](https://arxiv.org/abs/2004.09082))

Derivation from the [[AdS/QCD correspondence]]:

* [[Holger Bech Nielsen]], [[Ismail Zahed]], _The Cheshire Cat Principle from Holography_ ([doi:10.1142/9789814280709_0017](https://doi.org/10.1142/9789814280709_0017), [arXiv:0904.0429](https://arxiv.org/abs/0904.0429)), Chapter 17 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))


Chetan Nayak

* [webpage](https://www.microsoft.com/en-us/research/people/cnayak/)

## Selected writings

On [[2d CFT|2d]] [[logarithmic CFT]] for generalized [[Laughlin wavefunctions]]:

* {#GurarieFlohrNayak97} [[Victor Gurarie]], [[Michael Flohr]], [[Chetan Nayak]], *The Haldane-Rezayi Quantum Hall State and Conformal Field Theory*, Nucl. Phys. B **498** (1997) 513-538 $[$[arXiv:cond-mat/9701212](https://arxiv.org/abs/cond-mat/9701212), [doi:10.1016/S0550-3213%2897%2900351-9](https://doi.org/10.1016/S0550-3213%2897%2900351-9)$]$

On [[topological quantum computation]] with [[anyons]]:

* [[Chetan Nayak]], [[Steven H. Simon]], [[Ady Stern]], [[Michael Freedman]], [[Sankar Das Sarma]], _Non-Abelian Anyons and Topological Quantum Computation_, Rev. Mod. Phys. 80, 1083 (2008) ([arXiv:0707.1888] (http://arxiv.org/abs/0707.1889))

On [[anyons]] in [[topological superconductors]] ([[Majorana zero modes]]):

* [[Sankar Das Sarma]], [[Michael Freedman]], [[Chetan Nayak]], _Majorana Zero Modes and Topological Quantum Computation_, npj Quantum Information 1, 15001 (2015) ([nature:npjqi20151](https://www.nature.com/articles/npjqi20151))

category: people


Chevalley fundamental group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The term **Chevalley fundamental group** is sometimes used for the fundamental group of a space defined via the automorphisms of a universal cover, hence its _[[étale fundamental group]]_ (see there for more). In [[algebraic geometry]]/[[arithmetic geometry]] this is essentially the _[[absolute Galois group]]_

This terminology is used by Borceux and Janeldize in their book on [[Galois Theories]].  To quote from that source:

'' There are two classical definitions of the fundamental group of a topological space which give isomorphic groups for certain ''good'' spaces."

The Chevalley fundamental group $Aut(p)= Aut(E,p)$ is defined only for connected spaces $B$ which admit a universal covering map $p: E\to B$ with connected $E$, and, of course, depends on it, but again, different $p$ produce isomorphic groups.''

The term _[[algebraic fundamental group]]_ is also sometimes used for this, although more usually that term is reserved for Grothendieck's fundamental group of  a scheme.

## References

* [[Francis Borceux]], [[George Janelidze]], __Galois Theories__, Cambridge Studies in Advanced Mathematics __72__, Cambridge University Press, 2001.

* [[C. Chevalley]], _Theory of Lie groups_, Princeton University Press, 1946.

[[!redirects Chevalley fundamental groups]]


Chevalley group

__Chevalley groups__ are finite group analogues of complex semisimple Lie groups.

* Robert Steinberg, _Lectures on Chevalley groups_, Yale University Lecture Notes 1967. [pdf](http://www.ms.unimelb.edu.au/~ram/Resources/YaleNotes.pdf)

[[!redirects Chevalley group]]


Chevalley's theorem on constructible sets

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

An image of a variety under a regular map is not necessarily a variety. A standard example is to take the map $\mathbf{A}^2\to\mathbf{A}^2$ which takes $(x,y)\mapsto (x,x y)$; the image of the affine plane consists of all points $(x,y)$ with $x\neq 0$ union $(0,0)$. This set is clearly not locally closed, hence not a quasiaffine subvariety of $\mathbf{A}^2$.

However, the image of a variety under a regular map is always a [[constructible set]] (finite union of [[locally closed set]]s). More generally,

THEOREM. (Chevalley) If $f: X\to Y$ is a regular morphism of varieties and $S\subset X$ is a Zariski constructible set. Then $f(S)$ is also Zariski constructible.

More generally,

Theorem ([[EGA]] IV 1.8.4.) If $f:X\to Y$ is a finitely presented morphism of schemes. Then the image of any constructible subset of $X$ under $f$ is a constructible subset of $Y$.

## Related entries

* [[tame topology]]

## Links

* MathOverflow [chevalleys-theorem-on-constructible-sets](http://mathoverflow.net/questions/80707/chevalleys-theorem-on-constructible-sets)

The relation to the [[elimination of quantifiers]] for the theory of algebraically closed fields is discussed in

* D. Sustretov, blog: [model theory IV: algebraically closed fields and compact complex manifolds](http://shenme.de/blog/posts/2012-08-28-model-theory-algebraically-closed-fields-and-compact-complex-manifelds.html)

## Reference

* [[André Joyal|A. Joyal]], _Les Théorèmes de Chevalley-Tarski et Remarques sur l'Algèbre Constructive_ , Cah. Top. Géom. Diff. Cat. **XVI** (1975) pp.256-258. ([Proceedings Colloque Amiens 1975](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1975__16_3/CTGDC_1975__16_3_217_0/CTGDC_1975__16_3_217_0.pdf), 6.81 MB)

category: algebraic geometry

[[!redirects Chevalley-Tarski theorem]]


Chevalley-Eilenberg algebra

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

# Contents
* tic
{:toc}

## Idea

The _Chevalley-Eilenberg algebra_ $CE(\mathfrak{g})$ of a [[Lie algebra]] is a [[differential graded algebra]] of elements dual to $\mathfrak{g}$ whose [[differential]] encodes the Lie bracket on $\mathfrak{g}$.

The [[cochain cohomology]] of the underlying [[cochain complex]] is the [[Lie algebra cohomology]] of $\mathfrak{g}$.

This generalizes to a notion of Chevalley-Eilenberg algebra for $\mathfrak{g}$ an [[L-∞-algebra]], a [[Lie algebroid]] and generally an [[∞-Lie algebroid]].

## Grading conventions {#GradingConvention}

This differential-graded subject is somewhat notorious for a plethora of equivalent but different conventions on gradings and signs.

For the following we adopt the convention that for $V$ an $\mathbb{N}$-[[graded vector space]] we write

$$
  \begin{aligned}
    \wedge^\bullet V &:= Sym(V[1])
     \\
      & = k \oplus (V_0) \oplus (V_1 \oplus V_0 \wedge V_0) \oplus
     (V_2 \oplus V_1 \wedge V_0 \oplus V_0 \wedge V_0 \wedge V_0)
     \oplus \cdots
  \end{aligned}
$$

for the [[free construction|free]] graded-commutative algebra on the graded vector space obtained by shifting $V$ _up_ in degree by one.

Here the elements in the $n$th term in parenthesis are in degree $n$.

A plain vector space, such as the dual $\mathfrak{g}^*$ of the vector space underlying a Lie algebra, is regarded as a $\mathbb{N}$-graded vector space in degree 0. The quantity $\wedge^\bullet \mathfrak{g}^*$ is the ordinary [[Grassmann algebra]] over $\mathfrak{g}^*$, where elements of $\mathfrak{g}^*$ are generators of degree 1.

## Of Lie algebras {#OfLieAlgebra}

### Definition {#DefForLieAlg}

The _Chevalley-Eilenberg algebra_ $CE(\mathfrak{g})$ of a finite dimensional [[Lie algebra]] $\mathfrak{g}$ is the [[semifree dga|semifree]] graded-commutative [[dg-algebra]] whose underlying graded algebra is the [[Grassmann algebra]]

$$
  \wedge^\bullet \mathfrak{g}^*
  =
  k \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^* ) \oplus \cdots
$$

(with the $n$th skew-symmetrized power in degree $n$),
and whose [[differential]] $d$ (of degree +1) is defined on $\mathfrak{g}^*$ as the dual of the Lie bracket

$$
  d|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*,
$$

and extended uniquely as a graded [[derivation]] on $\wedge^\bullet \mathfrak{g}^*$.

That this differential indeed squares to 0, $d \circ d = 0$, is precisely the fact that the Lie bracket satisfies the [[Jacobi identity]].

If we choose a dual basis $\{t^a\}$ of $\mathfrak{g}^*$ and let $\{C^a{}_{b c}\}$ be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is

$$
  d t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
  \,,
$$

where here and in the following a sum over repeated indices is implicit.

This has a more or less evident generalization to infinite-dimensional Lie algebras.,

### Properties {#PropertiesForLieAlg}

One observes that for $\mathfrak{g}$ a vector space,  the graded-commutative dg-algebra structures on $\wedge^\bullet \mathfrak{g}^*$ are precisely in bijection with Lie algebra structures on $\mathfrak{g}$: the dual of the restriction of $d$ to $\mathfrak{g}^*$ defines a skew-linear bracket and the condition $d^2 = 0$ holds if and only if that bracket satisfies the Jacobi identity.

Moreover, morphisms if Lie algebras $\mathfrak{g} \to \mathfrak{h}$ are precisely in bijection with morphisms of dg-algebras $CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})$. And the $CE$-construction is functorial.

Therefore, if we write $dgAlg_{sf,1}$ for the category whose objects are [[semifree dga]]s on generators in degree 1, we find that the construction of CE-algebras from Lie algebras constitutes a canonical [[equivalence of categories]]

$$
  LieAlg \stackrel{CE(-)}{\underset{\simeq}{\to}} (dgAlg_{sf,1})^{op}
  \,,
$$

where on the right we have the [[opposite category]].

This says that in a sense the Chevalley-Eilenberg algebra is just another way of looking at (finite dimensional) Lie algebras.

There is an analogous statement not involving the dualization: Lie algebra structures on $\mathfrak{g}$ are also in bijection with the structure of a _[[differential graded coalgebra]]_ $(\vee^\bullet \mathfrak{g}, D)$ on the free graded-co-commutative coalgebra $\vee^\bullet \mathfrak{g}$ on $\mathfrak{g}$ with $D$ a [[derivation]] of degree -1 squaring to 0.

The relation between the differentials is simply dualization

$$
  (\vee^\bullet \mathfrak{g}, D) \leftrightarrow
  (\wedge^\bullet \mathfrak{g}^* , d )
$$

where for each $\omega \in \wedge^\bullet \mathfrak{g}^*$ we have

$$
  d \omega = \omega(D(-))
  \,.
$$

## Of $L_\infty$-algebras

The equivalence between Lie algebras and differential graded algebras/coalgebras discussed above  suggests a grand generalization by simply generalizing the [[Grassmann algebra]] over a [[vector space]] $\mathfrak{g}^*$ to the Grassmann algebra over a [[graded vector space]].

If $\mathfrak{g}$ is a graded vector space, then a differential $D$ of degree -1 squaring to 0 on $\vee^\bullet \mathfrak{g}$ is precisely the same as equipping $\mathfrak{g}$ with the structure of an [[L-∞ algebra]].

Dually, this corresponds to a general [[semifree dga]]

$$
  CE(\mathfrak{g}) := (\wedge^\bullet \mathfrak{g}^*, d = D^*)
  \,.
$$

This we may usefully think of as the Chevalley-Eilenberg algebra of the $L_\infty$-algebra $\mathfrak{g}$.

So _every_ commutative [[semifree dga]] (degreewise finite-dimensional) is the Chevaley-Eilenberg algebra of some [[L-∞ algebra]] of finite type.

This means that many constructions involving [[dg-algebra]]s are secretly about [[∞-Lie theory]]. For instance the [[Sullivan construction]] in [[rational homotopy theory]] may be interpreted in terms of [[Lie integration]] of $L_\infty$-algebras.

## Of Lie algebroids {#OfLieAlgebroids}

For $\mathfrak{a}$ a [[Lie algebroid]] given as

* a [[vector bundle]] $E\to X$

* with anchor map $\rho : E \to T X$

* and bracket $[-,-] \;\colon\; \Gamma(E)\wedge_{\mathbb{R}} \Gamma(E) \to \Gamma(E)$

the corresponding Chevalley-Eilenberg algebra is

$$
  CE(\mathfrak{a}) := \left(\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*, d\right)
  \,,
$$

where now the [[tensor product]]s and dualization is over the ring $C^\infty(X)$ of [[smooth function]]s on the base space $X$ (with values in the [[real number]]s). The differential $d$ is given by the formula

$$
  (d\omega)(e_0, \cdots, e_n)
  =
  \sum_{\sigma \in Shuff(1,n)} sgn(\sigma) \rho(e_{\sigma(0)})(\omega(e_{\sigma(1)}, \cdots, e_{\sigma(n)}))
  +
  \sum_{\sigma \in Shuff(2,n-1)} sign(\sigma)
   \omega([e_{\sigma(0)},e_{\sigma(1)}],e_{\sigma(2)}, \cdots, e_{\sigma(n)} )
  \,,
$$

for all $\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*$ and $(e_i \in \Gamma(E))$, where $Shuff(p,q)$ denotes the set of $(p,q)$-[[shuffle]]s $\sigma$ and $sgn(\sigma)$ the [[signature]] $\in \{\pm 1\}$ of the corresponding [[permutation]].

For $X = *$ the point we have that $\mathfrak{a}$ is a Lie algebra and this definition reproduces the above definition of the CE-algebra of a Lie algebra (possibly up to an irrelevant global sign).

## Of $\infty$-Lie algebroids

See [[∞-Lie algebroid]].

## Examples

### Of abelian Lie $n$-algebras

The CE-algebra of the Lie algebra of the [[circle group]] $\mathfrak{u}(1)$ is the graded-commutative dg-algebra on a single generator in degree 1 with vanishing differential.

More generally, the $L_\infty$-algebra $b^n \mathfrak{u}(1)$ is the one whose CE algebra is the commutative dg-algebra with a single generator in degree $n+1$ and vanishing differential.

### Of $\mathfrak{su}(2)$

The CE-algebra of $\mathfrak{su}(2)$ has three generators $x, y, z$ in degree one and differential

$$
  d x_1 = x_2 \wedge x_3
$$

and cyclically.

### Of the tangent Lie algebroid $T X$

For $X$ a [[smooth manifold]] and $T X$ its [[tangent Lie algebroid]], the corresponding CE-algebra is the [[de Rham algebra]] of $X$.

$$
  CE(T X) = (\wedge^\bullet_{C^\infty(X)} \Gamma(T^* X), d_{dR})
  \,.
$$

For $(v_i \in \Gamma(T X))$ [[vector field]]s and $\omega \in \Omega^n = \wedge^n_{C^\infty(X)} \Gamma(T X)^*$ a [[differential form]] of degree $n$,
the [formula for the CE-differential](#OfLieAlgebroids)

$$
  (d\omega)(v_0, \cdots, v_n)
  =
  \sum_{\sigma \in Sh(1,n)} sgn(\sigma) v_{\sigma(0)}(\omega(v_{\sigma(1)}, \cdots, v_{\sigma(n)}))
  +
  \sum_{\sigma \in Shuff(2,n-1)} sgn(\sigma)
   \omega([v_{\sigma(0)},v_{\sigma(1)}],v_{\sigma(2)}, \cdots, v_{\sigma(n)} )
  \,,
$$

is indeed that for the de Rham differential.

### Of the string Lie 2-algebra

For $\mathfrak{g}$ a semisimple Lie algebra with binary [[invariant polynomial]] $\langle -,-\rangle$ -- the [[Killing form]] -- , the CE-algebra of the [[string Lie 2-algebra]] is

$$
  CE(\mathfrak{string}) = (\wedge^\bullet( \mathfrak{g}^+ \oplus \mathbb{R}^*[1]), d_{string})
$$

where the differential restricted to $\mathfrak{g}^*$ is $[-,-]^*$ while on the new generator $b$ spanning $\mathbb{R}^*[1]$ it is

$$
  d b = \langle -, [-,-]\rangle \in \wedge^3 \mathfrak{g}^*
  \,.
$$

### Weil algebra

For $\mathfrak{g}$ a [[Lie algebra]], the CE-algebra of the [[Lie 2-algebra]] given by the [[differential crossed module]] $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ is the [[Weil algebra]] $W(\mathfrak{g})$ of $\mathfrak{g}$

$$
  CE(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})
  =
  W(\mathfrak{g})
  \,.
$$

### Lie algebra cohomology

[[Lie algebra cohomology]] of a $k$-[[Lie algebra]] $\mathfrak{g}$ with coefficients in the left $\mathfrak{g}$-module $M$ is defined as $H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)$. It can be computed as $Hom_{\mathfrak{g}}(V(\mathfrak{g}),M)$ (a similar story is for [[Lie algebra homology]]) where $V(\mathfrak{g})=U(\mathfrak{g})\otimes\Lambda^*(\mathfrak{g})$ is the [[Chevalley-Eilenberg chain complex]]. If $\mathfrak{g}$ is finite-dimensional over a field then $Hom_{\mathfrak{g}}(V(\mathfrak{g}),k) =  CE(\mathfrak{g}) = \Lambda^* \mathfrak{g}^*$ is the underlying complex of the Chevalley-Eilenberg algebra, i.e. the [[Chevalley-Eilenberg cochain complex]] with trivial coefficients.

A [[cocycle]] in degree n of the [[Lie algebra cohomology]] of a [[Lie algebra]] $\mathfrak{g}$ with values in the trivial module $\mathbb{R}$ is a morphism of [[L-∞ algebra]]s

$$
  \mathfrak{g} \to b^{n-1} \mathfrak{u}(1)
  \,.
$$

In terms of CE-algebras this is a [[dg-algebra]] morphism

$$
  CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathfrak{u}(1))
  \,.
$$

Since by the above example the dg-algebra on he right has a single generator in degree $n$ and vanishing differential, such a morphism is precisely the same thing as a degree $n$-element in $CE(\mathfrak{g})$, i.e. an element  $\omega \in \wedge^n \mathfrak{g}^*$ which is closed under the CE-differential

$$
  d_{CE} \omega = 0
  \,.
$$

This is what one often sees as the definition of a cocycle in [[Lie algebra cohomology]]. However, from the general point of view of [[cohomology]], it is better to think of the cocycle equivalently as the morphism $\mathfrak{g} \to b^{n-1}\mathfrak{u}(1)$.

### BRST complex

In [[physics]], the Chevalley-Eilenberg algebra $CE(\mathfrak{g}, N)$ of the [[action]] of a  [[Lie algebra]] or [[L-∞ algebra]] of a [[gauge group]] $G$ on space $N$ of fields is called the [[BRST complex]].

In this context

* the generators in $N$ in degree 0 are called **fields**;

* the generators $\in \mathfrak{g}^*$ in degree $1$ are called **ghosts**;

* the generators in degree $2$ are called **ghosts of ghosts**;

* etc.

If $N$ is itself a [[chain complex]], then this is called a [[BV-BRST formalism|BV-BRST complex]]

## Related concepts

* [[∞-Lie algebra cohomology]]

* [[super L-∞ algebra]]

* **Chevalley-Eilenberg algebra**

  * [[differential graded-commutative algebra]]

  * [[differential graded-commutative superalgebra]]

* [[Weil algebra]]

* [[invariant polynomial]]

## References

An elementary introduction for CE-algebras of Lie algebras is at the beginning of

* [[José de Azcárraga]], J. M. Izquierdo, J. C. Perez Bueno, _An introduction to some novel applications of Lie algebra cohomology and physics_ ([arXiv:physics/9803046](http://arxiv.org/abs/physics/9803046))

More details are in  of

* [[José de Azcárraga]], José M. Izquierdo, section 6.7 of _[[Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics]]_ , Cambridge monographs of mathematical physics, (1995)

See also almost any text on [[Lie algebra cohomology]] (see the list of references there).

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Chevalley-Eilenberg algebra in synthetic differential geometry

<div class="rightHandSide toc">
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</div>

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

+-- {: .standout}

[[Urs Schreiber]]. This is something I thought about in the context of my discussion at [[schreiber:Chevalley-Eilenberg algebra]].

=--

#Idea#

We give a description of the [[Chevalley-Eilenberg algebra]] of the [[Lie algebra]] of a [[Lie group]] as the [[∞-quantity]] of functions $C^\infty(\mathbf{B}G_e^{(1)})$ on the [[simplicial object|simplicial space]] of infinitesimal neighbourhoods of the identity in the sense of [[synthetic differential geometry]] in the simplicial smooth space $\mathbf{B}G$ that is the [[Lie ∞-groupoid]] incarnation of the [[delooping]] of the [[Lie group]].

The derivation is analogous to and usefully compared with how the deRham algebra of differential forms on a [[manifold]] $X$ is the [[∞-quantity]] of functions on the [[infinitesimal singular simplicial complex]] $X^{\Delta^\bullet_{diff}} \hookrightarrow \Pi(X)$ of $X$, as described at [[differential forms in synthetic differential geometry]].

We proceed entirely by using theorems and propositions from the book

* [[Anders Kock]], _Synthetic Geometry of Manifolds_ ([pdf](http://home.imf.au.dk/kock/SGM-final.pdf))

in particular section 6.8 combined with section 4.3. We effectively show that these statements are precisely the ones needed to unwrap what the normalized [[Moore complex|Moore cochain complex]] of the [[cosimplicial algebra]] $C^\infty(\mathbf{B} G_e^{(1)})$ in the [[monoidal Dold-Kan correspondence]] is like.

# Definitions and setup #

Let $G$ be a [[Lie group]] (by which we mean a finite dimensional Lie group). Write $\mathbf{B}G$ for the [[simplicial object|simplicial]] [[smooth space]] which in degree $k$ is the cartesian [[product]] $G^k$ with the standard face and degeneray map (see the examples at [[nerve]] for details).

Let $T$ be some [[topos]] that models the axioms of [[synthetic differential geometry]] and which has a [[full and faithful functor|full and faithful]] embedding [[Diff]] $\hookrightarrow T$.

Consider then $\mathbf{B}G$ as a simplicial object in $T$. As usual, we shall call objects in $T$ [[space]]s in the following.

Let $G_e^{(1)} \hookrightarrow G$ be the space that is the first infinitesimal neighbourhood of the neutral element $e$ in $G$. By definition this space is ismorphic to the [[infinitesimal space]]

$$
  D(n) = \{(d_1, \cdots, d_n) \in R^n | \forall i,j : d_i d_j = 0\}
$$

for $n$ the dimension of $D$. By the [[log-exp bijection in synthetic differential geometry]] this space is canonically identified with the [[vector space]] $g := Lie(G)$ underlying the [[Lie algebra]] of $G$.

Moreover, by the [[Kock-Lawvere axiom]] morphisms $f : D(n) \simeq G_e^{(1)} \to R$ are necessarily linear $f : d \mapsto f_0 \to f_1 \cdot d$, hence under the $log$--$exp$ bijection are nothing but elements in the dual vector space $g^*$.

Recall that the ordinary [[Chevalley-Eilenberg algebra]] of $g$ is the [[differential graded algebra]] whose underlying graded-commutative algebra is the [[Grassmann algebra]] $\wedge^\bullet g^* = \mathbb{R} \oplus g^* \oplus g^* \wedge g^* \oplus \cdots$.

So the subset of $C^\infty(G_e^{(1)})$ that vanishes at 0 is naturally isomorphic to the degree-$1$ part of the Chevalley--Eilenberg algebra.

Notice now that the multiplication on the group $G$ does not restrict to a multiplication on $G_e^{(1)}$ because the sum $d_1 + d_2$ of two elements that each square to 0 is does in general not square to 0, -- $(d_1 + d_2)^2 = 2 d_1 d_2$ but only its cube $(d_1 + d_2)^3 = 0 $ does. Therefore the group multiplication induces a composition

$$
 \cdot :  G_e^{(1)}\times G_2^{(1)} \to G_e^{(2)}
 \,.
$$

Consider therefore the space of "infinitesimal $1$-cells of $\mathbf{B}G$ whose composite is again an infinitesimal $1$-cell", i.e. the [[pullback]]

$$
  \array{
    (G \times G)_e^{(1)}
    &\to&
    G_e^{(1)} \times G_e^{(1)}
    \\
    \downarrow
    &&
    \downarrow^{\cdot}
    \\
    G_e^{(1)}
    &\hookrightarrow&
    G_e^{(2)}
  }
  \,.
$$

By item 3) of [theorem (6.8.1)](http://home.imf.au.dk/kock/SGM-final.pdf#page=226) this pullback is precisely the space of elements $(x,y) \in G_e^{(1)} \times G_e^{(1)}$ such that not only $x$ and $y$ are infinitesimal neighbours of the neutral element $e$, but also of each other.

By the [[Kock-Lawvere axiom]] (entirely analogous to the similar step in the derivation of simplicial [[differential forms in synthetic differential geometry]]) it should follow from this that maps $(G \times G)_e^{(1)} \to R$ that vanish on degenerate elements are in bijection with antisymmetric maps that are canonically identified with elements in $g^* \wedge_{\mathbb{R}} g^*$ (need to say this in more detail...)

Continuing in this manner (...details for higher degrees to be filled in...) we define the simplicial space

$$
  \mathbf{B}G_e^{(1)}
  :=
  (
    \cdots (G \times G)_e^{{1}}
    \stackrel{\to}{\stackrel{\to}{\to}}
    G_e^{(1)}
    \stackrel{\to}{\to}
    {*}
  )
  \,.
$$

The _normalized_ Moore cochain complex $N^\bullet(C^\infty(\mathbf{B}G_e^{(1)}))$ of the cosimplicial algebra

$$
  C^\infty(\mathbf{B}G_e^{(1)})
  :=
  (
    \cdots C^\infty((G \times G)_e^{{1}})
    \stackrel{\leftarrow}{\stackrel{\leftarrow}{\leftarrow}}
    C^\infty(G_e^{(1)})
    \stackrel{\leftarrow}{\leftarrow}
    {*}
  )
$$

is in degree $k$ given by the kernel of the joint degeneracy maps. As in the discussion at [[differential forms in synthetic differential geometry]] this picks out the functions that vanish on degenerate simplices. So from the above we get

$$
  N^\bullet(C^\infty(\mathbf{B}G_e^{(1)}))
  =
  (\cdots
    g^* \wedge g^*
    \stackrel{d = \sum_i (-1)^i d_i}{\leftarrow}
    g^*
    \stackrel{0}{\leftarrow}
    \mathbb{R}
  )
  \,.
$$

Recall that the differential of the [[Chevalley-Eilenberg algebra]] is on $g^*$ just the dual $[-,-]^* : g^* \to g^* \wedge g^*$ of the Lie bracket $[-,-] : g \otimes g \to g$.

We need to check that this is reproduced by the differential  of the Moore cochain complex, which is the alternating sum of the face maps $d = \sum_i (-1)^i d_i$. Let $f \in g^*$. Then we find for all $(x,y) \in (G \times G)_e^{(1)}$ that

$$
  (d f) (x,y) = f(p_1(x,y)) - f(x \cdot y) + f(p_2(x,y))
  = f (x) + f(y) - f(x \cdot y)
  \,.
$$

Now we use the crucial [formula (6.8.2)](http://home.imf.au.dk/kock/SGM-final.pdf#page=227) from Anders Kock's book, which says that the group product on the infinitesimal elements $x,y$ is given by

$$
  x \cdot y = x + y + \frac{1}{2}\{x,y\}
  \,,
$$

where the last term is the group commutator

$$
  \{x,y\}:= x \cdot y \cdot x^{-1} \cdot y^{-1}
  \,.
$$

So this is the term that remains in the formula for $d f$:

$$
  (d f)(x,y) = -\frac{1}{2} f(\{x,y\})
  \,.
$$

On that we apply [theorem 6.6.1](http://home.imf.au.dk/kock/SGM-final.pdf#page=222) of Kock's book, which says (in its third item) that under the $log$--$exp$ bijection by which we identified the infinitesimal neighbourhood $G_e^{(1)}$ (and functions on it) with the tangent space $T_e(G)$ (and linear functions on it) the group commutator maps to the Lie algebra commutator. So indeed under the identification of $f$ with an element in $g^*$ we find

$$
  d f = [-,-]^* f
  \,.
$$

This is indeed the differential of the [[Chevalley-Eilenberg algebra]].

(discussion needs to be completed: situation in higher degree and cup-product mapping to wedge product needs to be discussed...)

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Chevalley-Eilenberg chain complex

Given a $k$-Lie algebra $\mathfrak{g}$ over a commutative unital [[ring]] $k$ which is free as a $k$-module, the __Chevalley--Eilenberg chain complex__ is a particular [[projective resolution]] $V_*(\mathfrak{g})\to k$ of the trivial $\mathfrak{g}$-module $k$ in the [[abelian category]] of $\mathfrak{g}$-modules (what is the same as $U\mathfrak{g}$-modules, where $U\mathfrak{g}$ is the universal enveloping algebra of $\mathfrak{g}$). Graded components of the underlying $k$-module of this resolution is given by

$$
V_p(\mathfrak{g}) = U(\mathfrak{g})\otimes_k \Lambda^p{\mathfrak{g}}
$$

and it has the obvious $U\mathfrak{g}$-module structure by multiplication in the first tensor factor, because $\Lambda^p{\mathfrak{g}}$ is free as a $k$-module.

If $u \in U\mathfrak{g}$ and $x_i\in \mathfrak{g}$ then the differential is given by

$
d(u\otimes x_1 \wedge \cdots \wedge x_p)
 = \sum_{i = 1}^p (-1)^{i+1} u x_i \otimes x_1 \wedge \cdots \wedge \hat{x}_i\wedge \cdots \wedge x_p
+ \sum_{i\lt j} (-1)^{i+j} u\otimes [x_i, x_j] \wedge \cdots \wedge \hat{x}_i\cdots \wedge \hat{x}_j\cdots \wedge x_p
$
###Related entries
 * [[Lie algebra homology]],
* [[Lie algebra cohomology]],
* [[Chevalley-Eilenberg cochain complex]]

and

* [[Chevalley-Eilenberg algebra]].

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Chevalley-Eilenberg cochain complex

> for more see at _[[Chevalley-Eilenberg algebra]]_

***

# Idea #

A finite dimensional [[Lie algebra]] $g$ or degreewise finite-dimensional [[L-∞ algebra]] $g$ is encoded in a differential $D : \vee^\bullet g \to \vee^\bullet g$ on the cofree co-commutative coalgebra generated by $g$.

The dual of this is a [[differential graded algebra]] $(\wedge^\bullet g^*, d)$. The underlying [[cochain complex]] (forgetting the [[monoid]]al structure) is the __Chevalley-Eilenberg cochain complex__.

There is in fact a bijection between [[semifree dga|quasi-free]] [[cochain complex|cochain]] [[differential graded algebra]]s in non-negative degree and [[L-∞ algebra]]s.

The Chevalley-Eilenberg complex is usually defined a bit more generally for [[Lie algebra]]s equipped with a [[Lie module]] $g \to End V$. In the above language this more general cochain complex is the one underlying the [[Lie ∞-algebroid]] that encodes this action in the sense of [[Lie ∞-algebroid representation]]s.

# Lie algebra cohomology #

The [[chain homology and cohomology|cohomology]] of the Chevalley-Eilenberg cochain complex agrees with the [[Lie algebra cohomology]]  with trivial coefficients. The Lie algebra is however defined also for infinite-dimensional Lie algebras and arbitrary module $M$ coefficients. Namely the Lie algebra cohomology is $H^*_{Lie}(\mathfrak{g},M) = Ext_{U(\mathfrak{g})}(\mathfrak{g},M) = H^*(Hom_{U(\mathfrak{g})}(V(\mathfrak{g}),M))\cong H^*(Hom_k(\Lambda^*\mathfrak{g},M))$ where $U(\mathfrak{g})$ is the universal enveloping of $\mathfrak{g}$ and $V(\mathfrak{g}) = U(\mathfrak{g})\otimes \Lambda^* \mathfrak{g}$ (with the appropriate differential) is the [[Chevalley-Eilenberg chain complex]]. Now if $\mathfrak{g}$ is finite-dimensional then $Hom_{U(\mathfrak{g})}(V(\mathfrak{g}),M)\cong CE(\mathfrak{g},M)$ and $CE(\mathfrak{g}) = CE(\mathfrak{g},k) = \Lambda^* \mathfrak{g}^*$.

# References #

* MathOverflow: [definitions of Chevalley-Eilenberg complex](http://mathoverflow.net/questions/8460/chevalley-eilenberg-complex-definitions)

* C. Chevalley, S. Eilenberg, _Cohomology theory of Lie groups and Lie algebras_, Trans. Amer. Math. Soc. 63, (1948). 85--124.

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Chevalley–Eilenberg algebra > history

< [[Chevalley–Eilenberg algebra]]
[[!redirects Chevalley–Eilenberg algebra -- history]]


Chi Hong Chow

## Selected writings

On the [[Peterson isomorphism]] relating the [[quantum cohomology]] of [[flag varieties]] to a [[Pontryagin ring]]:

* [[Chi Hong Chow]], *Peterson-Lam-Shimozono's theorem is an affine analogue of quantum Chevalley formula* [[arXiv:2110.09985](https://arxiv.org/abs/2110.09985)]

* [[Chi Hong Chow]], *On D. Peterson's presentation of quantum cohomology of $G/P$* [[arXiv:2210.17382](https://arxiv.org/abs/2210.17382)]

reviewed in

* [[Jimmy Chow]], *Homology of based loop groups and quantum cohomology of flag varieties*, talk at *Western Hemisphere Virtual Symplectic Seminar* (2021) [[pdf](https://freemath.xyz/chowslides.pdf), [[Chow-QuantumCohomologyOfFlagVarieties.pdf:file]], video:[YT](https://www.youtube.com/watch?v=ovbQBj7t3eU)]

category: people

[[!redirects Jimmy Chow]]


Chi-Kwong Fok

* [personal page](https://sites.google.com/site/alexckfok/)

* [institute page](https://shanghai.nyu.edu/academics/faculty/directory/chi-kwong-fok)

## Selected writings

On [[twisted ad-equivariant K-theory]] of [[Lie groups]] generalized to [[KR-theory]]:

* [[Chi-Kwong Fok]], *The Real K-theory of compact Lie groups*, SIGMA 10 (2014), 022, ([arXiv:1308.3871](https://arxiv.org/abs/1308.3871), [doi:10.3842/SIGMA.2014.022](https://doi.org/10.3842/SIGMA.2014.022))

* [[Chi-Kwong Fok]], *Equivariant twisted Real K-theory of compact Lie groups*,   Journal of Geometry and Physics 124 (2018) 325-349 ([arXiv:1503.00957](https://arxiv.org/abs/1503.00957), [doi:10.1016/j.geomphys.2017.11.013](https://doi.org/10.1016/j.geomphys.2017.11.013))

* [[Alex Fok]], *Equvariant twisted KK-theory of noncompact Lie groups* (2022) $[$[talk](Center+for+Quantum+and+Topological+Systems#AlexFokMay22)$]$

category: people

[[!redirects Chi Kwong Fok]]
[[!redirects Alex Fok]]
[[!redirects Alex Fong]]