Kähler manifold

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
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[[!include synthetic differential geometry - contents]]
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#### Complex geometry
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[[!include complex geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _Kähler manifold_ is a [[smooth manifold]] compatibly equipped with

1. [[complex structure]];

1. [[Riemannian structure]];

1. [[symplectic structure]].

If the symplectic structure is not compatibly present, it is just a [[Hermitian manifold]].

| [[complex structure]] | + [[Riemannian structure]] | + [[symplectic structure]] |
|--|--|--|
[[complex structure]] | [[Hermitian structure]] | [[Kähler structure]] |

Where a Riemannian manifold is a real [[smooth manifold]] equipped with a nondegenerate smooth symmetric 2-form $g$ (the [[Riemannian metric]]), an __almost Kähler manifold__ is a [[complex manifold|complex holomorphic manifold]] equipped with a nondegenerate hermitian 2-form $h$ (the __Kähler $2$-form__).  The real [[cotangent bundle]] is replaced with the complex cotangent bundle, and symmetry is replaced with hermitian symmetry. An almost Kähler manifold is a __Kähler manifold__ if it satisfies an additional integrability condition.

The Kähler 2-form can be decomposed as $h = g+i\omega$; here $g$ is a [[Riemannian metric]] and $\omega$ a [[symplectic form]].

## Definition

### Linear Kähler structure
 {#LinearKählerStructure}

+-- {: .num_defn #KaehlerVectorSpace}
###### Definition
**([[Kähler vector space]])**

Let $V$ be a [[finite dimensional vector space|finite-dimensional]] [[real vector space]]. Then a _linear Kähler structure_ on $V$ is

1. a _[[linear complex structure]]_ on $V$, namely a [[linear map|linear]] [[endomorphism]]

   $$
     J \;\colon\; V \to V
   $$

   whose [[composition]] with itself is minus the [[identity morphism]]:

   $$
     J \circ J = - id_V
   $$

1. a skew-symmetric [[bilinear form]]

   $$
     \omega \in \wedge^2 V^\ast
   $$

such that

1. $\omega(J(-),J(-)) = \omega(-,-)$;

1. $g(-,-) \coloneqq \omega(-,J(-))$ is a [[Riemannian metric]], namely

   a non-degenerate positive-definite [[bilinear form]] on $V$

   (necessarily symmetric, due to the other properties: $g(w,v) = \omega(w,J(v)) = -\omega(J(v),w) = - \omega(J(J(v)), J(w)) = +\omega(v,J(w)) = g(v,w)$).

=--

(e.g. [Boalch 09, p. 26-27](#Boalch09))

Linear Kähler space structure may conveniently be encoded in terms of [[Hermitian space]] structure:

+-- {: .num_defn #HermitianForm}
###### Definition
**([[Hermitian form]] and [[Hermitian space]])**

Let $V$ be a [[real vector space]] equipped with a [[complex structure]] $J\colon V \to V$. Then a _[[Hermitian form]]_ on $V$ is

* a complex-valued real-[[bilinear form]]

  $$
    h \;\colon\; V \otimes V \longrightarrow \mathbb{C}
  $$

such that this is _symmetric sesquilinear_, in that:

1. $h$ is complex-linear in the first argument;

1. $h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$

where $(-)^\ast$ denotes [[complex conjugation]].

A Hermitian form is _positive definite_ (often assumed by default) if for all $v \in V$

1. $h(v,v) \geq 0$

1. $h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.

A [[complex vector space]] $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) _[[Hermitian space]]_.

=--

+-- {: .num_prop #BasicPropertiesOfHermitianForms}
###### Proposition
**(basic properties of [[Hermitian forms]])**

Let $((V,J),h)$ be a positive definite [[Hermitian space]] (def. \ref{HermitianForm}). Then

1. the [[real part]] of the [[Hermitian form]]

   $$
     g(-,-) \;\coloneqq\; Re(h(-,-))
   $$

   is a [[Riemannian metric]], hence a symmetric positive-definite real-[[bilinear form]]

   $$
     g \;\colon\; V \otimes V \to \mathbb{R}
   $$

1. the [[imaginary part]] of the [[Hermitian form]]

   $$
     \omega(-,-) \;\coloneqq\; -Im(h(-,-))
   $$

   is a [[symplectic form]], hence a non-degenerate skew-symmetric real-[[bilinear form]]

   $$
     \omega \;\colon\; V \wedge V \to \mathbb{R}
     \,.
   $$

hence

$$
  h = g - i \omega
  \,.
$$

The two components are related by

$$
  \label{RelationBetweennOmegaAndgOnHermitianSpace}
  \omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
$$

Finally

$$
  h(J(-),J(-)) = h(-,-)
$$

and so the  Riemannian metrics $g$ on $V$ appearing from (and fully determining) Hermitian forms $h$ via $h = g - i \omega$ are precisely those for which

$$
  \label{HermitianMetric}
  g(J(-),J(-)) = g(-,-)
  \,.
$$

These are called the _[[Hermitian metrics]]_.

=--

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

The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:

$$
  \begin{aligned}
    g(w,v)
    & \coloneqq
    Re(h(w,v))
    \\
    & =
    Re\left( h(v,w)^\ast\right)
    \\
    & =
    Re(h(v,w))
    \\
    & =
    g(v,w)
    \,.
  \end{aligned}
$$

That $h$ is invariant under $J$ follows from its sesquilinarity

$$
  \begin{aligned}
    h(J(v),J(w))
    &=
    i h(v,J(w))
    \\
    & =
    i (h(J(w),v))^\ast
    \\
    & =
    i (-i) (h(w,v))^\ast
    \\
    & =
    h(v,w)
  \end{aligned}
$$

and this immediately implies the corresponding invariance of $g$ and $\omega$.

Analogously it follows that $\omega$ is skew symmetric:

$$
  \begin{aligned}
    \omega(w,v)
    & \coloneqq
    -Im(h(w,v))
    \\
    & =
    -Im\left( h(v,w)^\ast\right)
    \\
    & =
    Im(h(v,w))
    \\
    & = - \omega(v,w)
    \,,
  \end{aligned}
$$

and the relation between the two components:

$$
  \begin{aligned}
    \omega(v,w)
    & =
    - Im(h(v,w))
    \\
    & =
    Re(i h(v,w))
    \\
    & =
    Re(h(J(v),w))
    \\
    & =
    g(J(v),w)
  \end{aligned}
$$

as well as

$$
  \begin{aligned}
    g(v,w)
    & =
    Re(h(v,w)
    \\
    & =
    Im(i h(v,w))
    \\
    & =
    Im(h(J(v),w))
    \\
    & =
    Im(h(J^2(v),J(w)))
    \\
    & =
    - Im(h(v,J(w)))
    \\
    & = \omega(v,J(w))
    \,.
  \end{aligned}
$$

=--

As a corollary:

+-- {: .num_prop #RelationBetweenKaehlerVectorSpacesAndHermitianSpaces}
###### Proposition
**(relation between [[Kähler vector spaces]] and [[Hermitian spaces]])**

Given a [[real vector space]] $V$ with a [[linear complex structure]] $J$, then the following are equivalent:

1. $\omega \in \wedge^2 V^\ast$ is a [[linear Kähler structure]] (def. \ref{KaehlerVectorSpace});

1. $g \in V \otimes V \to  \mathbb{R}$ is a [[Hermitian metric]] (eq:HermitianMetric)

where $\omega$ and $g$ are related by (eq:RelationBetweennOmegaAndgOnHermitianSpace)

$$
  \omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(w))
  \,.
$$

=--

### Kähler manifolds

(...)

### In terms of $G$-Structure
 {#InTermsOfGStructure}

A Kähler manifold is a [[integrability of G-structure|first-order integrable]] [[almost Hermitian structure]], hence a first order integrable [[G-structure]] for $G = U(n) \hookrightarrow GL(2n,\mathbb{R})$ the [[unitary group]].

By the fact (see at _[unitary group -- relation to orthogonal, symplectic and general linear group](unitary+group#RelationToOrthogonalSymplecticAndGeneralLinearGroup)_) that $U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})$ this means that a Kähler manifold structure is precisely a joint [[orthogonal structure]]/[[Riemannian manifold]] structure, [[symplectic manifold]] structure and [[complex manifold]] structure.

(e.g. [Moroianu 07, 11.1](#Moroianu07), [Verbitsky 09](#Verbitsky09))

## Examples
 {#Examples}

The archetypical elementary example is the following:

+-- {: .num_example #StandardAlmostKaehlerVectorSpaces}
###### Example
**(standard [[Kähler vector space]])**

Let $V \coloneqq \mathbb{R}^2$ be the 2-dimensional [[real vector space]] equipped with the [[complex structure]] $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical [[linear basis]] $(e_i)$ of $\mathbb{R}^2$, this is

$$
  J = (J^i{}_j)
  \coloneqq
  \left(
    \array{
      0 & -1
      \\
      1 & 0
    }
  \right)
  \,.
$$

Moreover let

$$
  \omega = (\omega_{i j}) \coloneqq \left( \array{0 & 1 \\ -1 & 0} \right)
$$

and

$$
  g = (g_{i j}) \coloneqq \left( \array{ 1 & 0 \\ 0 & 1}  \right)
  \,.
$$

Then $(V, J, \omega, g)$ is a [[Kähler vector space]] (def. \ref{KaehlerVectorSpace})

The corresponding [[Kähler manifold]] is $\mathbb{R}^2$ regarded as a [[smooth manifold]] in the standard way and equipped with the [[bilinear forms]] $J, \omega g$ extended as constant rank-2 [[tensors]] over this manifold.

If we write

$$
  x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
$$

for the standard [[coordinate functions]] on $\mathbb{R}^2$ with

$$
  z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$

and

$$
 \overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$

for the corresponding complex coordinates, then this translates to

$$
  \omega
  \in
  \Omega^2(\mathbb{R}^2)
$$

being the [[differential 2-form]] given by

$$
  \begin{aligned}
    \omega
    & =
    d x \wedge d y
    \\
    & =
    \tfrac{i}{2} d z \wedge d \overline{z}
  \end{aligned}
$$

and with [[Riemannian metric]] [[tensor]] given by

$$
  g = d x \otimes d x + d y \otimes d y
  \,.
$$

The [[Hermitian form]] is given by

$$
  \begin{aligned}
    h
    & =
    g - i \omega
    \\
    & =
    d z \otimes d \overline{z}
    \,.
  \end{aligned}
$$

=--

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

This is elementary, but, for the record, here is one way to make it fully explicit (we use [[Einstein summation convention]] and "$\cdot$" denotes [[matrix multiplication]]):

$$
  \begin{aligned}
    \omega_{i j'} J^{j'}{}_j
    & =
    \left(
       \array{
         0 & 1
         \\
         -1 & 0
       }
    \right)
      \cdot
    \left(
      \array{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \\
    &
    =
    \left(
      \array{
        1 & 0
        \\
        0 & 1
      }
    \right)
    \\
    & =
    g_{i j}
  \end{aligned}
$$

and similarly

$$
  \begin{aligned}
    \omega(J(-),J(-))_{i j}
    & =
    \omega_{i' j'} J^{i'}{}_{i} J^{j'}{}_{j}
    \\
    & =
    (J^t \cdot \omega \cdot J)_{i j}
    \\
    & =
    \left(
    \left(
      \array{
        0 & 1
        \\
        -1 & 0
      }
    \right)
    \cdot
    \left(
      \array{
        0 & 1
        \\
        -1 & 0
      }
    \right)
      \cdot
    \left(
      \array{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \right)_{i j}
    \\
    & =
    \left(
    \left(
      \array{
        -1 & 0
        \\
        0 & -1
      }
    \right)
    \cdot
    \left(
      \array{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \right)_{i j}
    \\
    & =
    \left(
      \array{
        0 & 1
        \\
        -1 & 0
      }
    \right)_{i j}
    \\
    & =
    \omega_{i j}
  \end{aligned}
$$

=--

+-- {: .num_example}
###### Example
**([[Fubini-Study metric]])**

There is a unique (up to a scalar) hermitian metric on  [[complex projective space]] (which may be normalized), the _[[Fubini-Study metric]]_.

All analytic subvarieties of a complex projective space are in fact [[algebraic variety|algebraic subvarieties]] and they inherit the Kähler structure from the projective space.

Examples include [[complex tori]] $\mathbb{C}^n/L$ where $L$ is a lattice in $\mathbb{C}^n$, [[K3-surfaces]], compact [[Calabi-Yau manifolds]], quadrics, products of projective spaces and so on.

=--

## Properties

### Relation to (almost) complex manifold

> The following based on [this MO comment](http://mathoverflow.net/a/73330/381) by [[Spiro Karigiannis]]

When $(X, J)$ is an [[almost complex manifold]], then there is a notion of smooth complex-valued [[differential forms]] of type $(p,q)$. A complex valued $2$-form $\omega$ is of type $(1,1)$ precisely if it satisfies

$$
  \omega(J v,J w) = \omega(v,w)
$$

for all smooth [[vector fields]] $v,w$ on $X$. Here $\omega$ is a *real* $2$-form of type $(1,1)$, if $\overline \omega = \omega$. Setting

$$
  g(v,w) = \omega(v, J w),
$$

defines a smooth symmetric rank $(2,0)$ [[tensor field]]. This is a
[[Riemannian metric]] precisely if it is fiberwise a [[positive definite bilinear form]]. If it $g(-,-) = \omega(-,J -)$ is hence a Riemannian metric, then $\omega(-,-)$ is called positive definite, too.

The triple of data $(J, \omega, g)$, where $J$ is an [[almost complex structure]], $\omega$ is a real positive $(1,1)$-[[differential form]], and $g$ is the associated [[Riemannian metric]] this way define an *[[almost Hermitian manifold]]*.

Now the condition for $X$ to be a Kähler is that $X$ be a [[complex manifold]] ($J$ is integrable) and that $d\omega = 0$.
Equivalently that for the [[Levi-Civita connection]] $\nabla$ of $G$ we have $\nabla \omega = 0$ or $\nabla J = 0$.

Hence given a [[complex manifold]] $X$, together with a  *closed* real $2$-form $\omega$, the only additional condition required to ensure that it defines a  Kähler metric is that it be a positive $(1,1)$-form.

### Relation to symplectic manifolds

Lifting a [[symplectic manifold]] structure to a Kähler manifold structure is also called choosing a _[[Kähler polarization]]_.

### Relation to Spin-structures

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

A spin structure on a [[compact topological space|compact]] [[Hermitian manifold]] ([[Kähler manifold]]) $X$ of complex [[dimension]] $n$ exists precisely if, equivalently

* there is a choice of [[square root]] $\sqrt{\Omega^{n,0}}$ of the [[canonical line bundle]] $\Omega^{n,0}$ (a "[[Theta characteristic]]");

* there is a trivialization of the [[first Chern class]] $c_1(T X)$ of the [[tangent bundle]].

=--

In this case one has:

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

There is a [[natural isomorphism]]

$$
  S_X \simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{n,0}}_X
$$

of the [[sheaf]] of [[sections]] of the [[spinor bundle]] $S_X$ on $X$ with the [[tensor product]] of the  [[Dolbeault complex]] with the corresponding [[Theta characteristic]];

Moreover, the corresponding [[Dirac operator]] is the [[Dolbeault-Dirac operator]] $\overline{\partial} + \overline{\partial}^\ast$.

=--

This is due to ([Hitchin 74](#Hitchin74)). A textbook account is for instance in ([Friedrich 74, around p. 79 and p. 82](#Friedrich74)).

### Hodge star operator
 {#HodgeStarOperator}

On a Kähler manifold $\Sigma$ of [[dimension]] $dim_{\mathbb{C}}(\Sigma) = n$ the [[Hodge star operator]] acts on the [[Dolbeault complex]] as

$$
  \star \;\colon\; \Omega^{p,q}(X) \longrightarrow \Omega^{n-q,n-p}(X)
  \,.
$$

(notice the exchange of the role of $p$ and $q$) See e.g. ([BiquerdHöring 08, p. 79](#BiquerdHoering08)).

### Hodge structure
 {#HodgeStructure}

The [[Hodge theorem]] asserts that for a compact Kähler manifold, the canonical $(p,q)$-grading of its [[differential forms]] descends to its [[de Rham cohomology]]/[[ordinary cohomology]]. The resulting structure is called a _[[Hodge structure]]_, and is indeed the archetypical example of such.

### As $\mathbb{C}$-Riemannian manifolds

[[!include normed division algebra Riemannian geometry -- table]]

## Related concepts

* [[bilinear form]], [[quadratic form]], [[sesquilinear form]]

* [[symplectic form]], [[Hermitian form]]

* **Kähler manifold**, [[hyper-Kähler manifold]], [[quaternionic Kähler manifold]]

  * [[Kähler potential]]

* [[Kähler-Einstein manifold]]

* [[symplectic manifold]]

* [[Lefschetz decomposition]]

* [[Kähler polarization]], [[almost Kähler geometric quantization]],

* [[Fedosov deformation quantization]]

* [[Hodge filtration]], [[Frölicher spectral sequence]]

* [[Hodge manifold]]

* [[Sasakian manifold]]

[[!include special holonomy table]]

## References

Kähler manifolds were first introduced and studied by P. A. Shirokov (cf. [a historical article](http://dx.doi.org/10.1070/RM1995v050n02ABEH002098)) and later independently by Kähler.

Textbook accounts:

* {#Voisin02} [[Claire Voisin]], section 3 of _[[Hodge theory and Complex algebraic geometry]] I,II_,  Cambridge Stud. in Adv. Math. __76, 77__, 2002/3

* [[Ana Cannas da Silva]], §16 in: *Lectures on Symplectic Geometry*, Lecture Notes in Mathematics **1764**, Springer (2008) [[doi:10.1007/978-3-540-45330-7](https://doi.org/10.1007/978-3-540-45330-7)]

Lecture notes include

* {#Moroianu07} [[Andrei Moroianu]], _Lectures on Kähler Geometry_, Cambridge University Press 2007 ([arXiv:math/0402223](https://arxiv.org/abs/math/0402223) [doi:10.1017/CBO9780511618666](https://doi.org/10.1017/CBO9780511618666), [pdf](https://moroianu.perso.math.cnrs.fr/tex/kg.pdf))

* {#Boalch09} Philip Boalch, *Noncompact complex symplectic and hyperkähler manifolds* (2009) [[pdf](https://webusers.imj-prg.fr/~philip.boalch/cours09/hk.pdf), [[Boalch-NoncompactSymplectic.pdf:file]]]

Discussion in terms of [[integrability of G-structure|first-order integrable]] [[G-structure]] include

* [Moroianu 07, 11.1](#Moroianu07)

* {#Verbitsky09} Misha Verbitsky, _Kähler manifolds_, lecture notes 2009 ([pdf](http://verbit.ru/MATH/TALKS/Unicamp-kahler-1.pdf))

Discussion of [[spin structures]] in Kähler manifolds is for instance in

* {#Friedrich97} [[Thomas Friedrich]], _Dirac operators in Riemannian geometry_, Graduate studies in mathematics 25, AMS (1997)

Discussion of [[Hodge theory]] on Kähler manifolds is in

* {#BiquerdHoering08} O. Biquard, A. Höring, _Kähler geometry and Hodge theory_, 2008 ([pdf](http://math.unice.fr/~hoering/hodge/hodge.pdf))

Discussion of [[Kähler orbifolds]]:

* Thalia D. Jeffres, _Singular Set of Some Kähler Orbifolds_, Transactions of the American Mathematical Society Vol. 349, No. 5 (May, 1997), pp. 1961-1971 ([jstor:2155355 ](https://www.jstor.org/stable/2155355 ))

* Akira Fujiki, _On primitively symplectic compact Kähler V-manifolds_, in:  [[Kenji Ueno]] (ed.), _Classification of Algebraic and Analytic Manifolds: Katata Symposium Proceedings 1982_, Birkhäuser 1983 (ISBN:9780817631376)

* [[Dominic Joyce]], Section 6.5.1 of: _Compact Manifolds with Special Holonomy_, Oxford Mathematical Monographs, Oxford University Press (2000) ([ISBN:9780198506010](https://global.oup.com/academic/product/compact-manifolds-with-special-holonomy-9780198506010?cc=de&lang=en&))

* Miguel Abreu, _Kähler Metrics on Toric Orbifolds_, J. Differential Geom. Volume 58, Number 1 (2001), 151-187 ([euclid:jdg/1090348285](https://projecteuclid.org/euclid.jdg/1090348285))

* Giovanni Bazzoni, Indranil Biswas, Marisa Fernández, Vicente Muñoz, Aleksy Tralle, _Homotopic properties of Kähler orbifolds_, In: Chiossi S., Fino A., Musso E., Podestà F., Vezzoni L. (eds.) _Special Metrics and Group Actions in Geometry_ Springer INdAM Series, vol 23. Springer (2017) ([arXiv:1605.03024](https://arxiv.org/abs/1605.03024), [doi:10.1007/978-3-319-67519-0_2](https://doi.org/10.1007/978-3-319-67519-0_2))

[[!redirects Kähler manifolds]]

[[!redirects Kaehler manifold]]
[[!redirects Kaehler manifolds]]

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[[!redirects Kahler 2-form]]
[[!redirects Kahler 2-formw]]

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[[!redirects Kähler 2-form]]

[[!redirects almost Kähler manifold]]

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[[!redirects Kähler orbifolds]]

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[[!redirects Kähler structures]]
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[[!redirects Kaehler structures]]

[[!redirects almost Kähler manifold]]
[[!redirects almost Kähler manifolds]]

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Kähler metric

the metric on a [[Kähler manifold]]


Kähler polarization

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Geometric quantization
+--{: .hide}
[[!include geometric quantization - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Symplectic geometry
+--{: .hide}
[[!include symplectic geometry - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Kähler polarization_ of a [[symplectic manifold]] is a [[polarization]] by a compatible [[Kähler manifold]] structure.

Given a [[prequantization]] of a Kähler polarized symplectic manifold by a [[holomorphic line bundle]], then the polarized sections are the _[[holomorphic sections]]_.

Hence the concept of Kähler polarization is that special case of _[[polarization]]_ which connects most intimately the [[symplectic geometry]] to [[complex analytic geometry]]. The generalization of this from [[complex analytic geometry]] to more general [[algebraic geometry]] is the concept of a _[[polarized algebraic variety]]_.

For more see at

* _[polarization -- Kähler polarizations](polarization#K%C3%A4hlerPolarization)_

* _[geometric quantization -- Quantum state space as space of polarized sections](http://ncatlab.org/nlab/show/geometric%20quantization#Polarizations)_

## Definition

### In terms of $G$-structures

In terms of [[G-structures]] this means that it is a lift from an [[integrability of G-structures|integrable]] $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$-[[G-structure]] ([[integrability of G-structure|integrable]] [[almost symplectic structure]]) to a first-order integrable $U(n) \hookrightarrow Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$-[[G-structure|structure]] ([[integrable G-structure|first-order integrable]] [[almost Hermitian structure]]).

### In terms of distributions in the complexified tangent bundle

(...)

$$
  \mathcal{P} \subset T_{\mathbb{C}} T X
$$

$$
  \mathcal{P} \cap \overline{\mathcal{P}} = 0
$$

$$
  \mathcal{P} = \left\{J v - i v \;|\; v \in T X \right\}
$$

## Related concepts

* [[real polarization]]

* [[p-convex polarization]]

* [[almost Kähler quantization]]

* [[spin^c quantization]]

* [[metaplectic quantization]]

## References

Discussion of the functoriality of Kähler polarization quantization with respect to the choice of [[metaplectic correction|metaplectically]] corrected Kähler structure is in section 3 of

* Lauridsen, _Aspects of quantum mathematics  -- Hitchin connections and the AJ conjecture_, PhD thesis Aarhus 2010 ([pdf](http://pure.au.dk/portal/files/41741925/imf_phd_2010_mrl.pdf))

[[!redirects Kähler polarizations]]


Kähler potential

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

On a [[Kähler manifold]] the [[Kähler form]] $\omega$ can locally be written as

$$
  \omega =\sqrt{-1} \partial \overline{\partial} K
$$

for some real-valued function $K$, where $\partial$ denotes the [[Dolbeault operator]]. This function $K$ is called the corresponding _Kähler potential_.

## Related concepts

* [[Poincaré lemma]]

[[!redirects Kähler potentials]]


Kähler vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #KaehlerVectorSpace}
###### Definition
**([[Kähler vector space]])**

Let $V$ be a [[finite dimensional vector space|finite-dimensional]] [[real vector space]]. Then a _linear Kähler structure_ on $V$ is

1. a _[[linear complex structure]]_ on $V$, namely a [[linear map|linear]] [[endomorphism]]

   $$
     J \;\colon\; V \to V
   $$

   whose [[composition]] with itself is minus the [[identity morphism]]:

   $$
     J \circ J = - id_V
   $$

1. a skew-symmetric [[bilinear form]]

   $$
     \omega \in \wedge^2 V^\ast
   $$

such that

1. $\omega(J(-),J(-)) = \omega(-,-)$;

1. $g(-,-) \coloneqq \omega(-,J(-))$ is a [[Riemannian metric]], namely

   a non-degenerate positive-definite [[bilinear form]] on $V$

   (necessarily symmetric, due to the other properties: $g(w,v) = \omega(w,J(v)) = -\omega(J(v),w) = - \omega(J(J(v)), J(w)) = \omega(w,J(w)) = g(v,w)$).

=--

(e.g. [Boalch 09, p. 26-27](#Boalch09))

## Properties

### Relation to Hermitian spaces

Linear Kähler space structure may conveniently be encoded in terms of [[Hermitian space]] structure:

+-- {: .num_defn #HermitianForm}
###### Definition
**([[Hermitian form]] and [[Hermitian space]])**

Let $V$ be a [[real vector space]] equipped with a [[complex structure]] $J\colon V \to V$. Then a _[[Hermitian form]]_ on $V$ is

* a complex-valued real-[[bilinear form]]

  $$
    h \;\colon\; V \otimes V \longrightarrow \mathbb{C}
  $$

such that this is _symmetric sesquilinear_, in that:

1. $h$ is complex-linear in the first argument;

1. $h(w,v) = \left(h(v,w) \right)^\ast$ for all $v,w \in V$

where $(-)^\ast$ denotes [[complex conjugation]].

A Hermitian form is _positive definite_ (often assumed by default) if for all $v \in V$

1. $h(v,v) \geq 0$

1. $h(v,v) = 0 \phantom{AA} \Leftrightarrow \phantom{AA} v = 0$.

A [[complex vector space]] $(V,J)$ equipped with a (positive definite) Hermitian form $h$ is called a (positive definite) _[[Hermitian space]]_.

=--

+-- {: .num_prop #BasicPropertiesOfHermitianForms}
###### Proposition
**(basic properties of [[Hermitian forms]])**

Let $((V,J),h)$ be a positive definite [[Hermitian space]] (def. \ref{HermitianForm}). Then

1. the [[real part]] of the [[Hermitian form]]

   $$
     g(-,-) \;\coloneqq\; Re(h(-,-))
   $$

   is a [[Riemannian metric]], hence a symmetric positive-definite real-[[bilinear form]]

   $$
     g \;\colon\; V \otimes V \to \mathbb{R}
   $$

1. the [[imaginary part]] of the [[Hermitian form]]

   $$
     \omega(-,-) \;\coloneqq\; -Im(h(-,-))
   $$

   is a [[symplectic form]], hence a non-degenerate skew-symmetric real-[[bilinear form]]

   $$
     \omega \;\colon\; V \wedge V \to \mathbb{R}
     \,.
   $$

hence

$$
  h = g - i \omega
  \,.
$$

The two components are related by

$$
  \label{RelationBetweennOmegaAndgOnHermitianSpace}
  \omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
$$

Finally

$$
  h(J(-),J(-)) = h(-,-)
$$

and so the  Riemannian metrics $g$ on $V$ appearing from (and fully determining) Hermitian forms $h$ via $h = g - i \omega$ are precisely those for which

$$
  \label{HermitianMetric}
  g(J(-),J(-)) = g(-,-)
  \,.
$$

These are called the _[[Hermitian metrics]]_.

=--

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

The positive-definiteness of $g$ is immediate from that of $h$. The symmetry of $g$ follows from the symmetric sesquilinearity of $h$:

$$
  \begin{aligned}
    g(w,v)
    & \coloneqq
    Re(h(w,v))
    \\
    & =
    Re\left( h(v,w)^\ast\right)
    \\
    & =
    Re(h(v,w))
    \\
    & =
    g(v,w)
    \,.
  \end{aligned}
$$

That $h$ is invariant under $J$ follows from its sesquilinarity

$$
  \begin{aligned}
    h(J(v),J(w))
    &=
    i h(v,J(w))
    \\
    & =
    i (h(J(w),v))^\ast
    \\
    & =
    i (-i) (h(w,v))^\ast
    \\
    & =
    h(v,w)
  \end{aligned}
$$

and this immediately implies the corresponding invariance of $g$ and $\omega$.

Analogously it follows that $\omega$ is skew symmetric:

$$
  \begin{aligned}
    \omega(w,v)
    & \coloneqq
    -Im(h(w,v))
    \\
    & =
    -Im\left( h(v,w)^\ast\right)
    \\
    & =
    Im(h(v,w))
    \\
    & = - \omega(v,w)
    \,,
  \end{aligned}
$$

and the relation between the two components:

$$
  \begin{aligned}
    \omega(v,w)
    & =
    - Im(h(v,w))
    \\
    & =
    Re(i h(v,w))
    \\
    & =
    Re(h(J(v),w))
    \\
    & =
    g(J(v),w)
  \end{aligned}
$$

as well as

$$
  \begin{aligned}
    g(v,w)
    & =
    Re(h(v,w)
    \\
    & =
    Im(i h(v,w))
    \\
    & =
    Im(h(J(v),w))
    \\
    & =
    Im(h(J^2(v),J(w)))
    \\
    & =
    - Im(h(v,J(w)))
    \\
    & = \omega(v,J(w))
    \,.
  \end{aligned}
$$

=--

As a corollary:

+-- {: .num_prop #RelationBetweenKaehlerVectorSpacesAndHermitianSpaces}
###### Proposition
**(relation between [[Kähler vector spaces]] and [[Hermitian spaces]])**

Given a [[real vector space]] $V$ with a [[linear complex structure]] $J$, then the following are equivalent:

1. $\omega \in \wedge^2 V^\ast$ is a [[linear Kähler structure]] (def. \ref{KaehlerVectorSpace});

1. $g \in V \otimes V \to  \mathbb{R}$ is a positive definite [[Hermitian metric]] (eq:HermitianMetric)

where $\omega$ and $g$ are related by (eq:RelationBetweennOmegaAndgOnHermitianSpace)

$$
  \omega(v,w)
  \;=\;
  g(J(v),w)
  \phantom{AAAAA}
  g(v,w)
  \;=\;
  \omega(v,J(v))
  \,.
$$

Hence Kähler vector spaces are equivalently the [[finite dimensional vector space|finite dimensional]] complex [[Hilbert spaces]].

=--

## Examples

The archetypical elementary example is the following:

+-- {: .num_example #StandardAlmostKaehlerVectorSpaces}
###### Example
**(standard [[Kähler vector space]])**

Let $V \coloneqq \mathbb{R}^2$ be the 2-dimensional [[real vector space]] equipped with the [[complex structure]] $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical [[linear basis]] $(e_i)$ of $\mathbb{R}^2$, this is

$$
  J = (J^i{}_j)
  \coloneqq
  \left(
    \array{
      0 & -1
      \\
      1 & 0
    }
  \right)
  \,.
$$

Moreover let

$$
  \omega = (\omega_{i j}) \coloneqq \left( \array{0 & 1 \\ -1 & 0} \right)
$$

and

$$
  g = (g_{i j}) \coloneqq \left( \array{ 1 & 0 \\ 0 & 1}  \right)
  \,.
$$

Then $(V, J, \omega, g)$ is a [[Kähler vector space]] (def. \ref{KaehlerVectorSpace}).

The corresponding [[Kähler manifold]] is $\mathbb{R}^2$ regarded as a [[smooth manifold]] in the standard way and equipped with the [[bilinear forms]] $J, \omega g$ extended as constant rank-2 [[tensors]] over this manifold.

If we write

$$
  x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
$$

for the standard [[coordinate functions]] on $\mathbb{R}^2$ with

$$
  z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$

and

$$
 \overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$

for the corresponding complex coordinates, then this translates to

$$
  \omega
  \in
  \Omega^2(\mathbb{R}^2)
$$

being the [[differential 2-form]] given by

$$
  \begin{aligned}
    \omega
    & =
    d x \wedge d y
    \\
    & =
    \tfrac{1}{2i} d z \wedge d \overline{z}
  \end{aligned}
$$

and with [[Riemannian metric]] [[tensor]] given by

$$
  g = d x \otimes d x + d y \otimes d y
  \,.
$$

The [[Hermitian form]] is given by

$$
  \begin{aligned}
    h
    & =
    g - i \omega
    \\
    & =
    d z \otimes d \overline{z}
  \end{aligned}
$$

=--

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

This is elementary, but, for the record, here is one way to make it fully explicit (we use [[Einstein summation convention]] and "$\cdot$" denotes [[matrix multiplication]]):

$$
  \begin{aligned}
    \omega_{i j'} J^{j'}{}_j
    & =
    \left(
       \array{
         0 & 1
         \\
         -1 & 0
       }
    \right)
      \cdot
    \left(
      \array{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \\
    &
    =
    \left(
      \array{
        1 & 0
        \\
        0 & 1
      }
    \right)
    \\
    & =
    g_{i j}
  \end{aligned}
$$

and similarly

$$
  \begin{aligned}
    \omega(J(-),J(-))_{i j}
    & =
    \omega_{i' j'} J^{i'}{}_{i} J^{j'}{}_{j}
    \\
    & =
    (J^t \cdot \omega \cdot J)_{i j}
    \\
    & =
    \left(
    \left(
      \array{
        0 & 1
        \\
        -1 & 0
      }
    \right)
    \cdot
    \left(
      \array{
        0 & 1
        \\
        -1 & 0
      }
    \right)
      \cdot
    \left(
      \array{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \right)_{i j}
    \\
    & =
    \left(
    \left(
      \array{
        -1 & 0
        \\
        0 & -1
      }
    \right)
    \cdot
    \left(
      \array{
        0 & -1
        \\
        1 & 0
      }
    \right)
    \right)_{i j}
    \\
    & =
    \left(
      \array{
        0 & 1
        \\
        -1 & 0
      }
    \right)_{i j}
    \\
    & =
    \omega_{i j}
  \end{aligned}
$$

=--

## Related concepts

* [[Kähler manifold]]

* [[Hilbert space]]

## References

Textbook account:

* [[Rolf Berndt]], p. 25 in: _An introduction to symplectic geometry_, Graduate Studies in Mathematics **26** (2001) [[ams:gsm-26](https://bookstore.ams.org/gsm-26), [pdf](http://www.ams.org/bookstore/pspdf/gsm-26-prev.pdf)]

Lecture notes:

* {#Boalch09} Philip Boalch, *Noncompact complex symplectic and hyperkähler manifolds* (2009) [[pdf](https://webusers.imj-prg.fr/~philip.boalch/cours09/hk.pdf), [[Boalch-NoncompactSymplectic.pdf:file]]]

* Konstantin Athanassopoulos, [p. 25](http://users.math.uoc.gr/~athanako/symplectic.pdf#page=27) of: *Notes on Symplectic Geometry*, lecture notes (2015) [[pdf](http://users.math.uoc.gr/~athanako/symplectic.pdf)]

In relation to [[Hodge theory]]:

* Wilmer Smilde, §3.2.2 in: *The Hodge Decomposition Theorem: An Alternative Approach* (2018) [[pdf](https://studenttheses.uu.nl/bitstream/handle/20.500.12932/29899/Bachelor%20Thesis%20-%20The%20Hodge%20Decomposition%20Theorem%20(Wilmer%20Smilde).pdf?sequence=2)]

For more see the references at *[[Kähler manifold]]*.

[[!redirects Kähler vector spaces]]

[[!redirects Kaehler vector space]]
[[!redirects Kaehler vector spaces]]

[[!redirects linear Kähler structure]]
[[!redirects linear Kähler structures]]

[[!redirects linear Kaehler structure]]
[[!redirects linear Kaehler structures]]

[[!redirects Kähler space]]
[[!redirects Kähler spaces]]

[[!redirects Kaehler space]]
[[!redirects Kaehler spaces]]


Kähler-Dirac operator

#Contents#
* table of contents
{:toc}

## Definition

For $(X,g)$ a [[Riemannian manifold]], the [[exterior differential]] is a [[differential operator]] on  (sections of the) [[exterior bundle]]. With respect to the induced [[inner product]] write $d^\ast$ for its [[adjoint operator]]. The _Kähler-Dirac operator_ is $d + d^\ast$.

## Properties

The Kähler-Dirac operator defines a canonical [[K-homology]] class on $X$.

## Related concepts

* [[Dolbeault-Dirac operator]]

* [[Spin^c Dirac operator]]


Kähler-Einstein manifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
=--
=--

## Idea

A _Kähler-Einstein manifold_ is a [[Riemannian manifold]] which is both a [[Kähler manifold]] as well as an [[Einstein manifold]].

## References

* Xiu-Xiong Chen, [[Simon Donaldson]], Song Sun, _Kähler-Einstein metrics and stability_ ([arXiv:1210.7494](http://arxiv.org/abs/1210.7494))

[[!redirects Kähler-Einstein manifolds]]

[[!redirects Kähler-Einstein metric]]
[[!redirects Kähler-Einstein metrics]]

[[!redirects Einstein-Kähler manifold]]
[[!redirects Einstein-Kähler manifolds]]

[[!redirects Einstein-Kähler metric]]
[[!redirects Einstein-Kähler metrics]]

[[!redirects Kahler-Einstein manifolds]]
[[!redirects Kahler-Einstein manifolds]]

[[!redirects Kahler-Einstein metric]]
[[!redirects Kahler-Einstein metrics]]

[[!redirects Einstein-Kahler manifold]]
[[!redirects Einstein-Kahler manifolds]]

[[!redirects Einstein-Kahler metric]]
[[!redirects Einstein-Kahler metrics]]


König's theorem

In [[set theory]], König's theorem, due to Julius König (Hungarian: Gyula Kőnig), is a basic result in [[cardinal]] arithmetic. It uses the [[axiom of choice]]. It is not to be confused however with [[König's lemma]], another result of combinatorial set theory, but using instead the axiom of [[dependent choice]] (and due to the son, Dénes Kőnig).

+-- {: .num_theorem}
###### Theorem
**(König)**
Let ${|X|}$ be the cardinality of a set $X$. If ${|A_i|} \lt {|B_i|}$ for all $i \in I$, then ${|\sum_i A_i|} \lt {|\prod_i B_i|}$.
=--

+-- {: .proof}
###### Proof
Suppose we have _proper_ inclusions $f_j: A_j \hookrightarrow B_j$. Choose basepoints $x_j \in B_j \setminus A_j$ and, letting $\prod_i B_i \stackrel{\pi_j}{\to} B_j$ be the product projection and $A_j \stackrel{i_j}{\to} \sum_i A_i$ the coproduct inclusion, define a map $f: \sum_j A_j \to \prod_j B_j$:

$$\array{
(\pi_j \circ f \circ i_k)(a) & = & f_j(a) & \; if \; a \in A_k \; and \; j = k \\
 & = & x_j & \; if \; a \in A_k \; and \; j \neq k
}$$

It is immediate that this $f$ is injective, and so ${|\sum_i A_i|} \leq {|\prod_i B_i|}$.

Now we prove the last inequality is strict. Suppose to the contrary that there exists a surjective function $f: \sum_i A_i \to \prod_i B_i$. The function $f_j = \pi_j \circ f \circ i_j: A_j \to B_j$ is not surjective, by the hypothesis ${|A_j|} \lt {|B_j|}$. Thus for each $j \in I$ we may choose $b_j \in B_j$ such that $b_j \notin Im(f_j)$. By supposition, there exists $a \in \sum_i A_i$ such that $f(a) = (b_j)_{j \in I}$. But $a \in A_j$ for some $j \in I$, whence $\pi_j f(a) = f_j(a)$, i.e., $b_j = f_j(a)$, and this contradicts how we chose $b_j$.
=--

One can see immediately how the axiom of choice enters the result: the product $\prod_i B_i$ might be empty otherwise, hence could not satisfy the required inequality. In fact one can _prove_ the axiom of choice from König's theorem, in the form of "every product of a family of inhabited sets is inhabited", by taking $A_i$ to be empty for each $i$.

The following important corollary deserves to be called _König's corollary_, since it is itself often referred to in the literature, but only ever as a follow-on from the above theorem. Note that we do not need to appeal to the full strength of König's theorem to prove this corollary.

+-- {: .num_cor}
###### Corollary
For any infinite cardinal $\kappa$, the cardinal $2^\kappa$ has [[cofinality]] greater than $\kappa$.
=--

+-- {: .proof}
###### Proof
We have $2^\kappa = 2^{\kappa \times \kappa} = (2^\kappa)^\kappa$. If $(\lambda_\alpha)_{\alpha \lt \kappa}$ is an increasing sequence with least upper bound $2^\kappa$, then we have surjection $\sum_{\alpha \lt \kappa} \lambda_\alpha \to 2^\kappa = (2^\kappa)^\kappa$ which immediately contradicts König's theorem.
=--

The appeal to the axiom of choice _in the form discussed above_ is unnecessary, since $2^\kappa = \prod_{i\in \kappa} 2$ is obviously inhabited. However, there are other ways in which choice-specific features make themselves known, for instance $2^\kappa$ is a cardinal (i.e. has a well-ordering). And the statement that every infinite cardinal $\kappa$ satisfies $\kappa \simeq \kappa \times \kappa$ is equivalent to the axiom of choice. And lastly, the definition of cofinality is very difficult in the absence of AC, see for instance [this blog post by Karagila](http://karagila.org/2015/cofinality-and-the-axiom-of-choice/).

[[!redirects König’s theorem]]
[[!redirects Konig's theorem]]

## Without the axiom of choice

However, it is possible to recover some of König's theorem in the absence of the axiom of choice. For sets $A$, $B$, define $(A \nsucceq B)$ to be the set of assignments $f \mapsto n(f)$ which for each partial function $f:A \rightharpoonup B$ specify an element $n(f) \in B \setminus im(f)$. This set being nonempty imiplies a strong form of "there are no partial surjections $A \twoheadrightarrow B$" (aka $A \ngeq^* B$), and the definition is as one might define $A \ngeq^* B$ using [[propositions as types]]. Conversely, if AC holds, then the $(A \nsucceq B)$ is nonempty iff $A \lt B$.

+-- {: .num_theorem}
###### Proposition
Given $n_i \in (A_i \nsucceq B_i)$ for each $i \in I$, then $\sum_i A_i \nsucceq \prod_i B_i$.
=--

+-- {: .proof}
###### Proof
Let $f: \sum_i A_i \rightharpoonup \prod_i B_i$. We define a member $m(f) \in \prod_i B_i$ as
$m(f)(i) = n_i(a \mapsto f(i,a)(i))$.
Now, note that it can't be the case that $m(f) = f(i,a)$ for any $i \in I$, $a \in A_i$, as then $f(i,a)(i) = m(f)(i) = n_i(a \mapsto f(i,a)(i)) \notin im(a \mapsto f(i,a)(i))$. This means that $m(f) \in \prod_i B_i \setminus \operatorname{im}(f)$, as required.
=--

Although the preconditions of the theorem are harder to attain, as $(A \nsucceq B)$ is nonempty much less often than $A \lt B$, it still holds in several useful cases:

+-- {: .num_theorem}
###### Proposition
The following conditions are each sufficient to construct an element $n$ of $A \nsucceq B$:

1. $m \in (A \nsucceq B')$ and an injection $i:B' \hookrightarrow B$

2. $m \in (A' \nsucceq B)$ and an injection $i:A \hookrightarrow A'$

3. $B = \mathcal{P}(A)$

4. $B$ has a wellorder $\lt$, and there is no partial surjection $A \to B$

5. An element $b \in B$ and, for each total function $f:A \to B$, a specified $m(f) \in B \setminus im(f)$

=--

+-- {: .proof}
###### Proof
1. For $f:A \rightharpoonup B$, $f$ restricts to a partial function $f':A \rightharpoonup B'$ ($f'(a) = b$ iff $f(a) = i(b)$), and $m(f')$ gives an element of $B'\setminus im(f')$, so taking $n(f') = i(m(f'))$ gives $n(f') = i(m(f')) \in B \setminus im(f)$

2. For $f:A \rightharpoonup B$, $f$ extends to a partial function $f':A' \rightharpoonup B$ (defined as $f'(i(a)) = f(a)$), and $m(f') \in B \setminus im(f) = B \setminus im(f')$, so take $n(f) = m(f')$.

3. Immediate from $1 \nsucceq 2$ and the above choiceless variant of König's theorem, but can also be proven directly. Let $f:A \rightharpoonup \mathcal{P}(A)$ be a partial function, and let $n(f) = \{a \in A \mid a \in dom(f) and a \notin f(a)\}$. Then if $f(a) = n(f)$, we have that $a \in n(f)$ iff $a \in dom(f) and a \notin f(a)$, so $a \notin dom(f)$, contradicting $f(a) = n(f)$. Therefore, $n(f) \in \mathcal{P}(A) \setminus im(f)$.

4. For $f:A \rightharpoonup B$, let $n(f) = min \{ \alpha \in B \mid \alpha \notin im(f) \}$. Then $n(f)$ is uniquely defined as $B$ is wellordered and the set is nonempty (otherwise f is a partial surjection), and $n(f) \in B \setminus im(f)$.

5. For $f:A \rightharpoonup B$, extend it to a total function $f':A \to B$ by letting $f'(x) = f(x)$ where defined and $f'(x) = b$ otherwise. Then $m(f') \in B \setminus im(f') \subseteq B \setminus im(f)$, so let $n(f) = m(f')$.

=--


Künneth theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _Künneth theorem_ is a statement relating the [[homology]] or [[cohomology]] of two objects $X$ and $Y$ with that of their [[product]] $X \times Y$.

In good situations it identifies the (co)homology of a product with the tensor product of the (co)homologies: $H^\bullet(X \times Y, E) \simeq H^\bullet(X,E) \otimes H^\bullet(Y,E)$. But in general this simple relation receives corrections by [[Tor]]-groups.

## Statement

### In ordinary homology
 {#InOrdinaryHomology}

We discuss the Künneth theorem in [[ordinary homology]].

Let $R$ be a [[ring]] and write $\mathcal{A} = R$[[Mod]] for its [[category]] of [[modules]].
(For instance $R = \mathbb{Z}$ the [[integers]], in which case $ \mathbb{Z} Mod \simeq$ [[Ab]] is the category of [[abelian groups]]. )

For $N_1, N_2 \in R Mod$ two [[modules]], write $N_1 \otimes N_2 \in R Mod$ for their [[tensor product of modules]]. Similarly for $C_\bullet, C'_\bullet \in Ch_\bullet(R Mod)$ two [[chain complexes]] of $R$-modules, write $(C \otimes_R C')_\bullet$ for their [[tensor product of chain complexes]]. Finally write $Tor^R_1(N_1,N_2)$ for the first [[Tor]]-module of $N_1$ with $N_2$.

We discuss the Künneth theorem over $R$ in stages, starting with important special cases and then passing to more general statements.

1. [Over a field](#InOrdinaryHomologyOverAField)

1. [Over a principal ideal domain](#OverAPrincipalIdealDomain)

1. [Over a general ring](#OverGeneralRings)

All these versions hold for chain homology and tensor products of general chain complexes. But under the [[Eilenberg-Zilber theorem]] all these statements apply directly in particular to the [[singular homology]] of [[topological spaces]] and their products. This is discussed below in

* [For singular homology of products of topological spaces](#InOrdinaryHomologyForTopologicalSpaces)

#### Over a field
 {#InOrdinaryHomologyOverAField}

Let $R = k$ be a [[field]].

+-- {: .num_theorem #InordinaryHomologyOverAField}
###### Theorem

For $R = k$ a [[field]], given two [[chain complexes]] of $k$-[[vector spaces]] $C_\bullet,C'_\bullet \in Ch_\bullet(k Vect)$, for each $n \in \mathbb{N}$ there is an [[isomorphism]]

$$
   \oplus_k \left(
     H_k\left(C_\bullet\right) \otimes_k H_{n-k}\left(C'_\bullet\right)
   \right)
     \stackrel{\simeq}{\to}
   H_n\left(
     C_\bullet \otimes_k C'_\bullet
   \right)
$$

between the [[chain homology]] of the [[tensor product of chain complexes]] and the [[tensor product of abelian groups]] of chain homologies.

=--

For a proof see the proof of theorem \ref{InordinaryHomology} below, of which this is a special case.

#### Over a principal ideal domain
 {#OverAPrincipalIdealDomain}

Let now $R$ be a [[ring]] which is a [[principal ideal domain]]. This may be a [[field]] as above, or for instance it may be the ring $R = \mathbb{Z}$ of [[integers]], in which case $R$-modules are equivalently just [[abelian groups]].

+-- {: .num_theorem #InordinaryHomology}
###### Theorem

For $R$ a [[principal ideal domain]], given a [[chain complex]] $C_\bullet \in Ch_\bullet(R Mod)$ of [[free modules]] over $R$ and given any other chain complex $C'_\bullet \in Ch_\bullet(R Mod)$, then for each $n \in \mathbb{N}$ there is a [[short exact sequence]] of the form

$$
   0
    \to
   \oplus_k \left(
     H_k\left(C_\bullet\right) \otimes_R H_{n-k}\left(C'_\bullet\right)
   \right)
     \to
   H_n\left(
     C_\bullet \otimes_R C'_\bullet
   \right)
     \to
   \oplus_k
     Tor_1^R\left(H_k\left(C_\bullet\right), H_{n-k-1}\left(C'_\bullet\right)\right)
     \to
   0
  \,.
$$

=--

This appears for instance as ([Hatcher, theorem 3B.5](#Hatcher)).

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

In the special case that $C'$ is concentrated in degree 0, this is the [[universal coefficient theorem]] in ordinary homology.

=--

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

In particular if all the [[Tor]]-groups on the right vanish, then the theorem asserts an [[isomorphism]]

$$
   \oplus_k \left(
     H_k\left(C_\bullet\right) \otimes_R H_{n-k}\left(C'_\bullet\right)
   \right)
     \stackrel{\simeq}{\to}
   H_n\left(
     C_\bullet \otimes_R C'_\bullet
   \right)
  \,.
$$

This is the case (assuming the [[axiom of choice]]) notably if $R$ is a [[field]] (since every module over a field is a [[free module]] -- every  [[vector space]] has a [[basis of a free module|basis]] -- and every free module is a [[flat module]]).

=--

+-- {: .proof}
###### Proof
**of theorem \ref{InordinaryHomology}**

Notice that since $C_k$ is assumed to be free, hence a [[direct sum]] of $R$ with itself, since the [[tensor product of modules]] distributes over direct sums, and since [[chain homology]] respects direct sums, we have

\[
  \label{A}
  H_n(C_k \otimes_R C')
   \simeq
  C_k \otimes_R H_{n-k}(C'_\bullet)
  \,.
\]

First consider now the special case that all the [[differentials]] of $C_\bullet$ are [[zero morphism|zero]], so that $H_k(C_\bullet) = C_k$. In this case (eq:A) yields $H_n(C_k \otimes_R C') \simeq H_k(C_\bullet) \otimes_R H_{n-k}(C'_\bullet)$ and therefore

$$
  \begin{aligned}
    H_n(C \otimes_R C')
    &\simeq
    H_n(\oplus_k C_k \otimes_R C' ))
    \\
    & \simeq
    \oplus_k H_n( C_k \otimes_R C' )
    \\
    & \simeq
    \oplus_k H_k(C_\bullet) \otimes_R H_{n-k}(C'_\bullet)
  \end{aligned}
  \,.
$$

Since $H_k(C) = C_k$ is a [[free module]] by assumption, it has no [[Tor]]-terms (by the discussion there) and hence this is the statement to be shown.

Now let $C_\bullet$ be a general chain complex of free modules. Notice that for each $n$ the [[cycle]]-[[chain]]-[[boundary]]-[[short exact sequence]]

$$
  0 \to Z_{n-k} \hookrightarrow C_{n-k} \stackrel{\partial_{n-k-1}}{\to} B_{n-1} \to 0
$$

[[split exact sequence|splits]] due to the assumption that $C_n$ is a [[free module]], and hence (as discussed at _[[split exact sequence]]_) that it exhibits a [[direct sum]] decomposition $C_n \simeq Z_n \oplus B_{n-1}$. Since the [[tensor product of modules]] distributes over direct sum, it follows that tensoring with any $C'_k$ yields another [[short exact sequence]]

$$
  0
   \to
  Z_{n-k} \otimes_R C'_k
   \to
  C_{n-k} \otimes_R C'_k
    \to
  B_{n-k-1} \otimes_R C'_k
    \to
  0
  \,.
$$

This means that if we regard the graded modules $Z_\bullet$ and $B_\bullet$ of chains and of boundaries as chain complexes with zero-differentials, then we have a short exact sequence of chain complexes

$$
  0 \to Z_\bullet \otimes_R C'_\bullet \to C_\bullet \otimes_R C'_\bullet \to B_{\bullet-1}\otimes_R C'_\bullet  \to 0
  \,.
$$

This induces its [[homology long exact sequence]] of the form

$$
  \cdots
   \to
  H_n( Z \otimes_R C')
   \to
  H_n( C \times_R C' )
   \to
  H_{n-1}( B \otimes_R C' )
   \to
  H_{n-1}( Z \otimes_R C' )
   \to
  \cdots
  \,.
$$

Here the terms involving the complexes $B$ and $Z$ of boundaries and cycles may be evalutated, since these have zero differentials, via the special case discussed at the beginning of this proof to yield the [[long exact sequence]]

$$
  \cdots
  \stackrel{i_n \otimes_R C'}{\to}
  \oplus_k ( Z_k \otimes_R H_{n-k}(C') )
  \to
  H_n(C \otimes_R C')
  \to
  \oplus_{k}( B_k \otimes_R H_{n-k-1}(C') )
   \stackrel{i_{n-1}}{\to}
  \oplus_k ( Z_k \otimes_R H_{n-k-1}(C') )
   \to
   \cdots
  \,,
$$

where $i_n \coloneqq H_n(i \otimes C')$ is the morphism induced from the inclusion $i \colon B_\bullet \hookrightarrow Z_\bullet$ of boundaries into cycles.

This means that by quotienting out an image on the left and a kernel on the right, we obtain a [[short exact sequence]]

$$
  0
   \to
  coker(i_n)
   \to
  H_n(C \otimes_R C')
   \to
  ker(i_{n-1})
   \to
  0
  \,.
$$

Since the [[tensor product of modules]] is a [[right exact functor]] it commutes with the [[cokernel]] on the left, as does the formation of [[direct sums]], and so we have

$$
  coker\left( \oplus_k \left(
   B_k \otimes_R H_{n-k}(C')
   \stackrel{i_k \otimes H_{n-k}\left(C'\right)}{\to}
   Z_k \otimes_R H_{n-k}\left(C'\right)
  \right)\right)
  \simeq
  \oplus_k \left(
    \left(
      coker\left(B_k \stackrel{i_k}{\to} Z_k\right)
      \otimes_R
      H_{n-k}(C')
    \right)
  \right)
  \simeq
  \oplus_k
  \left(
    H_k\left(C\right) \otimes_R H_{n-k}\left(C'\right)
  \right)
  \,.
$$

This is the left term in the short exact sequence to be shown. For the right term the analogous argument does not quite go through, because tensoring is not in addition a [[left exact functor]], in general. The failure to be so is precisely measured by the [[Tor]]-module:

Notice that by the assumption that $C_n$ is free and using the fact (discussed at _[[principal ideal domain]]_) that over our $R$ the submodules $B_n, Z_n \hookrightarrow C_n$ are themselves free modules, the defining [[short exact sequence]] $0 \to B_n \stackrel{i_n}{\to} Z_n \to H_n(C) \to 0$ exhibits a [[projective resolution]] of $H_n(C)$.  Therefore by definition of [[Tor]] we have

$$
  Tor_1(H_k(C), H_{n-k}(C'))
  \simeq
  ker\left(
   i_k \otimes H_{n-k}(C')
  \right)
  \,.
$$

This identifies the term on the right of the exact sequence to be shown.

=--

#### Over general rings
 {#OverGeneralRings}

For $R$ any ring, there is a [[spectral sequence]] converging to the homology of the tensor product, whose second page involves all the [[Tor]]-modules: the _[[Künneth spectral sequence]]_

$$
  E_{p,q}^2
   \coloneqq
  \oplus_{k_1 + k_2 = q}
  Tor_p^{R Mod}( H_{k_1}(C), H_{k_2}(C'))
  \Rightarrow
  H_{p+q}(C \otimes C')
$$

(...)

For instance ([Williams, section 4.2](#Williams)).

### In ordinary cohomology
 {#InOrdinaryCohomology}

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

Let $R$ be a [[ring]] and let $X,Y$ be [[topological spaces]]. If the [[ordinary cohomology|ordinary]] [[cohomology ring]] $H^k(Y,R)$ is a [[finitely generated module|finitely generated]] [[free module]] over $R$, then the comparison map ("cross product")

$$
  H^\bullet(X,R)
  \otimes_R
  H^\bullet(Y,R)
   \longrightarrow
  H^\bullet(X\times Y, R)
$$

(from the [[tensor product]] of [[cohomology rings]] to the cohomology ring of the [[product space]]) is an [[isomorphism]].

=--

e.g. [Hatcher (2001), theorem 3.15](#Hatcher), and in more generality: [Spanier (1966), section 5.5, theorem 11](#Spanier77), [Weibel (1994), Thm. 3.6.1](#Weibel94)

### In Generalised cohomology theories

The Künneth theorem for [[generalised cohomology theories]] is a special case of the [[universal coefficient theorem]].  Let $E$ be a [[ring spectrum]], $X$ and $Y$ two [[spectra]].  Then we define a new [[cohomology theory]] as the $E$-[[module spectrum]] $G \coloneqq F(Y, E)$ where $F(-,-)$ is the [[function spectrum]] (that is, the [[internal hom]] in the category of [[spectra]]).  The ([[reduced cohomology|reduced]]) cohomology of $X$ in this cohomology theory is thus given by:

$$
G^*(X) = [X,F(Y,E)] = [X \wedge Y,E] = E^*(X \wedge Y)
$$

A [[universal coefficient theorem]] gives a way of computing $G^*(X)$ from knowledge of $E^*(X)$ and $G^*(pt)$.  In this case, $G^*(pt) = E^*(Y)$ so our initial data is $E^*(X)$ and $E^*(Y)$.

For more details, see the page on the _[[universal coefficient theorem]]_.

## Examples

### For singular homology of products of topological spaces
 {#InOrdinaryHomologyForTopologicalSpaces}

All of these statements have their analogs for [[singular homology]] of [[topological spaces]] $X , Y \in $ [[Top]]: by the [[Eilenberg-Zilber theorem]] there is a [[quasi-isomorphism]]

$$
  C_\bullet(X \times Y)
  \simeq_{qi}
  C_\bullet(X) \otimes C_\bullet(Y)
$$

between the [[singular chain complex]] of the [[product space]] and the [[tensor product of chain complexes]] of the separate singular chain complexes. Hence in particular there are isomorphisms of [[singular homology]]

$$
  H_n(X \times Y) \simeq
  H_n(C_\bullet(X) \otimes C_\bullet(Y))
  \,.
$$

Using this in the above statements of the Künneth theorem yields directly the Künneth theorem for singular homology of topological spaces.

## Related concepts

* [[Eilenberg-Zilber theorem]]

* [[universal coefficient theorem]]

## References

### In ordinary (co)homology

The original articles are

* H. Künneth, _Über die Bettischen Zahlen einer Produktmannigfaltigkeit_ Math. Ann. , 90 (1923) pp. 65–85

* H. Künneth, _Über die Torsionszahlen von Produktmannigfaltigkeiten_ Math. Ann. , 91 (1924) pp. 125–134

Textbook accounts:

* {#Spanier66} [[Edwin Spanier]], _Algebraic topology_, McGraw Hill (1966), Springer (1982) [[doi:10.1007/978-1-4684-9322-1](https://link.springer.com/book/10.1007/978-1-4684-9322-1)]

* {#Hatcher} [[Allen Hatcher]], Section [3.B](http://www.math.cornell.edu/~hatcher/AT/ATch3.4.pdf) of: *[Algebraic topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)* (2001)

* {#Weibel94} [[Charles Weibel]], Section 3.6 in: *[[An introduction to homological algebra]]*,  Cambridge Studies in Adv. Math. **38**, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)]

Lecture notes:

* Rob Thompson, _Products and the Künneth theorem_ ([pdf](http://math.hunter.cuny.edu/~rthompso/topology_notes/chapter%20nine.pdf))

* {#Williams} [[Brandon Williams]], Section 4.2 in _Spectral sequences_ ([pdf](http://www.math.sunysb.edu/~mbw/notes/orals/Spectral%20Sequences.pdf))

* [[James Milne]], section 22 of: _[[Lectures on Étale Cohomology]]_

### In generalized (co)homology

* [[Stefan Schwede]], part II, section 6.2 of _[[Symmetric spectra]]_ (2012)

See also the corresponding references at _[[universal coefficient theorem]]_.

In the context of [[parameterized spectra]]

* {#MaySigurdsson06} [[Peter May]], J. Sigurdsson, section 22.7 of _[[Parametrized Homotopy Theory]]_, 2006

[[!redirects Künneth formula]]

[[!redirects Kuenneth theorem]]
[[!redirects Kuenneth formula]]
[[!redirects Kunneth theorem]]
[[!redirects Kunneth formula]]
[[!redirects Künneth isomorphism]]

[[!redirects Künneth theorem for complexes]]


L-complete module

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

#Contents#
* table of contents
{:toc}

##Idea

For a [[commutative ring]] $R$ and a [[finitely generated module|finitely generated]] [[ideal]] $I$, the [[category]] of $I$-[[adic completion|adically complete modules]] is not [[abelian category|abelian]]. The category of _$L$-complete modules_ is, in some precise sense, the smallest abelian [[subcategory]] of [[RMod]] which contains the $I$-[[adic completion|adically complete modules]]. Roughly speaking, it is obtained by formally adjoining [[cokernels]] of [[morphisms]] between $I$-[[adic completion|adically complete modules]].

## Motivation and definition##

Let $R$ be a [[commutative ring]] and $I \subset R$ a [[finitely generated module|finitely generated]] [[ideal]]. The $I$-[[adic completion]] of an $R$-[[module]] $M$ is the [[inverse limit]]

$$
  \hat M
  =
  \lim_{\leftarrow} M/I^n M
  \,.
$$

Let $i:M\rightarrow \hat M$ be the canonical map.

\begin{definition}
An $R$-module $M$ is said to be

1. _quasi-complete_ if $i$ is [[surjective]]

1. _separated_ if $i$ is [[injective]]

1. _complete_ if $i$ is [[bijective]].

\end{definition}

The completion functor is idempotent (warning: this is not true in general if $I$ is not finitely generated), hence induces an [[idempotent monad]] on the category [[RMod]], and is [[left adjoint]] to the inclusion of the category $\widehat{R Mod}$ of complete modules into all modules. Hence, $\widehat{R Mod}$ is a [[reflective subcategory]] of $R Mod$. It is not, however, an [[abelian subcategory]]: the [[quotient]] of two complete modules is not, in general, complete.

A key observation is that the completion functor is not [[right exact functor|right exact]] when seen as an [[endofunctor]] of $R Mod$. Hence, let $L$ be the 0th [[left derived functor]] of the completion functor, which is right exact by definition.

\begin{definition}
An $R$-module $M$ is called $L$-complete if the canonical map
$$M\rightarrow L(M)$$
is an isomorphism.
\end{definition}
\begin{proposition}
Let $M$ be an $R$-module.

* If $M$ is finitely generated, then $\hat M=L(M)$.
* If $M$ is complete, then it is $L$-complete.
* The canonical map $M\rightarrow \hat M$ factor through the canonical map $M\rightarrow L(M)$, and the map $L(M)\rightarrow \hat M$ is surjective. In particular, an $L$-complete module is always quasi-complete.

\end{proposition}

Let $(m_i)_{i\in \mathbb{N}}$ be a sequence of elements in $M$ such that for all $n$, all but finitely many of the $m_i$'s belong to $I^n$. Then $M$ is complete if and only if for every such sequence, the partial sums $\sum_{i\leq k} m_i$ have a well-defined limit in $M$. The above proposition shows an $L$-complete needs not to be separated, hence, this limit might not exists. However, informally speaking, $L$-complete modules are those for which we can still make sense of some sort of limit.

\begin{proposition}([Salch 20](#Salch20))

The [[category]] of $L$-complete modules is the smallest abelian [[full subcategory]] of [[RMod]] which contains the $I$-[[adic completion|adically complete modules]].

\end{proposition}

##References##

* {#Salch20} [[Andrew Salch]], Approximation of subcategories by abelian subcategories ([arXiv:2011.01827](https://arxiv.org/abs/2011.01827))

* [[Charles Rezk]], _Analytic completion_ ([pdf](http://www.math.uiuc.edu/~rezk/analytic-paper.pdf))

[[!redirects L-complete modules]]


L-finite category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
=--
=--

# $L$-finite categories
* table of contents
{: toc}

## Definition
 {#Definition}

\begin{proposition}\label{CharacterizationsOfLFiniteLimits}
**(characterizations of L-finite limits)** \linebreak
A [[category]] $C$ is **$L$-finite** if the following equivalent conditions hold, which are all equivalent:

* The [[terminal object]] of the [[functor category]] $[C,Set]$ to [[Set]] is ($\omega$-)[[compact object|compact]].

* $C$-[[limits]] [[commutativity of limits and colimits|commute]] with [[filtered colimits]] in [[Set]].

* $C$ has an [[initial functor|initial]] [[finitely generated category|finitely generated]] [[subcategory]].

* $C$ admits an [[initial functor]] from a [[finite category]].[^1]

* $C$-limits lie in the [[saturated class of limits|saturation]] of the class of [[finite limits]].

\end{proposition}
([Paré 1990, p. 740 (10 of 16), around Prop. 7](#Pare90))

\begin{remark}\label{RelationTkKFiniteness}
**(relation to K-finite sets)** \linebreak
The notion of L-finite category (Def. \ref{CharacterizationsOfLFiniteLimits}) is a sort of [[vertical categorification|categorification]] of the notion of [[K-finite set]]:

* A set $X$ is $K$-finite if the [[top element]] $1 \in \Omega^X$ belongs to the closure of the [[singletons]] under finite [[unions]].

* A category $C$ is $L$-finite if the terminal object $1\in Set^C$ belongs to the closure of the [[representable functor|representables]] under [[finite colimits]].

\end{remark}

In [Paré 1990, p. 741 (11 0f 16)](#Pare90) this observation is attributed to [[Richard Wood]].

## Examples

\begin{example}\label{CategoriesWithInitialObjectAreLFinite}
**(categories with initial objects are L-finite)** \linebreak
  Any category $\mathcal{C}$ with an [[initial object]] $\varnothing \,\in\, \mathcal{C}$ is L-finite, with the inclusion of the [[terminal category]] mapping to this initial object
$\{\varnothing\} \xhookrightarrow{\;} \mathcal{C}$
being an [[initial functor]] (by [this exp.](final+functor#InclusionOfATerminalObjectIsFinal)) as required by Def. \ref{CharacterizationsOfLFiniteLimits}.
\end{example}

## Related concepts

* [[finite category]]

* [[filtered colimit]]

* [[sound doctrine]]

## References

* {#Pare90} [[Robert Paré]], Section 3 of: *Simply connected limits*.  Can. J. Math., Vol. XLH, No. 4, 1990, pp. 731-746 ([doi:10.4153/CJM-1990-038-6](https://doi.org/10.4153/CJM-1990-038-6))

[^1]: There is a typo in [Paré Prop. 7](#Pare90) in the statement of this equivalence: it says "[[final functor|final]]" instead of "[[initial functor|initial]]".

[[!redirects L-finite category]]
[[!redirects L-finite categories]]
[[!redirects L-finite limit]]
[[!redirects L-finite limits]]
[[!redirects L-finite colimit]]
[[!redirects L-finite colimits]]


L-function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_$L$-functions_ are certain [[meromorphic functions]] generalizing the
[[Riemann zeta function]]. They are typically defined on parts of the [[complex plane]] by a [[power series]] expressions -- called the _$L$-series_ --  which [[convergence|converges]] in that region,  and then meromorphically extended to all of the [[complex plane]] by [[analytic continuation]].

The most canonically defined class of examples of L-functions are the _[[Artin L-functions]]_ defined for any [[Galois representation]] $\sigma \colon Gal \longrightarrow GL_n(\mathbb{C})$ as the [[Euler products]] of, essentially, the [[characteristic polynomials]] of all the [[Frobenius homomorphisms]] acting via $\sigma$.

Another common example of an L-function comes from [[modular forms]]. If we have a cusp form $f$ of weight $k$, with Fourier expansion

$$f(z)=\sum_{n=1}^{\infty}a_{n}e^{2\pi i n z}$$

the associated L-function $L_{f}(s)$ is given by

$$L_{f}(s)=\sum_{n}^{\infty}a_{n}n^{-s}.$$

The L-function $L_{f}(s)$ may also be obtained as the [[Mellin transform]] of $f$. This is an example of an [[automorphic L-function]], which more generally is not as easy to construct.

Most other kinds of L-functions are such as to reproduces these [[Artin L-functions]] from more "arithmetic" data:

1. for 1-dimensional [[Galois representations]] $\sigma$ (hence for $n = 1$) [[Artin reciprocity]] produces for each $\sigma$ a [[Dirichlet character]], or more generally a [[Hecke character]] $\chi$, and therefrom is built a _[[Dirichlet L-function]]_ or _[[Hecke L-function]]_ $L_\chi$, respectively, which equals the corresponding [[Artin L-function]] $L_\sigma$;

1. for general $n$-dimensional [[Galois representations]] $\sigma$ the [[conjecture]] of [[Langlands correspondence]] states that there is an [[automorphic representation]] $\pi$ corresponding to $\sigma$ and an _[[automorphic L-function]]_ $L_\pi$ built from that, which equalso the [[Artin L-function]] $L_\sigma$.

L-functions typically satisfy [[analogy|analogs]] of all the special properties enjoyed by the [[Riemann zeta function]], such as satisfying a
"[[functional equation]]" which asserts invariance under modular transformations of the parameter.

The generalized [[Riemann conjecture]] is concerned with zeros of the _[[Dedekind zeta function]]_ for which the L-series (the [[Dirichlet L-function]]) is complicated from the classical Riemann case by the presence of the additional parameter, the Dirichlet character.

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## Examples

* [[Dirichlet L-function]]

* [[Artin L-function]]

* [[Hecke L-function]]

* [[Hasse-Weil L-function]]

## Related entries

* [[special values of L-functions]]

* [[S-duality]], [[automorphic form]], [[Langlands program]],

* [[automorphic L-function]]

* [[Beilinson conjecture]]

* [[motivic L-function]]

* [[explicit formulae (L-function)]]

## References

* {#Gelbhart84} [[Stephen Gelbart]], section C starting on p. 14 (190) of _An elementary introduction to the Langlands program_,  Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))

* E. Kowalski, first part of _Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures_ ([pdf](http://www.math.ethz.ch/~kowalski/lectures.pdf))

* {#GoldfeldHundley11} [[Dorian Goldfeld]], [[Joseph Hundley]], chapter 2 of _Automorphic representations and L-functions for the general linear group_, Cambridge Studies in Advanced Mathematics 129, 2011 ([pdf](https://www.maths.nottingham.ac.uk/personal/ibf/text/gl2.pdf))

* wikipedia [L-function](http://en.wikipedia.org/wiki/L-function), [Dirichlet L-function](http://en.wikipedia.org/wiki/Dirichlet_L-function), [special values of L-function](http://en.wikipedia.org/wiki/Special_values_of_L-functions), [generalized Riemann hypothesis](http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis), [Artin L-function](http://en.wikipedia.org/wiki/Artin_L-function), [equivariant L-function](http://en.wikipedia.org/wiki/Equivariant_L-function), [functional equation](en.wikipedia.org/wiki/Functional_equation_(L-function)), [modularity theorem](http://en.wikipedia.org/wiki/Modularity_theorem)

* [[Terrence Tao]]'s blog: [Distinguished Lecture Series III: Shou-wu Zhang, "Triple L-series and effective Mordell conjecture"](http://terrytao.wordpress.com/2007/05/04/distinguished-lecture-series-iii-shou-wu-zhang-%E2%80%9Ctriple-l-series-and-effective-mordell-conjecture%E2%80%9D/)

Some history is in

* {#Cogdell12} James W. Cogdell, _L-functions and non-abelian class eld theory, from
Artin to Langlands_, 2012 ([pdf](https://people.math.osu.edu/cogdell.1/Artin-www.pdf))

[[!redirects L-functions]]


L-infinity algebra > history

< [[L-infinity algebra]]
[[!redirects L-infinity algebra -- history]]


L-infinity algebras in physics

### General

The concept of [[L-∞ algebras]] as [[graded vector spaces]] equipped with $n$-ary brackets satisfying a generalized [[Jacobi identity]] is due to:

* {#Stasheff92} [[Jim Stasheff]], _Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras_, in _Quantum groups_ Number 1510 in Lecture Notes in Math. Springer, Berlin, 1992 ([doi:10.1007/BFb0101184](https://doi.org/10.1007/BFb0101184)).

* {#LadaStasheff92} [[Tom Lada]], [[Jim Stasheff]],  _Introduction to sh Lie algebras for physicists_, Int. J. Theo. Phys. **32** (1993) 1087-1103 \[doi:10.1007/BF00671791, [arXiv:hep-th/9209099](http://arxiv.org/abs/hep-th/9209099)\]

* {#LadaMarkl94} [[Tom Lada]], [[Martin Markl]], _Strongly homotopy Lie algebras_, Communications in Algebra **23** 6  (1995) \[doi:10.1080/00927879508825335, [arXiv:hep-th/9406095](http://arxiv.org/abs/hep-th/9406095)\]

* {#Kontsevich97} [[Maxim Kontsevich]], Section 4.3 of: *Deformation quantization of Poisson manifolds, I*,  Lett. Math. Phys. **66** (2003) 157-216 ([arXiv:q-alg/9709040](https://arxiv.org/abs/q-alg/9709040), [doi:10.1023/B:MATH.0000027508.00421.bf](https://doi.org/10.1023/B:MATH.0000027508.00421.bf))

At least [Stasheff 92](#Stasheff92) was following [Zwiebach 92](#Zwiebach92), who had observed that the [[n-point functions]] in [[closed string field theory]] equip the [[BRST complex]] of the [[closed string|closed]] [[bosonic string]] with $L_\infty$-algebra structure (see further reference [there](string+field+theory#ReferencesHomotopyAlgebra)). Zwiebach, in turn, was following the [[BV-formalism]] due to [Batalin-Vilkovisky 83](#BatalinVilkovisky83), [Batakin-Fradkin 83](#BatakinFradkin83).

See also at _[L-infinity algebra -- History](L-infinity-algebra#History)_.

Discussion in terms of [[cofibrant resolutions]] of the [[Lie operad]]:

* {#KrizMay95} [[Igor Kriz]], [[Peter May]], p. 28 of: _Operads, algebras, modules and motives_, Astérisque __233__, Société Mathématique de France (1995) ([pdf](http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf), [numdam:AST_1995__233__1_0](http://www.numdam.org/item/?id=AST_1995__233__1_0))

* {#LodayVallette12} [[Jean-Louis Loday]], [[Bruno Vallette]], Sec. 3.2.12 and onwards in: *Algebraic Operads*, Grundlehren der mathematischen Wissenschaften **346**, Springer 2012 ([ISBN 978-3-642-30362-3](https://www.springer.com/gp/book/9783642303616), [pdf](http://irma.math.unistra.fr/~loday/PAPERS/LodayVallette.pdf))

A historical survey is

* {#Stasheff16} [[Jim Stasheff]], _Higher homotopy structures, then and now_, talk at _[Opening workshop](https://www.mpim-bonn.mpg.de/node/6356)_ of _[Higher Structures in Geometry and Physics](https://www.mpim-bonn.mpg.de/node/5883)_, MPI Bonn 2016 ([[StasheffHomotopyStructuresReview.pdf:file]], [arXiv:1809.02526](https://arxiv.org/abs/1809.02526))

See also

* {#Daily04} Marilyn Daily, _$L_\infty$-structures_, PhD thesis, 2004 ([web]( http://www.lib.ncsu.edu/resolver/1840.16/5282))

* Klaus Bering, [[Tom Lada]], _Examples of Homotopy Lie Algebras_ Archivum Mathematicum ([arXiv:0903.5433](http://arxiv.org/abs/0903.5433))

Comprehensive survey with emphasis on [[L-infinity algebra cohomology]]:

* Ben Reinhold, *$L_\infty$-algebras and their cohomology*, Emergent Scientist **3** 4 (2019) [[doi:10.1051/emsci/2019003](https://doi.org/10.1051/emsci/2019003)]

Review for the special case of [[Lie 2-algebras]] with emphasis on the perspective of [[categorification]]:

* [[John Baez]], [[Alissa Crans]], _Higher-dimensional algebra VI: Lie 2-algebras_, [TAC](http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html) 12, (2004), 492--528. ([arXiv:math/0307263](http://arxiv.org/abs/math/0307263))

### As models for rational homotopy types

That $L_\infty$-algebras are models for [[rational homotopy theory]] is implicit in [Quillen 69](rational+homotopy+theory#Quillen69) (via their [equivalence with dg-Lie algebras](model+structure+on+dg-Lie+algebras#RectificationResolution)) and was made explicit in [Hinich 98](#Hinich98). Exposition is in

* {#BuijsFelixMurillo12} [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], section 2 of _$L_\infty$-rational homotopy of mapping spaces_ ([arXiv:1209.4756](https://arxiv.org/abs/1209.4756)),  published as _$L_\infty$-models of based mapping spaces_,  J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524.

and genralization to non-[[connected topological space|connected]] rational spaces is discussed in

* {#BuijsMurillo12} [[Urtzi Buijs]], [[Aniceto Murillo]], _Algebraic models of non-connected spaces and homotopy theory of $L_\infty$-algebras_, Advances in Mathematics 236 (2013): 60-91. ([arXiv:1204.4999](https://arxiv.org/abs/1204.4999))

### $L_\infty$-algebras in physics
 {#ReferencesInPhysics}

The following lists, mainly in chronological order of their discovery, [[L-∞ algebra]] structures appearing in [[physics]], notably in [[supergravity]], [[BV-BRST formalism]], [[deformation quantization]], [[string theory]], higher [[Chern-Simons theory]]/[[AKSZ sigma-models]] and [[local field theory]].

For more see also at _[[higher category theory and physics]]_.

#### In supergravity

In their equivalent [[formal dual]] guise of [[Chevalley-Eilenberg algebras]] (see [above](#ReformulationInTermsOfSemifreeDGAlgebra)), $L_\infty$-algebras of [[finite type]] -- in fact [[super L-∞ algebras]] -- appear in pivotal role in the [[D'Auria-Fré formulation of supergravity]] at least since

* {#DAuriaFre}  [[Riccardo D'Auria]], [[Pietro Fré]], _[[GeometricSupergravity.pdf:file]]_, Nuclear Physics B201 (1982) 101-140

In the [[supergravity]] literature these CE-algebras are referred to as "FDA"s. This is short for "free differential algebra", which is a slight misnomer for what in mathematics are called [[semifree dga]]s (or sometimes "quasi-free" dga-s).

The translation of [[D'Auria-Fré formulation of supergravity|D'Auria-Fré formalism]] to explicit ([[super L-∞ algebra|super]]) $L_\infty$-algebra language is made in

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], example 5 in section 6.5.1, p. 54 of _[[schreiber:L-infinity algebra connections|$L_\infty$ algebra connections and applications to String- and Chern-Simons n-transport]]_, in: *Quantum Field Theory*, Birkhäuser (2009) 303-424 \[arXiv:0801.3480, [doi:10.1007/978-3-7643-8736-5_17](https://doi.org/10.1007/978-3-7643-8736-5_17)\]

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_, International Journal of Geometric Methods in Modern Physics Volume 12, Issue 02 (2015) 1550018 ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

connecting them to the [[higher WZW terms]] of the [[Green-Schwarz sigma models]] of fundamental [[super p-branes]] ([[schreiber:The brane bouquet]]).

Further exposition of this includes

* [[Urs Schreiber]],  _[Homotopy Lie n-algebras in Supergravity](https://www.physicsforums.com/insights/homotopy-lie-n-algebras-supergravity/)_, PhysicsForums-Insights 2015

See also at _[[supergravity Lie 3-algebra]]_, and _[[supergravity Lie 6-algebra]]_.

Notice that there is a _different_ concept of "Filipov [[n-Lie algebra]]" suggested in ([Bagger-Lambert 06](BLG+model#BaggerLambert06)) to play a role in the description of the [[conformal field theory]] in the [[near horizon limit]] of [[black p-branes]], notably the [[BLG model]] for the conformal [[worldvolume]] theory on the [[M2-brane]] .

A realization of thse "Filippov $3$-Lie algebras" as 2-term $L_\infty$-algebras ([[Lie 2-algebras]]) equipped with a binary [[invariant polynomial]] ("metric Lie 2-algebras") is in

* Sam Palmer, [[Christian Saemann]], section 2 of _M-brane Models from Non-Abelian Gerbes_, JHEP 1207:010, 2012 ([arXiv:1203.5757](http://arxiv.org/abs/1203.5757))

* {#SaemannRitter13} [[Patricia Ritter]], [[Christian Saemann]], section 2.5 of _Lie 2-algebra models_, JHEP 04 (2014) 066 ([arXiv:1308.4892](http://arxiv.org/abs/1308.4892))

based on

* {#MFFMER08} [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Elena Méndez-Escobar]], [[Patricia Ritter]], _On the Lie-algebraic origin of metric 3-algebras_, Commun.Math.Phys.290:871-902,2009 ([arXiv:0809.1086](http://arxiv.org/abs/0809.1086))

See also

* [[José Figueroa-O'Farrill]], section _Triple systems and Lie superalgebras_ in _M2-branes, ADE and Lie superalgebras_, talk at IPMU 2009 ([pdf](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf))

[[!include C-field gauge algebra -- references]]

#### In BV-BRST formalism
 {#ReferencesBVBRSTFormalism}

The introduction of [[BV-BRST complexes]] as a model for the [[derived critical locus]] of the [[action functionals]] of [[gauge theories]] is due to

* {#BatalinVilkovisky81} [[Igor Batalin]], [[Grigori Vilkovisky]],  _Gauge Algebra and Quantization_, Phys. Lett. B 102 (1981) 27–31. doi:10.1016/0370-2693(81)90205-7

* {#BatalinVilkovisky83} [[Igor Batalin]], [[Grigori Vilkovisky]], _Feynman rules for reducible gauge theories_, Phys. Lett. B 120 (1983) 166-170.
doi:10.1016/0370-2693(83)90645-7

* {#BatakinFradkin83} [[Igor Batalin]], [[Efim Fradkin]], _A generalized canonical formalism and quantization of reducible gauge theories_, Phys. Lett. B122 (1983) 157-164.

* {#BatalinVilkovisky83b} [[Igor Batalin]], [[Grigori Vilkovisky]],  _Quantization of Gauge Theories with Linearly Dependent Generators_, Phys. Rev. D 28 (10): 2567–258 (1983) doi:10.1103/PhysRevD.28.2567.  Erratum-ibid. 30 (1984) 508 doi:10.1103/PhysRevD.30.508

as reviewed in

* {#HenneauxTeitelboim92} [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_, Princeton University Press 1992. xxviii+520 pp.

* [[Joaquim Gomis]], J. Paris, S. Samuel, _Antibrackets, Antifields and Gauge Theory Quantization_ ([arXiv:hep-th/9412228](http://arxiv.org/abs/hep-th/9412228))

The understanding that these [[BV-BRST complexes]] mathematically are the [[formal dual]] [[Chevalley-Eilenberg algebra]] of a [[derived L-∞ algebroid]] originates around

* {#Stasheff96} [[Jim Stasheff]], _Homological Reduction of Constrained Poisson Algebras_, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 ([arXiv:q-alg/9603021](http://arxiv.org/abs/q-alg/9603021), [Euclid](https://projecteuclid.org/euclid.jdg/1214459757))

* {#Stasheff97} [[Jim Stasheff]], _The (secret?) homological algebra of the Batalin-Vilkovisky approach_ ([arXiv:hep-th/9712157](http://arxiv.org/abs/hep-th/9712157))

Discussion in terms of homotopy [[Lie-Rinehart pairs]] is due to

* {#Kjeseth01} [[Lars Kjeseth]], _Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs_, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. ([Euclid](https://projecteuclid.org/euclid.hha/1140370269))

The [[L-∞ algebroid]]-structure is also made explicit in ([def. 4.1 of v1](http://arxiv.org/abs/0910.4001v1)) of ([Sati-Schreiber-Stasheff 09](#SatiSchreiberStasheff09)).

The extraction of [[L-infinity algebras|$L_\infty$-algebras]] from the formal neighbourhood of a [[derived critical locus]] is maybe first made explicit in:

* [[Maxim Grigoriev]], [[Dmitry Rudinsky]], *Notes on the $L_\infty$-approach to local gauge field theories* $[$[arXiv2303.08990](https://arxiv.org/abs/2303.08990)$]$

#### In string field theory

The first _explicit_ appearance of $L_\infty$-algebras in theoretical physics is the $L_\infty$-algebra structure on the [[BRST complex]] of the [[closed string|closed]] [[bosonic string]] found in the context of closed bosonic [[string field theory]] in

* {#Zwiebach92} [[Barton Zwiebach]], _Closed string field theory: Quantum action and the B-V master equation_ , Nucl.Phys. B390 (1993) 33 ([arXiv:hep-th/9206084](http://arxiv.org/abs/hep-th/9206084))

* [[Jim Stasheff]], _Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space_ Talk given at the _Conference on Topics in Geometry and Physics_ (1992) ([arXiv:hep-th/9304061](http://arxiv.org/abs/hep-th/9304061))

Generalization to open-closed bosonic string field theory yields [[L-∞ algebra]] interacting with [[A-∞ algebra]]:

* [[Hiroshige Kajiura]], _Homotopy Algebra Morphism and Geometry of Classical String Field Theory_ (2001) ([arXiv:hep-th/0112228](http://arxiv.org/abs/hep-th/0112228))

* {#KajiuraStasheff04} [[Hiroshige Kajiura]], [[Jim Stasheff]], _Homotopy algebras inspired by classical open-closed string field theory_, Comm. Math. Phys. 263 (2006) 553–581 (2004) ([arXiv:math/0410291](http://arxiv.org/abs/math/0410291))

* {#Markl} [[Martin Markl]], _Loop Homotopy Algebras in Closed String Field Theory_ (1997) ([arXiv:hep-th/9711045](http://arxiv.org/abs/hep-th/9711045))

See also

* [[Jim Stasheff]], _Higher homotopy algebras: String field theory and Drinfeld's quasiHopf algebras_, proceedings of _International Conference on Differential Geometric Methods in Theoretical Physics_, 1991 ([spire](https://inspirehep.net/record/327712))

For more see at _[string field theory -- References -- Relation to A-infinity and L-infinity algebras](string%20field%20theory#ReferencesHomotopyAlgebra)_.

#### In deformation quantization

The general solution of the [[deformation quantization]] problem of [[Poisson manifolds]] due to

* [Kontsevich 97](#Kontsevich97)

makes crucial use of [[L-∞ algebra]]. Later it was understood that indeed [[L-∞ algebras]] are equivalently the universal model for infinitesimal [[deformation theory]] (of anything), also called [[formal moduli problems]]:

* {#Hinich98} [[Vladimir Hinich]], _DG coalgebras as formal stacks_ ([arXiv:9812034](http://arxiv.org/abs/math/9812034))

* {#Lurie} [[Jacob Lurie]], _[[Formal Moduli Problems]]_

* {#Pridham} [[Jonathan Pridham]], _Unifying derived deformation theories_, Adv. Math. 224 (2010), no.3, 772-826 ([arXiv:0705.0344](http://arxiv.org/abs/0705.0344))

#### In heterotic string theory

Next it was again $L_\infty$-algebras of [[finite type]] that drew attention. It was eventually understood that the [[string structures]] which embody a refinement of the [[Green-Schwarz anomaly cancellation]] mechanism in [[heterotic string theory]] have a further smooth refinement as [[G-structures]] for the [[string 2-group]], which is the [[Lie integration]] of a [[Lie 2-algebra]] called the _[[string Lie 2-algebra]]_. This is due to

* {#BCSS} [[John Baez]], [[Alissa Crans]], [[Urs Schreiber]], [[Danny Stevenson]], _From loop groups to 2-groups_, _Homotopy, Homology and Applications_ **9** (2007), 101-135.  ([arXiv:math.QA/0504123](http://arxiv.org/abs/math.QA/0504123))

* {#Henriques} [[André Henriques]], _Integrating $L_\infty$ algebras_, Compos. Math. __144__  (2008), no. 4, 1017--1045 ([doi](http://dx.doi.org/10.1112/S0010437X07003405),[math.AT/0603563](http://arxiv.org/abs/math.AT/0603563))

and the relation to the Green-Schwarz mechanism is made explicit in

* {#SatiSchreiberStasheff09} [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Twisted Differential String and Fivebrane Structures]]_, Communications in Mathematical Physics, 2012, Volume 315, Issue 1, pp 169-213  ([arXiv:0910.4001](http://arxiv.org/abs/0910.4001))

This article also observes that an analogous situation appears in [[dual heterotic string theory]] with the _[[fivebrane Lie 6-algebra]]_ in place of the string Lie 2-algebra.

#### Higher Chern-Simons field theory and AKSZ sigma-models

Ordinary [[Chern-Simons theory]] for a simple gauge group is all controled by a [[Lie algebra cohomology|Lie algebra 3-cocycle]]. The generalization of Chern-Simons theory to [[AKSZ-sigma models]] was understood to be encoded by [[symplectic Lie n-algebroids]] (later re-popularized as "[[shifted symplectic structures]]") in

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

* [[Pavol Ševera]], _[[Some title containing the words "homotopy" and "symplectic", e.g. this one]]_, based on a talk at "Poisson 2000", CIRM Marseille, June 2000; ([arXiv:0105080](http://arxiv.org/abs/math/0105080))

* [[Dmitry Roytenberg]], _On the structure of graded symplectic supermanifolds and Courant algebroids_ in _Quantization, Poisson Brackets and Beyond_ , [[Theodore Voronov]] (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 ([arXiv](http://arxiv.org/abs/math/0203110))

* {#Roytenberg} [[Dmitry Roytenberg]], _AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories_ Lett.Math.Phys.79:143-159,2007 ([arXiv:hep-th/0608150](http://arxiv.org/abs/hep-th/0608150)).

The globally defined AKSZ action functionals obtained this way were shown in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:AKSZ Sigma-Models in Higher Chern-Weil Theory]]_, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 ([arXiv:1108.4378](http://arxiv.org/abs/1108.4378))

to be a special case of the higher Lie integration process of

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]]_, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 ([arXiv:1011.4735](http://arxiv.org/abs/1011.4735))

Further exmaples of non-symplectic $L_\infty$-Chern-Simons theory obtained this way include [[7-dimensional Chern-Simons theory]] on [[differential string structure|string 2-connections]]:

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane|Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory]]_, Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229–321

#### In local prequantum field theory

Infinite-dimensional $L_\infty$-algebras that behaved similar to [[Poisson bracket]] [[Lie algebras]] -- _[[Poisson bracket Lie n-algebras]]_ -- were noticed

* {#Rogers10} [[Chris Rogers]], _$L_\infty$ algebras from multisymplectic geometry_ ,  Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50  ([arXiv:1005.2230](http://arxiv.org/abs/1005.2230), [journal](http://link.springer.com/article/10.1007%2Fs11005-011-0493-x)).

* {#Rogers11} [[Chris Rogers]], _Higher symplectic geometry_ PhD thesis (2011) ([arXiv:1106.4068](http://arxiv.org/abs/1106.4068))

In

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles ]]_, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

these were shown to be the infinitesimal version of the symmetries of [[prequantum n-bundles]] as they appear in [[local prequantum field theory]], in higher generalization of how the [[Poisson bracket]] is the Lie algebra of the [[quantomorphism group]].

These also encode a homotopy refinement of the [[Dickey bracket]] on [[Noether theorem|Noether]] [[conserved currents]] which for [[Green-Schwarz sigma models]] reduces to Lie $n$-algebras of [[BPS charges]] which refine [[super Lie algebras]] such as the   [[M-theory super Lie algebra]]:

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Lie n-algebras of BPS charges]]_ ([arXiv:1507.08692](http://arxiv.org/abs/1507.08692))

* [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Lie n-algebras of higher Noether currents ]]_

This makes concrete the suggestion that there should be $L_\infty$-algebra refinements of the [[Dickey bracket]] of [[conserved currents]] in [[local field theory]] that was made in

* [[Glenn Barnich]], [[Ronald Fulp]], [[Tom Lada]], [[Jim Stasheff]], _The sh Lie structure of Poisson brackets in field theory_ ([arXiv:hep-th/9702176](http://arxiv.org/abs/hep-th/9702176))

Comprehesive survey and exposition of this situation is in

* [[Urs Schreiber]], _[[schreiber:Higher Prequantum Geometry]]_, in [[Gabriel Catren]], [[Mathieu Anel]] (eds.), _[New Spaces for Mathematics and Physics](https://ncatlab.org/nlab/show/New+Spaces+for+Mathematics+and+Physics)_, 2016

#### In perturbative quantum field theory

Further identification of [[L-∞ algebras]]-[[structure]] in the [[Feynman amplitudes]]/[[S-matrix]] of [[Lagrangian field theory|Lagrangian]] [[perturbative quantum field theory]]:

* {#Froeb18} [[Markus Fröb]], _Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories_ [Communications in Mathematical Physics](https://dx.doi.org/10.1007/s00220-019-03558-6) 2019 (online first) ([arXiv:1803.10235](https://arxiv.org/abs/1803.10235))

* {#Arvanitakis19} [[Alex Arvanitakis]], _The $L_\infty$-algebra of the S-matrix_ ([arXiv:1903.05643](https://arxiv.org/abs/1903.05643))

#### In double field theory

In [[double field theory]]:

* [[Andreas Deser]], [[Jim Stasheff]], _Even symplectic supermanifolds and double field theory_, Communications in Mathematical Physics November 2015, Volume 339, Issue 3, pp 1003-1020 ([arXiv:1406.3601](http://arxiv.org/abs/1406.3601))

* [[Olaf Hohm]], [[Barton Zwiebach]], _$L_\infty$ Algebras and Field Theory_ ([arXiv:1701.08824](https://arxiv.org/abs/1701.08824))

### Related expositions

* [[fiber bundles in physics]]

* [[higher category theory and physics]]

* [[string theory FAQ]]

* [[motives in physics]]

* [[Hilbert's sixth problem]]

* [[model theory and physics]]

* [[motivation for sheaves, cohomology and higher stacks]]

* [[applications of (higher) category theory]]

* [[motivation for higher differential geometry]]

* [[motivation for cohesion]]


L-infinity algebroid > history

< [[L-infinity algebroid]]
[[!redirects L-infinity algebroid -- history]]


L-infinity Lie algebroid > history

< [[L-infinity Lie algebroid]]
[[!redirects L-infinity Lie algebroid -- history]]


L-infinity operad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Higher algebra
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[[!include higher algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _$L_\infty$-operad_ is a cofibrant [[resolution]] of the [[Lie operad]] in the [[model structure on operads]] in the [[category of chain complexes]].

Its [[algebras over an operad]] are [[L-infinity algebras]].

## Related concepts

* [[A-∞ operad]]

* [[E-∞ operad]]

* [[model structure for L-∞ algebras]]

## References
 {#References}

### In chain complexes

Discussion as an operad in [[chain complexes]]:

Original articles include

* [[Takashi Kimura]], [[Jim Stasheff]], [[Alexander Voronov]], around Def. 2.1 of: _On operad structures of moduli spaces and string theory_, Comm. Math. Phys. Volume 171, Number 1 (1995), 1-25 ([euclid:1104273401](https://projecteuclid.org/euclid.cmp/1104273401))

Review includes

* Jeroen Maes, around Def. 5.6.4 of _Derived homotopy algebras_, PhD thesis 2016  ([web](https://idus.us.es/xmlui/handle/11441/48473))

See also

* [[Igor Kriz]], [[Peter May]], around p. 28 of _Operads,  algebras, modules and motives_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf))

### In spectra
 {#ReferencesInSpectra}

Discussion as an operad in [[spectra]] (in [[stable homotopy theory]]) is in

* [[Gijs Heuts]], around def. 4.12 of _Lie algebras and $v_n$-periodic spaces_ ([arXiv:1803.06325](https://arxiv.org/abs/1803.06325))

based on

* Michael Ching, _Bar constructions for topological operads and the Goodwillie derivatives of the identity,  Geom. Topol. , 9:833–933, 2005 ([arXiv:math/0501429](https://arxiv.org/abs/math/0501429))

* Michael Ching, _Bar-cobar duality for operads in stable homotopy theory_, Journal of Topology, 2012 ([arXiv:1009.5034](https://arxiv.org/abs/1009.5034))

[[!redirects L-infintiy operads]]

[[!redirects L-∞ operad]]
[[!redirects L-∞-operad]]
[[!redirects L-∞ operads]]
[[!redirects L-∞-operads]]

[[!redirects homotopy Lie operad]]
[[!redirects homotopy Lie operads]]

[[!redirects strong homotopy Lie operad]]
[[!redirects strong homotopy Lie operads]]