Leray-Hirsch theorem

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## Idea

The Leray-Hirsch theorem states sufficient fiberwise condition for the [[ordinary cohomology]] of the total space of a [[fiber bundle]] with [[coefficients]] in a [[commutative ring]] to be [[free module]] over the [[cohomology ring]] of the base space.

An important consequence is the [[Thom isomorphism]].

## Statement

Let

\[
  \label{TheFiberBundles}
  \array{
     F &\stackrel{\iota}{\hookrightarrow}& Y
     \\
     && \downarrow^{\mathrlap{p}}
     \\
     && X
  }
\]

be an $F$-[[fiber bundle]] (in [[Top]]) of [[topological spaces]] that admit the [[mathematical structure|structure]] of finite [[CW-complexes]].

### In ordinary cohomology
 {#StatementInOrdinaryCohomology}

Let $R$ be a [[commutative ring]] and write $H^\bullet(-;\,R)$ for the [[cohomology rings]] of [[ordinary cohomology]] with [[coefficients]] in $R$.

If there exists

* a [[finite set]] of elements

  $$
    \alpha_i \in H^\bullet(Y,R)\;,\;\;\;\; i \in \{1, 2, \cdots, n\}
  $$

  in the [[ordinary cohomology]] of $Y$ with [[coefficients]] in $R$,

such that

* for each point $x \in X$ the restriction ([[pullback in cohomology|pullback]] along $\iota$) of the $\alpha_i$ to the [[fiber]] $F_x \hookrightarrow Y$

  1. is $R$-[[linearly independent]]

  1. their $R$-[[linear span]] is [[isomorphism|isomorphic]] to the [[cohomology group]] $H^\bullet(F,R)$ of the fiber

     $$
       H^\bullet(F;\, R)
       \simeq
       R
       \langle
         \iota_x^\ast \alpha_1, \cdots, \iota_x^\ast \alpha_n
       \rangle
       \;\;\;\;
       \in R Mod
      $$

  (i.e. a [[free module]] over $R$)

then:

1. the $\{\alpha_1, \cdots, \alpha_n\}$ themselves are $H^\bullet(X;\,R)$-[[linearly independent]],

1. their $H^\bullet(X;\,R)$-[[linear span]] gives the [[cohomology group]] of the total space $Y$:

   $$
     H^\bullet(Y;\,R)
     \;\simeq\;
     H^\bullet(X;\,R)
     \langle
       \alpha_1, \cdots,\alpha_n
     \rangle
     \;\;\;\;\;
     \in
     \;
     H^\bullet(X;\, R) Mod
     \,,
   $$

   via the [[isomorphism]]

   $$
     H^\bullet(X;\, R)
       \otimes_R
     H^\bullet(F;\, R)
       \stackrel{\simeq}{\longrightarrow}
     H^\bullet(Y;\, R)
   $$

   given by [[pullback in cohomology|pulling back]] classes from the base space and there forming their [[cup product]] with these generators on the total space:

   $$
     \underset{i,j}{\sum} c_i \otimes \iota^\ast(\alpha_j)
       \mapsto
     \underset{i,j}{\sum}
     p^\ast(c_i) \cup \alpha_j
     \,.
   $$

\linebreak

### In generalized cohomology
 {#StatementInGeneralizedCohomology}

The statement generalizes verbatim from [[ordinary cohomology]] to any [[multiplicative cohomology theory|multiplicative]] [[Whitehead-generalized cohomology theory]] $E$ ([Conner-Floyd 66, theorem 7,4](#ConnerFloyd66), attributed there to [[Albrecht Dold]], review in [Tamaki-Kono 06, Section 3.1](#TamakiKono06)):

Let $E$ be a [[multiplicative cohomology theory|multiplicative]] [[Whitehead-generalized cohomology theory]] and write

* $E^\bullet(-)$ for its [[cohomology rings]];

* $E_{-\bullet} \;\coloneqq \; E^\bullet(\ast)$ for its [[ground ring]]

If there exists

* a [[finite set]] of elements

  \[
    \label{GeneratorsInELerayHirschTheorem}
    \alpha_i \;\in\; E^\bullet(Y)\;,\;\;\;\; i \in \{1, 2, \cdots, n\}
  \]

  in the [[ordinary cohomology]] of the total space $Y$,

such that

* for each point $x \in X$ the restriction ([[pullback in cohomology|pullback]] along $\iota$) of the $\alpha_i$ to the [[fiber]] $F_x \hookrightarrow Y$

  1. is $E_{-\bullet}$-[[linearly independent]]

  1. their $E_{-\bullet}$-[[linear span]] is [[isomorphism|isomorphic]] to the [[cohomology group]] $E^\bullet(F)$ of the fiber

     \[
       \label{AssumptionOnFiberCohomologyInELerayHirschTheorem}
       E_{-\bullet}(F;)
       \simeq
       E_{-\bullet}
       \langle
         \iota_x^\ast \alpha_1, \cdots, \iota_x^\ast \alpha_n
       \rangle
       \;\;\;\;
       \in E_{-\bullet} Mod
      \]

   (i.e. a [[free module]] over $E_{-\bullet}$)

then:

1. the $\{\alpha_1, \cdots, \alpha_n\}$ themselves are $E^\bullet(X)$-[[linearly independent]],

1. their $E^\bullet(X)$-[[linear span]] gives the [[cohomology group]] of the total space $Y$:

   $$
     E^\bullet(Y)
     \;\simeq\;
     E^\bullet(X)
     \langle
       \alpha_1, \cdots,\alpha_n
     \rangle
     \;\;\;\;\;
     \in
     \;
     E^\bullet(X) Mod
     \,,
   $$

   via the [[isomorphism]]

   $$
     E^\bullet(X)
       \otimes_{E_{-\bullet}}
     E^\bullet(F)
       \stackrel{\simeq}{\longrightarrow}
     E^\bullet(Y)
   $$

   given by [[pullback in cohomology|pulling back]] classes from the base space and there forming their [[cup product]] with these generators on the total space:

   $$
     \underset{i,j}{\sum} c_i \otimes \iota^\ast(\alpha_j)
       \mapsto
     \underset{i,j}{\sum}
     p^\ast(c_i) \cup \alpha_j
     \,.
   $$

## Examples

### Complex-oriented cohomology of the twistor fibration
 {#ComplexOrientedCohomologyOfTwistorFibration}

Let $E$ be a [[Whitehead-generalized cohomology theory]] equipped with [[complex oriented cohomology|complex orientation]] in the form of a [[first Chern class|first]] [[Conner-Floyd-Chern class]]

$$
  c^E_1
  \;\in\;
  {\widetilde E}{}^2\big( \mathbb{C}P^\infty \big)
    \longrightarrow
  {\widetilde E}{}^2\big( \mathbb{C}P^n \big)
  \,.
$$

Then, for $n \in \mathbb{N}$, the $E$-[[cohomology ring]] of the [[complex projective space]] $\mathbb{C}P^n$ is (see [there](complex+oriented+cohomology+theory#CohomologyRingOfBU1))

$$
  E^\bullet
  \big(
    \mathbb{C}P^n
  \big)
  \;\simeq\;
  E_{-\bullet}
  \big[
    c^E_1
  \big]
  \big/
  \big(
    c^E_1
  \big)^{n+1}
  \;\;\;
  \in
  \;
  E_{-\bullet} Algebras
  \,,
$$

whence the [[cohomology group]] is

\[
  \label{ECohomologyGroupOfComplexProjectiveSpace}
  E^\bullet
  \big(
    \mathbb{C}P^n
  \big)
  \;\simeq\;
  E_{-\bullet}
  \big\langle
    1,\,
    c^E_1,\,
    \big(c^E_1\big)^2,\,
    \cdots
    ,\,
    \big(c^E_1\big)^n
  \big\rangle
  \;\;\;
  \in
  \;
  E_{-\bullet} Modules
  \,.
\]

For each $n = 2 k + 1$ these are [[Riemann sphere]]
$\mathbb{H}^\times/\mathbb{C}^\times = \mathbb{C}P^1$-[[fiber bundles]]

$$
  \array{
    \mathbb{C}P^1
    &
    \longrightarrow
    &
    \mathbb{C}^{2k+1}
    &
    =
    &
    \big(
      \mathbb{C}^{2k+2} \setminus \{0\}
    \big)
    \big/
    \mathbb{C}^\times
    \\
    &&
    \big\downarrow
    &&
    \big\downarrow
    {}^{
       \mathrlap{
       v \cdot \mathbb{C}^\times
         \mapsto
       v \cdot \mathbb{H}^\times
       }
    }
    \\
    &&
    \mathbb{H}P^k
    &
    =
    &
    \big(
      \mathbb{H}^{k+1} \setminus \{0\}
    \big)
    \big/
    \mathbb{H}^\times
  }
$$

over [[quaternionic projective space]] $\mathbb{H}P^{k}$,
whose [[fiber]]-inclusion is ([[homotopy|homotopic]] to) the canonical inclusion $\mathbb{C}P^1 \hookrightarrow \mathbb{C}P^n$ (see [there](complex+projective+space#Definition)).

E.g. for $k = 1$ this is also known as the _[[twistor fibration]]_; while for  $k = \infty$ this is the fibration of [[classifying spaces]]

$$
  \array{
    SU(2)/\mathrm{U}(1)
    &\longrightarrow&
    B \mathrm{U}(1)
    \\
    &&
    \big\downarrow
     {}^{\mathrlap{
       B\big( z \mapsto diag(z,z^\ast)  \big)
     }}
    \\
    &&
    B SU(2)
    \,.
  }
$$

Therefore, by (eq:ECohomologyGroupOfComplexProjectiveSpace), the assumption (eq:AssumptionOnFiberCohomologyInELerayHirschTheorem) of the $E$-Leray-Hirsch theorem ([above](#StatementInGeneralizedCohomology)) is met if we take the classes (eq:GeneratorsInELerayHirschTheorem) to be the [[cup product|cup powers]] $(c^E_1)^n$. Now the $E$-Leray-Hirsch theorem says that:

\begin{xymatrix@R=14pt@C=30pt}
    \mathbb{C}P^1
    \ar[rr]
    &&
    \mathbb{C}P^3
    \ar[rr]
    &&
    \mathbb{H}P^1
    \\
    E^\bullet
    \big(
      \mathbb{C}P^1
    \big)
    \ar@{=}[d]
    \ar@{<<-}[rr]
    &
    &
    E
    {}^{\bullet}
    \big(
      \mathbb{C}P^3
    \big)
    \ar@{=}[d]
    \ar@{<-^{)}}[rr]
    &&
    E
    {}^{\bullet}
    \big(
      \mathbb{H}P^1
    \big)
    \ar@{=}[d]
    \\
    E^\bullet(\ast)
    \big\langle
      1, c^E_1
    \big\rangle
    \ar@/_2pc/@{<<-}[ddrr]
    &&
    E^\bullet(\ast)
    \big\langle
      1, c^E_1, (c^E_1)^2, (c^E_1)^3
    \big\rangle
    &&
    E^\bullet(\ast)
    \big\langle
      1,\,
      \scalebox{.7}{$\tfrac{1}{2}$} p^E_1
    \big\rangle
    \ar@/^2pc/@{^{(}->}[ddll]
    \\
    {\phantom{A}} \atop {\phantom{A}}
    \\
    &&
    E^\bullet(\ast)
    \big\langle
      1,\,
      c^E_1,\,
      \scalebox{.7}{$\tfrac{1}{2}$} p^E_1,\,
      \scalebox{.7}{$\tfrac{1}{2}$} p^E_1
      \cdot c^E_1
    \big\rangle
    \ar[uu]
      ^-{\simeq}
      _-{
        \scalebox{.7}{$
          \begin{array}{ccc}
            1
              &
            c^E_1
              &
            ...
            \\
            \scalebox{.65}{
               \rotatebox{90}{$\mapsto$}
            }
            &
            \scalebox{.65}{
               \rotatebox{90}{$\mapsto$}
            }
            &
            \scalebox{.65}{
               \rotatebox{90}{$\mapsto$}
            }
            \\
            1
            &
            c^E_1
            &
            \scalebox{.7}{$\tfrac{1}{2}$} p^E_1,\,
          \end{array}
        $}
      }
\end{xymatrix}

## Related concepts

* [[Thom isomorphism]]

## References

### For ordinary cohomology

Review of the theorem for [[ordinary cohomology]]:

* [[Alan Hatcher]], _[Algebraic topology](https://www.math.cornell.edu/~hatcher/AT/ATpage.html)_, 2002, theorem 4D.1 on p. 432 ([pdf](https://www.math.cornell.edu/~hatcher/AT/ATch4.4.pdf))

* {#Ebert12} [[Johannes Ebert]], section 2.3 of _A lecture course on Cobordism Theory_, 2012 ([pdf](http://wwwmath.uni-muenster.de/u/jeber_02/skripten/bordism-skript.pdf))

### For generalized cohomology

Discussion for [[Whitehead-generalized cohomology|Whitehead-generalized]] [[multiplicative cohomology theories]]:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Theorem 7.4 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__, Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Section 3.1 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))


Leron Borsten

* [webpage](https://www.dias.ie/index.php?option=com_contact&view=contact&id=181:dr-leron-borsten&catid=205&Itemid=30&lang=en)

## Selected writings

On [[Freudenthal magic square|magic squares]] of [[magic supergravities]]  and [[U-duality]] via the [[classical double copy]]:

* {#BorstenDuffHughesNagy13} [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic square from Yang-Mills squared*, Phys. Rev. Lett. **112** (2014) 131601  [[arXiv:1301.4176](http://arxiv.org/abs/1301.4176)]

* {#ABDHN} [[Alexandros Anastasiou]], [[Leron Borsten]], [[Michael Duff]], [[Mia J. Hughes]], [[Silvia Nagy]], *A magic pyramid of supergravities*, JHEP (2014) 178 [[arXiv:1312.6523](http://arxiv.org/abs/1312.6523)]

On the [[classical double copy]] ([[KLT relations]], relating [[gravity]] to [[Yang-Mills theory]]) via [[L-infinity algebras|$L_\infty$-algebras]]:

* [[Leron Borsten]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Tommaso Macrelli]], [[Christian Saemann]], [[Martin Wolf]], _Double copy from homotopy algebras_, Fortschritte der Physik, **69** 8-9 (2021) 2100075 $[$[arXiv:2102.11390](https://arxiv.org/abs/2102.11390), [doi:10.1002/prop.202100075](https://doi.org/10.1002/prop.202100075)$]$

Understanding [[tensor hierarchies]] via [[higher gauge theory]] for [[EL-infinity algebras|$E L_\infty$-algebras]] with [[adjusted Weil algebras]]:

* [[Leron Borsten]], [[Hyungrok Kim]], [[Christian Saemann]].  *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies* [[arXiv:2106.00108](https://arxiv.org/abs/2106.00108)]

On the work of [[Mike Duff]]:

* [[Leron Borsten]], [[Alessio Marrani]], [[Christopher N. Pope]], [[Kellogg Stelle]], *Introduction to the special issue dedicated to Michael J. Duff FRS on the occasion of his 70th birthday*, Proceedings of the Royal Society A,
**478** (2022) 2259 [[doi:10.1098/rspa.2022.0166](https://doi.org/10.1098/rspa.2022.0166)]

On [[higher gauge theory]]:

* [[Leron Borsten]], [[Mehran Jalali Farahani]], [[Branislav Jurčo]], [[Hyungrok Kim]], [[Jiří Nárožný]], [[Dominik Rist]], [[Christian Saemann]], [[Martin Wolf]], *Higher Gauge Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2401.05275](https://arxiv.org/abs/2401.05275)]

## Related entries

* [[super Yang-Mills theory]]

* [[division algebra and supersymmetry]]

* [[Freudenthal magic square]], [[magic supergravity]]

* [[M-theory supersymmetry algebra]]

category: people


Les Dérivateurs

_Les Dérivateurs_ is a manuscript of some 2000 pages left behind by [[Alexander Grothendieck]] when he disappeared in 1991, dealing with the fundamentals of [[homotopy theory]]. It can be considered as a sequel to the more famous [[Pursuing Stacks]], but has a less conversational style.

Portions have been LaTeXed and are available at [Maltsiniotis' derivators page](https://webusers.imj-prg.fr/~georges.maltsiniotis/groth/Derivateursengl.html). More accessible are Maltsiniotis' [introductory notes](http://www.math.jussieu.fr/~maltsin/ps/m.ps) (Postscript).

category: reference

[[!redirects Les Derivateurs]]


letter theory

* [[A-theory]]

* [[B-theory]]

* [[C-theory]]

* [[D-theory]]

* [[E-theory]]

* [[F-theory]]

* [[G-theory]]

* [[H-theory]]

* [[I-theory]]

* [[J-theory]]

* [[K-theory]]

* [[L-theory]]

* [[M-theory]]

* [[N-theory]]

* [[O-theory]]

* [[P-theory]]

* [[Q-theory]]

* [[R-theory]]

* [[S-theory]]

* [[T-theory]]

* [[U-theory]]

* [[V-theory]]

* [[W-theory]]

* [[X-theory]]

* [[Y-theory]]

* [[Z-theory]]

See also: [[Cohomology Theory Database]]


Lev Borisov

* [webpage](http://sites.math.rutgers.edu/~borisov/)

## related $n$Lab entries

* [[motivic integration]]

category: people


Lev Landau

_Lev Landau_ (Лев Давидович Ландау) was one of the leading theoretical physicists of 20th century; leading an original school of theoretical physics in Harkov, Soviet Union. He died early as a  consequence of injuries in a traffic accident.

* wikipedia: [en](http://en.wikipedia.org/wiki/Lev_Landau), [ru](http://ru.wikipedia.org/wiki/%D0%9B%D0%B0%D0%BD%D0%B4%D0%B0%D1%83,_%D0%9B%D0%B5%D0%B2_%D0%94%D0%B0%D0%B2%D0%B8%D0%B4%D0%BE%D0%B2%D0%B8%D1%87)

See also [[Landau-Lifschitz]].

## Selected writings

Introducing [[Landau-Ginzburg models]] in [[superconductivity]]:

* [[Vitaly Ginzburg]], [[Lev Landau]], _On the Theory of Superconductivity_,  reprinted In: _On Superconductivity and Superfluidity_ Springer (2009) ([doi:1007/978-3-540-68008-6_4](https://doi.org/10.1007/978-3-540-68008-6_4))

## Related $n$Lab entries

* [[Landau's Fermi liquid theory]]

category: people

[[!redirects Lev Landau]]
[[!redirects Лев Ландау]]
[[!redirects Lev Lavidovich Landau]]
[[!redirects Lev Lavidovič Landau]]
[[!redirects Лев Давидович Ландау]]

[[!redirects Л. Д. Ландау]]


Lev Lipatov

* [webpage](http://thd.pnpi.spb.ru/~lipatov)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Lev_Lipatov)

## related $n$Lab entries

* [[perturbative quantum field theory]]

* [[perturbation theory]]

* [[asymptotic series]]

category: people


Lev Pontrjagin

\begin{imagefromfile}
    "file_name": "LevSemenovichPontryagin.jpg",
    "float": "right",
    "width": "200",
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

**Lev Semёnovič Pontrjagin** (Лев Семёнович Понтрягин) was an influential Soviet mathematician working in Moscow. His main works till the late 1940s were in [[general topology]], especially the study of [[topological groups]] (including study of [[Pontrjagin duality]]) and [[algebraic topology]] in which he also explored applications of [[differentiable manifolds]] for computation of [[homotopy groups]] (the [[Pontrjagin theorem]]); including the introduction of unstable framed [[cobordism]] (unstable [[Cohomotopy]]) in that work. After the success of the French mathematical school in the late 1940s in introducing new methods of [[sheaf and topos theory|sheaf theory]], which Pontrjagin did not like as being less direct than the more intuitive geometric methods he was master of, and after some pressure from the government, Pontrjagin switched into [[applied mathematics]] where he did some fundamental work, especially in [[optimization theory]].

Pontrjagin was a full member of the (Soviet) Academy of Sciences. He was blind since the age of 14.

* wikipedia [in German](http://de.wikipedia.org/wiki/Lew_Semjonowitsch_Pontrjagin), [in English](http://en.wikipedia.org/wiki/Lev_Pontryagin), [in Russian](http://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BD%D1%82%D1%80%D1%8F%D0%B3%D0%B8%D0%BD,_%D0%9B%D0%B5%D0%B2_%D0%A1%D0%B5%D0%BC%D1%91%D0%BD%D0%BE%D0%B2%D0%B8%D1%87), [[joyalscatlab:Pontryagin, Lev|entry at joyalscatlab]]

## Selected writings

Introducing [[Pontrjagin duality]]:

* {#Pontrjagin34} [[Lev Pontrjagin]], *Theory of topological commutative groups*, Annals of Mathematics Second Series, Vol. 35, No. 2 (Apr., 1934), pp. 361-388 ([doi:10.2307/1968438](https://doi.org/10.2307/1968438))

  Russian translation: Uspekhi Mat. Nauk, 1936, no. 2, 177–195 ([mathnet:umn8882](http://mi.mathnet.ru/eng/umn8882))

Introducing [[Pontryagin's theorem]] -- the [[bijection]] between [[cobordism classes]] of [[normally framed submanifolds]] of a [[closed manifold|closed]] [[smooth manifold]] and its [[Cohomotopy]]-sets, established via the  [[Pontryagin-Thom collapse]]/[[Cohomotopy charge]] -- and its application to the computation of

the [[first stable homotopy group of spheres]]:

* [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, _Communication I_, Doklady Akademii Nauk SSSR 19(3) (1938), 147-149

and the [[second stable homotopy group of spheres]]:

* {#Pontrjagin38} [[Lev Pontrjagin]], _[[Classification of continuous maps of a complex into a sphere]]_, _Communication II_, Dokl. Akad. Nauk SSSR 19 (1938), 361-363

  > (this article contains a famous mistake, see [[Michael Hopkins|Hopkins]]'s talk at [Atiyah's 80th Birthday conference](https://www.maths.ed.ac.uk/~v1ranick/atiyah80.htm), see [slide 8](https://www.maths.ed.ac.uk/~v1ranick/hopkins.pdf#page=8), [9:45](https://empg.maths.ed.ac.uk/Videos/Atiyah80/Hopkins.mov))

* [[Lev Pontryagin]], _[[Homotopy classification of mappings of an (n+2)-dimensional sphere on an n-dimensional one]]_, Doklady Akad. Nauk SSSR (N.S.) 19 (1950), 957–959, (Russian) ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont3.pdf))

  > (this article fixes the mistake)

all three of which are available in English translation in [Gamkrelidze 86](#Gamkrelidze86).

A comprehensive account of this [[cobordism theory]]/[[stable homotopy theory]] via [[Pontryagin's theorem]] (now mostly: "[[Pontryagin-Thom theorem]]") is then given in:

* {#Pontrjagin55} [[Lev Pontrjagin]], _[[Smooth manifolds and their applications in homotopy theory]]_, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955, AMS Translation Series 2, Vol. 11, 1959 ([doi:10.1142/9789812772107_0001](https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf))

Introducing the [[Pontrjagin product]]:

* [[Lev Pontrjagin]], _Homologies in compact Lie groups_, Rec. Math. [Mat. Sbornik] N.S., 1939 Volume 6(48), Number 3, Pages 389–422 ([mathnet:5835](http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5835&option_lang=eng))

{#IntroducingPontrjaginClasses} Introducing [[Pontrjagin classes]]:

* [[Л. С. Понтрягин]], _Характеристические циклы многообразий_, ДАН, XXXV, № 2 (1942).

* [[Л. С. Понтрягин]], _Характеристические циклы дифференцируемых многообразий_, Матем. сб., 21(63):2 (1947), 233–284.  [MathNet.Ru PDF](http://mi.mathnet.ru/msb6237 ).  English translation: [[Lev Pontrjagin]], _Characteristic cycles on differentiable manifolds_, Mat. Sbornik N. S. 21(63) (1947), 233-284; A.M.S. Translation 32 (1950).

Selected articles:

* {#Gamkrelidze86} [[Revaz Gamkrelidze]] (ed.), _L. S. Pontryagin -- Selected Works_, Volume 1: _Selected Research Papers_,  [Classics of Soviet Mathematics](https://www.routledge.com/Classics-of-Soviet-Mathematics/book-series/CRCCLASOVMTH), Gordon and Breach 1986, CRC Press 2019 ([ISBN:9782881241055](https://www.routledge.com/Selected-Research-Papers/Pontryagn/p/book/9782881241055))

## Related $n$Lab entries

* [[Pontrjagin class]]

* [[Pontrjagin-Thom theorem]]

category: people

[[!redirects Lev Pontrjagin]]
[[!redirects L. S. Pontrjagin]]
[[!redirects Lev Pontryagin]]
[[!redirects Lev Pontrâgin]]
[[!redirects Лев Понтрягин]]
[[!redirects Лев Семёнович Понтрягин]]
[[!redirects Лев Семенович Понтрягин]]
[[!redirects Л. С. Понтрягин]]
[[!redirects Pontrjagin]]
[[!redirects Pontryagin]]
[[!redirects Pontrâgin]]
[[!redirects Понтрягин]]


Lev Rozansky

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=206359)

## Selected writings

On the [[universal Vassiliev invariant]] of the [[unknot]] ([[Wheels theorem]]):

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))

Introducing [[Rozansky-Witten theory]]:

* {#RozanskyWitten96} [[Lev Rozansky]], [[Edward Witten]], _Hyper-Kähler geometry and invariants of 3-manifolds_, Selecta Math., New Ser. __3__ (1997), 401–458 ([arXiv:hep-th/9612216](https://arxiv.org/abs/hep-th/9612216), [doi:10.1007/s000290050016](https://doi.org/10.1007/s000290050016), [MR98m:57041](http://www.ams.org/mathscinet-getitem?mr=98m:57041))

On [[Rozansky-Witten theory]] as a [[boundary field theory]]:

* [[Anton Kapustin]], [[Lev Rozansky]], [[Natalia Saulina]],
_Three-dimensional topological field theory and symplectic algebraic geometry I_ ([arXiv:0810.5415](https://arxiv.org/abs/0810.5415))

category: people


level

* [[level (Chern-Simons theory)]]

[[!redirects levels]]

category: disambiguation


level (Chern-Simons theory)

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###Context###
#### $\infty$-Chern-Simons theory
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[[!include infinity-Chern-Simons theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[local Lagrangian]] of [[Chern-Simons theory]] gives rise to an [[action functional]] which is [[gauge invariance|gauge invariant]] only for certain discrete choices of the scale (a global prefactor) of the Lagrangian. This is called the **level** of the [[theory (physics)|theory]]. The fact that it takes values in a [[discrete group|discrete]] [[subgroup]] of the [[real numbers]] is called **level quantization** of the [[theory (physics)|theory]].

## Properties

### As a characteristic class

Formally, for [[gauge group]] being some [[compact Lie group]] $G$, the level is identified with a choice of [[universal characteristic class]], hence an element

$$
  [c] \in H^4(B G, \mathbb{Z})
$$

in the [[integral cohomology]] of the [[classifying space]] of $G$. Under the [[Chern-Weil homomorphism]] every such $c$ corresponds to an [[invariant polynomial]] $\langle -,-\rangle_c$ on the [[Lie algebra]]. (The point being here that while every scalar multiple of an invariant polynomial is itself again an invariant polynomial, only a lattice of these does corespond to a class in [[integral cohomology]].)

If $G$ is furthermore connected and simply connected, then the [[local Lagrangian]] of Chern-Simons theory is the [[Chern-Simons form]] $CS_c$ of this specific invariant polynomial $\langle -,-\rangle_c$.

Generally, for $c \;\colon\; B G \to B^3 U(1) \simeq K(\mathbb{Z},4)$ a [[universal characteristic map]] the [[extended Lagrangian]] of  the corresponding [[3d Chern-Simons theory]] is a [[differential characteristic class|differential refinement]] $\hat \mathbf{c}$ of a lift $\mathbf{c}$ through [[geometric realization of cohesive infinity-groupoids|geometric realization]] of [[smooth infinity-groupoids]]

$$
  \array{
    \mathbf{B}G_{conn}
    &\stackrel{\hat \mathbf{c}}{\to}&
    \mathbf{B}^3 U(1)_{conn}
    \\
    \downarrow && \downarrow &&& \mathbf{H}
    \\
    \mathbf{B}G
    &\stackrel{\mathbf{c}}{\to}&
    \mathbf{B}^3 U(1)
    \\
    && &&& \downarrow^{\mathrlap{{\vert \Pi(-)\vert}}}
    \\
    B G &\stackrel{c}{\to}& B^3 U(1) &&& L_{whe} Top
  }
  \,.
$$

Hence $\hat \mathbf{c}$ is the "differentially refined level" of the theory. Notice that in terms of this the statement that the [[action functional]] is [[gauge invariance|gauge invariant]] for "given level" is the statement that for $\Sigma_3$ a [[closed manifold]] of [[dimension]] 3, the [[transgression]] of the [[extended Lagrangian]] $\hat \mathbf{c}$ to the [[moduli stack]] fo [[field (physics)|fields]] on $\Sigma_3$ is a map of [[smooth ∞-groupoid|smooth stacks]]

$$
  \exp\left(
    2 \pi i
    \int_{\Sigma_3} [\Sigma_3, \hat \mathbf{c}]
  \right)
  \;\colon\;
  [\Sigma_3, \mathbf{B}G_{conn}]
  \to
  U(1)
  \,.
$$

Analogous reasoning and termionology applies to [[higher dimensional Chern-Simons theory]] and generally to [[∞-Chern-Simons theory]].

### Level renormalization
 {#LevelRenormalization}

Upon [[quantization of 3d Chern-Simons theory|quantization]], the [[level (Chern-Simons theory)|Chern-Simons level]] receives a [[renormalization]] which shifts it by the [[dual Coxeter number]] of the [[gauge group]] (see the references [below](#ReferencesLevelRenormalization)).

## Related concepts

* [[su(2)-anyon]]

* [[quantization of 3d Chern-Simons theory]]

[[!include extended prequantum field theory - table]]

## References

### General

For traditional accounts see at  _[Chern-Simons theory - References](#Chern-Simons+theory)_.

Introductory discussion is in the section _[Physics in Higher Geometry: Motivation and Survey](geometry%20of%20physics#PhysicsMotivationAndSurvey)_ at

* _[[geometry of physics]]_.

### Level renormalization
 {#ReferencesLevelRenormalization}

On the [[renormalization]] of the Chern-Simons level (shift by the [[dual Coxeter number]]):

* [[Luis Alvarez-Gaumé]], J. M. F. Labastida, A. V. Ramallo, *A note on perturbative Chern-Simons theory*, Nuclear Physics B **334** 1 (1990) 103-124 (doi:10.1016/0550-3213(90)90658-Z)

* [[Mikhail A. Shifman]], *Four-dimension aspect of the perturbative renormalization in three-dimensional Chern-Simons theory*, Nuclear Physics B **352** 1 (1991) 87-112 (doi:10.1016/0550-3213(91)90130-P)

* M. Asorey, F. Falceto, J. L. Lopez, G. Luzon, *Universalty and Ultraviolet Regularizations of Chern-Simons Theory*, Nucl.Phys. B **429** (1994) 344-374 ([arXiv:hep-th/9403117](https://arxiv.org/abs/hep-th/9403117), doi:10.1016/0550-3213(94)00331-9)

[[!redirects Chern-Simons level]]


level of a topos

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#### Topos Theory
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=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

For $\mathbf{H}$ a [[topos]], then an [[essential subtopos]]  $\mathbf{H}_l \hookrightarrow \mathbf{H}$ is called a _level_ of $\mathbf{H}$. This is equivalently the inclusion of the [[right adjoint]] of an [[adjoint cylinder]]/[[adjoint modality]].

The terminology is due to the fact ([Kelly-Lawvere 89](#KL89)) that the [[essential geometric morphism|essential]] [[subtoposes]] of a topos, or more generally the essential localizations of a suitably complete category, form a [[complete lattice]].

If for two levels $\mathbf{H}_{1} \hookrightarrow \mathbf{H}_2$ the second one includes the [[modal types]] of the [[idempotent comonad]] of the first one, and if it is minimal with this property, then [[Lawvere]] speaks of "[[Aufhebung]]" (see there for details) of the [[unity of opposites]] exhibited by the first one.

## Examples

* Every [[open subtopos]] is a level.

* For $\mathbf{H} = $ [[sSet]] the topos of [[simplicial sets]], then the tower of [[simplicial coskeleton|n-coskeleton]]-inclusions for all $n$ is a sequence of levels (the extra left adjoint exhibiting the essentialness being the [[simplicial skeleton]]). The [[Aufhebung]] in this example is discussed in detail in ([KRRZ 11](#KRRZ11)).

* The lowest level is the inclusion of the trivial subtopos as the [[terminal object]]. See also at _[[unity of opposites]]_ the section _[Werden](http://ncatlab.org/nlab/show/adjoint+modality#Werden)_.

* Often the level above this will be [[cohesion]] and the level above that, if it exists, [[differential cohesion]].

  Here differential cohesion itself typically comes in a countable tower of levels, given by the $n$th order [[infinitesimal objects]], for each $n$.

* The [[Hegelian taco]] abstracts a certain stacking of three levels viewed as a diagram in a [[2-category]] into a [[graphic monoid]].

* An essential localization $i_!\dashv i^\ast \dashv i_\ast :\mathcal{A}\to\mathcal{B}$ where $i_!\simeq i_\ast$ is called a **quintessential localization** (cf. [Johnstone 1996](#JS96) and the discussion at [[quality type]]).

## Properties

The lattice of essential localizations of a category is contained in the lattice of all localizations. While the suprema of localizations coincide in both lattices, the infinimum in the lattice of all localizations of even a pair of essential localizations need _not_ be an essential localization  ([Kelly-Lawvere 89](#KL89)).

Recall (e.g. from [Borceux 1994](#Borceux1), p.106) that [[adjunction|adjunctions]] compose: if $G\dashv F:\mathcal{A}\to\mathcal{B}$ and $K\dashv H:\mathcal{B}\to\mathcal{C}$ then $G\circ K \dashv H\circ F:\mathcal{A}\to\mathcal{C}$. Applying this twice to a pair of essential localizations yields:

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

Given two essential localizations $i_!\dashv i^\ast \dashv i_\ast :\mathcal{A}\to\mathcal{B}$ and $q_!\dashv q^\ast \dashv q_\ast :\mathcal{B}\to\mathcal{C}$ , hence in the situation:

 $$
\mathcal{A}\overset{\hookrightarrow}{\underset{\hookrightarrow}{\to}} \mathcal{B}\overset{\hookrightarrow}{\underset{\hookrightarrow}{\to}}\mathcal{C}
$$

there results a composed essential localization $q_!\cdot i_!\dashv i^\ast \cdot q^\ast \dashv q_\ast\cdot i_\ast :\mathcal{A}\to\mathcal{C}$. $\qed$
=--

Here , of course, $i^\ast \cdot q^\ast \dashv q_\ast\cdot i_\ast$ is just the adjunction involved in the composition $q\cdot i$ of the two geometric morphisms $q$ and $i$ (provided the categories involved are toposes), the important thing is that the essentialities $q_!$ and $i_!$ compose as well.

Since this just states that inclusions as well as essential geometric morphism are closed under composition and therefore so are essential inclusions, our primary interest here is to extract an explicit formula for the modalities corresponding to the composition:

The corresponding [[adjoint modalities]] are $(q_!\cdot i_!)\cdot (i^\ast \cdot q^\ast) \dashv  (q_\ast\cdot i_\ast)\cdot  (i^\ast \cdot q^\ast) :\mathcal{C}\to\mathcal{C}$.

## Related concepts

* [[quality type]]

* [[subtopos]]

* [[open subtopos]]

* [[essential sublocale]]

* [[essential subtopos]]

* [[essential geometric morphism]]

* [[Aufhebung]]

* [[Hegelian taco]]

## References

* {#SGA4} [[Michael Artin|M.Artin]], [[Alexander Grothendieck|A.Grothendieck]], [[J. L. Verdier]] (eds.), _Théorie des Topos et Cohomologie Etale des Schémas - SGA 4_ , Springer LNM **269** Heidelberg 1972. (sec. IV 7.6., pp.414-416)

* {#Borceux1}[[Francis Borceux|F. Borceux]], _Handbook of Categorical Algebra vol. 1_ , Cambridge UP 1994.

* {#JS96}[[Peter Johnstone|P. Johnstone]], _Remarks on Quintessential and Persistent Localizations_ , TAC **2** no.8 (1996) pp.90-99. ([pdf](http://www.tac.mta.ca/tac/volumes/1996/n8/n8.pdf))

* {#KL89}[[G. M. Kelly]], [[F. W. Lawvere]], _On the Complete Lattice of Essential Localizations_ , Bull.Soc.Math. de Belgique **XLI** (1989) 289-319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

* {#KRRZ11} C. Kennett, [[Emily Riehl|E. Riehl]], M. Roy, M. Zaks, _Levels in the toposes of simplicial sets and cubical sets_ , JPAA **215** no.5 (2011) pp.949-961. ([arXiv:1003.5944](http://arxiv.org/abs/1003.5944))

* {#LW11}[[Rory Lucyshyn-Wright|R. Lucyshyn-Wright]], _Totally Distributive Toposes_ , arXiv.1108.4032 (2011). ([pdf](http://arxiv.org/pdf/1108.4032v3))

* {#MM19}[[Francisco Marmolejo|F. Marmolejo]], [[Matías Menni|M. Menni]], *Level $\epsilon$* , arXiv:1909.12757 (2019). ([abstract](https://arxiv.org/abs/1909.12757))

* {#VT01}[[Enrico Vitale|E. M. Vitale]], _Essential Localizations and Infinitary Exact Completions_ , TAC **8** no.17 (2001) pp.465-480. ([pdf](http://www.tac.mta.ca/tac/volumes/8/n17/n17.pdf))


level set

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#### Topology
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[[!include topology - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

Given a [[function]] $f \colon X \to K$ where $K$ is a [[ring]] or [[field]] or [[abelian group]] or possibly anything else, and given an [[element]] $c \in K$, then the _level set_ of $f$ at $c$ in $X$ is the [[preimage]] $f^{-1}(c) \in X$.

For $c = 0$ this is the [[zero locus]].

## Related concepts

* [[Morse theory]]

* [[Reeb graph]]

* [[Hamiltonian]]

* [[moment map]]

* [[well group]]

[[!redirects level sets]]


level structure

Meanings include:

* _[[level structure on an elliptic curve]]_

* (...)

[[!redirects level structures]]


level structure on an elliptic curve

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###Context###
#### Elliptic cohomology
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[[!include elliptic cohomology -- contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Over the [[complex numbers]] an [[elliptic curve]] (hence a [[complex manifold|complex]] [[torus]]) may be, and often is, presented as a [[quotient]] of the [[complex plane]] by a framed [[lattice]], determined by a point $\tau$ in the [[upper half plane]] $\mathfrak{h}$. But indeed this data determines a [[framed elliptic curve]], namely the underlying [[curve]] $\Sigma$ together with the "edges along which it is glued" by this construction. These are equivalently a choice of [[ordering|ordered]] [[basis]]

$$
  (a_1,a_2)\in H_1(\Sigma,\mathbb{Z})
$$

of the first [[ordinary homology]] of $\Sigma$ with [[integer]] [[coefficients]] (and vanishing [[intersection number]]).  The [[special linear group]] $SL_2(\mathbb{Z})$ naturally acts on this data (by [[Möbius transformations]] on $\tau$) and the [[quotient]] [[projection]] exhibits an infinite [[covering]] ([[atlas]]) of the  [[moduli stack of elliptic curves]] over the complex numbers

$$
  \mathfrak{h} \longrightarrow \mathfrak{h}//SL_2(\mathbb{C}) \simeq \mathcal{M}_{ell}(\mathbb{C})
  \,.
$$

The concept of _level structure_ on an elliptic curve is a structure weaker than that of a [[framed elliptic curve|framing]] which analogously gives a [[finite number|finite]] [[covering]]. Instead of considering cycles in integral homology, a _level $n$-structure_ for [[natural number]] $B$ is given by cycles in homology with [[coefficients]] just in the [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ (e.g. [Hain 08, def. 4.6](#Hain08)).

On such level-$n$ data now acts instead just the group $SL_2(\mathbb{Z}/n\mathbb{Z})$. The [[kernel]] of the [[projection]] maps is called the _level $n$-subgroup_ (an example of a _[[congruence subgroup]]_)

$$
  \Gamma(n) \to SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/n\mathbb{Z})
  \,.
$$

There is then a [[moduli space]] of complex elliptic curves equipped with level $n$-structure

$$
  \mathcal{M}_{ell}(\mathbb{C})[n] \simeq \mathfrak{h}//\Gamma_n
$$

called the _[[modular curve]]_, and this is now a finite cover (of rank the [[order of a group|order]] of the [[finite group]] $SL_2(\mathbb{Z}/n\mathbb{Z})$) of the actual [[moduli stack of elliptic curves|moduli stack of complex elliptic curves]]

$$
  \mathcal{M}_{ell}[n](\mathbb{C})
  \longrightarrow
  \mathcal{M}_{ell}[n](\mathbb{C})//SL_2(\mathbb{Z}/n\mathbb{Z})
  \simeq
  \mathcal{M}_{ell}(\mathbb{C})
  \,.
$$

## Definition

### Congruence subgroups

Let $n \in \mathbb{N}$ be a [[natural number]]. Write

$$
  p_n \;\colon\; SL_2(\mathbb{Z}) \to SL_2(\mathbb{Z}/\mathbb{Z}_n)
$$

for the projection from the [[special linear group]] induced by the [[quotient]] [[projection]] $\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$ to the [[cyclic group]].

The [[congruence subgroups]] of the [[special linear group]] $SL_2(\mathbb{Z})$ (essentially the [[modular group]]) are defined as follows.

The **principal congruence subgroup** is

$$
  \Gamma(n) \coloneqq ker(p_n) = p_n^{-1}\left(\left\{\array{ 1 & 0 \\ 0 & 1}\right\}\right)
$$

The other two are

$$
  \Gamma_0(n) \coloneqq p_n^{-1}\left(\left\{\array{ \ast & \ast \\ 0 & \ast}\right\}\right)
$$

$$
  \Gamma_1(n) \coloneqq p_n^{-1}\left(\left\{\array{ 1 & \ast \\ 0 & \ast}\right\}\right)
$$

### Over general rings

(e.g [Voloch, def. 1.1](#Voloch)  [Ando 00, section 1.4](#Ando00), [Ando-Hopkins-Strickland 02, section 15.2](#AndoHopkinsStrickland02), [Hill-Lawson 13, section 3.6](#HillLawson13))

## Properties

### Relation to spin structures on elliptic curves
 {#RelationToSpinStructures}

For [[elliptic curves]] over the [[complex numbers]] ([[complex manifold|complex]] oriented pointed [[tori]]) the congruence subgroup $\Gamma_0(2)$ has the interpretation as being precisely the subgroup of the [[modular group]] which preserves one of the "NS-R" [[spin structures]].

In detail, the elliptic curve $\Sigma$, being [[framed manifold|framed]] has a canonical [[spin structure]]  given by the trivial [[double cover]]. The space of all spin structures is a [[torsor]] over $H^1(\Sigma, \mathbb{Z}/2\mathbb{Z}) \simeq [\pi_1(\sigma), \mathbb{Z}/2\mathbb{Z}] \simeq [\mathbb{Z} \times \mathbb{Z}, \mathbb{Z}/2\mathbb{Z}] \simeq (\mathbb{Z}/2\mathbb{Z})^2$. In terms of this action the canonical one is labeled $(0,0)$ and then there are three more, labeled $(1,0)$, $(0,1)$ and $(1,1)$. The [[modular group]] acts on these via the quotient map $p_2 \;\colon\;  SL_2(\mathbb{Z}) \to SL_{2}(\mathbb{Z}/2\mathbb{Z})$. Hence it preserves $(0,0)$ and mixes the other three spin structures. Precisely $\Gamma_0(2)$ preserves $(1,0)$ (and an isomorphic subgroup of course preserves $(0,1)$). The principal congruence subgroup $\Gamma(2)$ is the one which preserves all four spin structures jointly.

In terms of [[type II string theory]] the spin structure $(1,0)$ is called the "NS-R boundary condition" for the spinors. The [[partition function]] of the [[type II superstring]] "in the NS-R sector" is therefore (at best, indeed it is, being the universal [[Ochanine elliptic genus]]) a [[modular form]] not for the full [[modular group]], but for $\Gamma_0(2)$ ([Witten 87a, below (13)](Witten87a)). For more on this see at _[Witten genus -- Modularity -- For the type II string](Witten+genus#ModularityForTypeIISuperstring)_. The homotopy-theoretic refinement of this involves [[tmf0(2)]], see at _[[spin orientation of Ochanine elliptic cohomology]]_.

## Properties

### Topological modular forms with level structure

The construction of [[topological modular forms]] ([[tmf]]) may be generalized to curves with level structure ([Mahowald-Rezk 09](#MahowaldRezk09)). A systematic kind of "[[modular equivariant elliptic cohomology]]" in this sense is discussed in ([Hill-Lawson 13](#HillLawson13)).

## References

### Over the complex numbers

The principal congruence subgroups are discussed for instance in

* {#Hain08} [[Richard Hain]], section 4.2 of _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

The relation of $\Gamma_1(2)$ to [[spin structures]] is discussed for instance in

* {#Freed87} [[Daniel Freed]], pages 24-25 of _On determinant line bundles_, 1987 ([pdf](http://www.ma.utexas.edu/users/dafr/detsur.pdf))

* {#Witten87a} [[Edward Witten]],  _Elliptic Genera And Quantum Field Theory_ , Commun.Math.Phys. 109 525 (1987) ([Euclid](http://projecteuclid.org/euclid.cmp/1104117076))

### Over general base rings

The concept of level structure on an elliptic curve is due to

* Nicholas M. Katz, [[Barry Mazur]], _Arithmetic moduli of elliptic curves_, Princeton University Press, Princeton, NJ, 1985

Review of the definition includes

* {#Voloch} [[Felipe Voloch]], _Modular curves -- 1. Level structures_ ([pdf](https://www.ma.utexas.edu/users/voloch/Ellnotes/feb23.pdf))

General discussion is in

* {#Greicius09} [[Aaron Greicius]], _Elliptic curves with surjective adelic Galois representations_ ([arXiv:0901.2513](http://arxiv.org/abs/0901.2513))

* David Zywina, _Elliptic curves with maximal Galois action on their torsion points_ ([arXiv:0809.3482](http://arxiv.org/abs/0809.3482))

Discussion of the corresponding [[moduli stack]] and its [[tmf]]$(n)$-spectrum is in

* {#Ando00} [[Matthew Ando]], section 1.4 of _Power operations in elliptic cohomology and representations of loop groups_ Transactions of the American
Mathematical Society 352, 2000, pp. 5619-5666. ([JSTOR](http://www.jstor.org/stable/221905), [pdf](http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf))

* {#AndoHopkinsStrickland02} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], part 3 of _The sigma orientation is an H-infinity map_. American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 ([arXiv:math/0204053](http://arxiv.org/abs/math/0204053))

* {#MahowaldRezk09} [[Mark Mahowald]] [[Charles Rezk]], _Topological modular forms of level 3_, Pure Appl. Math. Quar. 5 (2009) 853-872 ([pdf](http://www.math.uiuc.edu/~rezk/tmf3-paper-final.pdf))

* {#HillLawson13} [[Michael Hill]], [[Tyler Lawson]], _Topological modular forms with level structure_ ([arXiv:1312.7394](http://arxiv.org/abs/1312.7394))

Specifically Level-2 structure in this context is discussed in

* {#Stojanoska11} [[Vesna Stojanoska]], _Duality for Topological Modular Forms_ ([arXiv:1105.3968](http://arxiv.org/abs/1105.3968))

* {#Behrens05} [[Mark Behrens]], section 1.3 of _A modular description of the K(2)-local sphere at the prime 3_ ([arXiv:math/0507184](http://arxiv.org/abs/math/0507184))

[[!redirects level structures on an elliptic curve]]

[[!redirects level structures on elliptic curves]]

[[!redirects level N-structure]]
[[!redirects level N structure]]
[[!redirects level N-structures]]
[[!redirects level N structures]]

[[!redirects level-n structure]]
[[!redirects level-n structures]]

[[!redirects level-n structure]]
[[!redirects level-n structures]]

[[!redirects level n-structure]]
[[!redirects level n structure]]
[[!redirects level n-structures]]
[[!redirects level n structures]]

[[!redirects level n subgroup]]
[[!redirects level n subgroups]]

[[!redirects level n-structure on an elliptic curve]]
[[!redirects level n-structure on elliptic curves]]

[[!redirects elliptic curve with level-n structure]]
[[!redirects elliptic curves with level-n structure]]

[[!redirects elliptic curve with level structure]]
[[!redirects elliptic curve with level structures]]

[[!redirects elliptic curves with level structure]]
[[!redirects elliptic curves with level structures]]


Levi-Civita connection

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

#Contents#
* table of contents
{: toc}

## Idea

The **Levi-Civita connection** is the unique symmetric [[connection on a bundle|connection]] on the [[tangent bundle]] of a [[Riemannian manifold]] or [[pseudo-Riemannian manifold]] that is compatible with the [[Riemannian metric|metric]] or [[pseudo-Riemannian metric|pseudo-metric]].  The [[curvature]] and [[geodesic]]s on a pseudo-Riemannian manifold are taken with respect to this connection.  The existence and uniqueness of the Levi-Civita connection is called the _[[fundamental theorem of Riemannian geometry]]_.

## Definition

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

For $(X,g)$ a [[Riemannian manifold]], the **Levi-Civita connection** $\nabla_g$ on $X$ is the unique [[connection on a bundle|connection]] on the [[tangent bundle]] $T X$ that

1. the [[covariant derivative]] of the metric vanishes, $\nabla_{g} g = 0$;

1. $\nabla_g$ has vanishing [[torsion of a metric connection|torsion]].

=--

We say in detail what this means in "[[first order formulation of gravity|first order formalism]]"/[[Cartan geometry]].

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

For $(X,g)$ a $(d-k,k)$-[[dimension]]al [[Riemannian manifold]] (for $k = 0$) or [[pseudo-Riemannian manifold]] (for $k = 1$), the **Levi-Civita connection** $\nabla_g$ on $X$ is the unique $ISO(d-k,k)$-[[connection on a bundle]] $\nabla$ (for $ISO(d-k)$ the [[Poincare group]]) such that

1. **metric compatibility**:  the [[vielbein]] $e$ gives the metric: $g = e \cdot e$;

1. **torsion freeness**: the curvature of the vielbein (the [[torsion]]) vanishes: $T = 0$.

=--

More in detail, locally on a patch $U_i \subset X$ the $ISO(d-k,k)$-connection $\nabla$ is given by a [[Poincare Lie algebra]]-[[Lie algebra valued 1-form|valued 1-form]]

$$
  (e, \omega) : T U_i \to \mathfrak{iso}(d-k,k) \simeq \mathbb{R}^d \ltimes \mathfrak{so}(d-k,k)
$$

with

1. $e_i$ an $\mathbb{R}^d$-valued form -- the [[vielbein]];

1. $\omega_i$ a [[special orthogonal Lie algebra]]-valued form -- the "[[spin connection]]".

The [[curvature]] 2-form of this similarly decomposes into

1. the [[torsion]] $T^a := F_{e}^a = d e^a + \omega^a{}_b \wedge e^b$,

   (this equation is also called the **first Cartan structure equation**)

1. the [[Riemann curvature]] $R_{g}{}^a{}_b := F_{\omega}^a{}_b = d \omega^a{}_b + \omega^a{}_c \wedge \omega^c{}_d$.

   (this equation is also called the **second Cartan structure equation**)

The [[Bianchi identity]] satisfied by this curvature is

1. $d F_e^a + \omega^a{}_b \wedge F_e^b = F_\omega^a{}_b \wedge e^b$;

1. $d F_\omega^a{}_b + \omega^a{}_c \wedge F_\omega^c{}_b - F_\omega^a{}_c \wedge \omega^c{}_b = 0$.

The metric compatibility condition in the definition of Levi-Civita connection says that

$$
  g = e^a \otimes e_a
  \,.
$$

The torsion-freeness condition says that

$$
  F_e = 0
  \,.
$$

## In terms of Christoffel symbols

The Levi-Civita connection may be discussed in terms of its components -- called [[Christoffel symbol]]s -- given by the canonical local trivialization of the [[tangent bundle]] over a coordinate patch. This has been the historical route and is still widely used in the literature.

**Metric compatibility**

Here a metric $g$ is **compatible** with the connection $\nabla$ or _preserved_ by it (here thought of in its incarnation as a [[covariant derivative]]) if and only if $\nabla_X g = 0$ for all $X$, which is equivalent to the preservation of the metric inner product of tangent vectors under [[parallel transport|parallel translation]].  Since
\[ X(g(X_1,X_2))  = (\nabla_X g)(X_1,X_2) + g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2) ,\]
by the fact that covariant differentiation commutes with contractions and satisfies the derviative identity, compatibility is equivalent to
\begin{equation}  X (g (X_1,X_2)) = g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2),\end{equation}
for all  $X,X_1, X_2$.

**Uniqueness and existence on $\mathbb{R}^n$**

Now assume $M \subset \mathbb{R}^n$ and we have such a connection associated to $g$.
Then the connection is uniquely determined by its [[Christoffel symbols]], which we can determine in terms of $g$ by a bit of elementary algebra.
In other words, we just need to compute $\nabla_{\partial_i} \partial_j$.
Now
\[ \partial_k g( \partial_i, \partial_j) = g( \nabla_{\partial_k} \partial_i, \partial_j) + g(  \partial_i, \nabla_{\partial_k}\partial_j).\]
We can get two other equations by cyclic permutation:
\[ \partial_i g( \partial_j, \partial_k) = g( \nabla_{\partial_i} \partial_j, \partial_k) + g(  \partial_j, \nabla_{\partial_i}\partial_k)\]
\[ \partial_j g( \partial_k, \partial_i) = g( \nabla_{\partial_j} \partial_k, \partial_i) + g(  \partial_k, \nabla_{\partial_j}\partial_i)\]
So let $S_{i j} := \nabla_{\partial_i} \partial_j = \nabla_{\partial_j} \partial_i$, by symmetry.
Let $T_{i j k} := \partial_i g( \partial_j, \partial_k)$; these are smooth real functions.
These equations can be written
\[ T_{k i j} = g( S_{i k}, \partial_j) + g( S_{j k}, \partial_i) \]
\[ T_{i j k} = g( S_{i j}, \partial_k) + g( S_{i k}, \partial_j) \]
\[ T_{j k i} = g( S_{j k}, \partial_i) + g( S_{i j}, \partial_k) \]
These are three linear equations in the unknowns
$g( S_{i k}, \partial_j),   g( S_{j k}, \partial_i), g( S_{i j}, \partial_k)$.  The system is nonsingular, so we get a unique solution, and consequently by nondegeneracy a unique possibility for the $S_{i j}$.

Incidentally, we have in fact shown the uniqueness assertion of the general theorem, since that is local.

We shall now prove existence in this restricted case.  Choose $S_{i j}$ to satisfy the system of three equations outlined above where $i \lt j \lt k$.  Then set $S_{j i} := S_{i j}$, and we have a connection $\nabla$ with $\nabla_{\partial_i} \partial_j := S_{i j}$ since the vector fields $\partial_i$ are a frame (i.e. a basis at each tangent space on $M$).
It is symmetric, since the torsion $T$ vanishes (by $S_{i j}=S_{j i}$) on pairs $(\partial_i,\partial_j)$, and hence identically, since it is a tensor.

We must check for compatibility.
The difference of the two terms in (1) vanishes when $X,X_1,X_2$ are of the form $\partial_i$.  The vanishing holds generally because the difference of the two sides, which is $(\nabla_X g)(X_1,X_2)$, is a tensor.  Hence compatibility follows.

**Uniqueness and existence in the general case**

We have already shown the uniqueness assertion, since that is local.  Connections restrict to connections on open subsets.

We have proved the existence of $\nabla$ when $M$ is an open submanifold of $\mathbb{R}^n$ (though not necessarily with the canonical metric $\sum_{i=1}^n d x_i \otimes d x_i$).  In general, cover $M$ by open subsets $U_i$ diffeomorphic to an open set in $\mathbb{R}^n$.  We get connections $\nabla_i$ on $U_i$ compatible with $g|_{U_i}$.

We claim that $\nabla_i|_{U_i \cap U_j} = \nabla_j|_{U_i \cap U_j}$.  This is an easy corollary of uniquness.  So we can patch the connections together to get the one  Levi-Civita connection on $M$.

## Applications

### In physics

In the [[physics]], the theory of [[general relativity]] models the field of [[gravity]] in terms of the Levi-Civita connection on a [[Lorentzian manifold]]. See there for more details.

## Related concepts

* [[Weitzenböck connection]]

* [[connection on a bundle]]

  * [[parallel transport]], [[holonomy]]

* [[principal connection]]

  * [[affine connection]], **Levi-Civita connection**, [[Cartan connection]]

* [[connection on a 2-bundle]]

* [[connection on an infinity-bundle]]

  * [[higher parallel transport]]

## References

A discussion in terms of [[synthetic differential geometry]] is in

* [[Gonzalo Reyes]], _General Relativity:

  _Metrics, connections and curvature_ ([pdf](http://po-start.com/reyes/wp-content/uploads/2007/01/metrics.pdf))

  _The Riemann-Christoffel tensor_ ([pdf](http://po-start.com/reyes/wp-content/uploads/2007/01/the-riemann-christoffel-tensor.pdf))

  _Affine connections, parallel
transport and sprays_ ([pdf](http://po-start.com/reyes/wp-content/uploads/2009/01/affineconnections.pdf))

[[!redirects Levi-Civita connections]]
[[!redirects metric connecitons]]


Levi-Civita symbol

#Contents#
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## Idea

For $n \in \mathbb{N}$, and $\sigma \in Sym(n)$ a [[permutation]] of $n$ elements, the _Levi-Civita symbol_ or _antisymmetric symbol_

$$
  \epsilon_{\sigma(1), \sigma(2), \cdots, \sigma(n)}
  \;\in\;
  \{\pm 1\}
$$

is ([[tensor]]-style notation for) the [[signature of a permutation|signature]] of the permutation.

Together with the [[Kronecker delta]]-notation, the Levi-Civita symbol is convenient in expressions subject to [[Einstein summation convention]], then serving essentially as a combinatorial form of [[string diagram]]/[[Penrose notation]] in [[tensor categories]].

## Related concepts

* [[signature of a permutation]]

* [[Einstein summation convention]]

* [[Kronecker delta]]

## References

See also

* Wikipedia, _[Levi-Civita symbol](https://en.wikipedia.org/wiki/Levi-Civita_symbol)_

[[!redirects Levi-Civita symbols]]

[[!redirects antisymmetric symbol]]
[[!redirects antisymmetric symbols]]


Levi-Civita tensor

__Levi-Civita__ tensor is a totally antisymmetric tensor of rank $k$ in $k$-dimensional vector space.

In the space $k^n$ where $k$ is the ground field, in the basis basis $e_i = (0,0,\ldots, 1,0,\ldots, 0)$ where $1$ is in the $i$-th position, one defines the components of the Levi-Civita tensor $\epsilon_{i_1,\ldots,i_n}$ to be zero if at least two indices are the same, $1$ if $(i_1,\ldots,i_n)$ is an even permutation of $(1,\ldots, n)$ and $-1$ if it is an odd permutation of $(1,\ldots,n)$. This has a generalization for an arbitrary [[Riemannian manifold]]. There are many useful identities in calculating with Levi-Civita tensor.

* wikipedia [Levi-Civita symbol](http://en.wikipedia.org/wiki/Levi-Civita_symbol)
* Weinstein's wolfram world: [Permutation Tensor](http://mathworld.wolfram.com/PermutationTensor.html)
* Mathematics Wiki [Levi-Civita symbol](http://www.mathematics.thetangentbundle.net/wiki/Levi-Civita_symbol)


Levin-Wen model

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## Idea

The Levin-Wen model is a [[model (in theoretical physics)]] for certain phenomena in 2-dimensional [[solid state physics]]. It can be regarded as a  [[3d TQFT]].

## Related concepts

* The [[4d TQFT]] analog of the Levin-Wen model is the [[Walker-Wang model]].

* [[topological order]]

## References

The original article is

* [[Michael Levin]], [[Xiao-Gang Wen]], *String-net condensation: A physical mechanism for topological phases*, Phys.Rev. B **71** (2005) 045110 $[$[arXiv:cond-mat/0404617](http://arxiv.org/abs/cond-mat/0404617)$]$

A discussion of the Levin-Wen model as an [[extended TQFT|extended]] [[3d TQFT]] formulated using [[tensor categories]] is in

* [[Alexei Kitaev]], [[Liang Kong]], _Levin-Wen models and tensor categories_ ([pdf](https://ochicken.top/Library/Physics/Quantum_Computation_and_Quantum_Information/2011.12%20Liang%20Kong_%20Levin-Wen_model_and_tensor_categories.pdf))

[[!redirects Levin-Wen models]]


Lex

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## Idea

**$Lex$** is a widely used name for the [[2-category]] of:

* [[small category|small]]$\;$[[finitely complete categories]],

* [[left exact functor|finite-limit-preserving functors]] between these, and

* [[natural transformations]] between those.

In [[categorical logic]], such categories are also known as *finite limits theories* or *[[essentially algebraic theories]]*.  Finite-limit-preserving functors are also known as *[[left exact functors]]* or **lex functors**, and these give $Lex$ its name.

## Related concepts

[[!include categories of categories - contents]]

## References

A study of [[descent]] in $Lex$:

* [[Marek Zawadowski]], _Lax descent theorems for left exact categories_ (1995) ([pdf](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-6bb9d2b7-234c-4e9f-8a17-24ecbae7658a/c/rm34601.pdf))

category: category

[[!redirects LexCat]]


lex category > history

< [[lex category]]
[[!redirects lex category -- history]]


lex total category

[[!redirects empty 112]]
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## Idea

> Lex-total categories are the good notion of topos. ([[Ross Street]], [1981](#Street81) p.201)

A **lex total category** is a category whose [[Yoneda embedding]] is a [[localization]] thereby generalizing [[Giraud's theorem|Giraud's characterization]] of [[Grothendieck topos|Grothendieck toposes]] as localizations of [[presheaf topos|presheaf toposes]] hence they can be viewed as a candidate for a notion of "topos".

## Definition

Recall that a locally small category $\mathcal{C}$ is called [[total category|total]] if the [[Yoneda embedding]] $Y:\mathcal{C}\hookrightarrow Set^{\mathcal{C}^{op}}$ has a left adjoint $L$.

+-- {: .num_defn}
###### Definition
A locally small category $\mathcal{C}$ is called _lex total_ if the [[Yoneda embedding]] $Y:\mathcal{C}\hookrightarrow Set^{\mathcal{C}^{op}}$ has a [[left exact]] (=finite limit preserving) left adjoint $L$.

=--

## Examples

In particular, [[totally distributive category|totally distributive categories]] where $L$ has a further left adjoint $W$ are lex total.

Below we will see that all [[Grothendieck toposes]] are lex total.

## Properties

+-- {: .num_prop}
###### Theorem
A lex total category $\mathcal{E}$ is a [[Grothendieck topos]] iff $\mathcal{E}$ has a small set of [[generator|generators]].
=--

The result was announced by Walters on the [[Isle of Thorns]] in 1976, a proof can be found in Street ([1981](#Street81), p.206). More detailed information on this characterization in particular concerning the size issues involved and the algebraic perspective it avails can be found at [[Grothendieck topos]] or the [blog posts](#Walters14) by Bob Walters.

## Related entries

* [[total category]]

* [[totally distributive category]]

## Link

* {#Walters14}[[Bob Walters]], _Lex total categories and Grothendieck toposes I-IV_ , series of blog posts 2014. ([link](http://rfcwalters.blogspot.com/2014/07/lex-total-categories-and-grothendieck.html))

## References

* {#LW11} [[Rory Lucyshyn-Wright]], _Totally distributive toposes_ , arXiv:1108.4032 (2011). ([abstract](https://arxiv.org/abs/1108.4032))

* [[Ross Street]], _The family approach to total cocompleteness and toposes_ , Trans. A. M. S. **284** (1978) pp.355-369.

* {#Street81}[[Ross Street|R. Street]], _Notions of topos_ , Bull. Austral. Math. Soc. **23** no.2 (1981) pp.199-208.

* {#SW78}[[Ross Street|R. Street]], [[Bob Walters|R. Walters]], _Yoneda structures on 2-Categories_ , JA **50** no.2 (1978) pp.350-379.

* [[R. J. Wood]], _Some remarks on total categories_ , JA **75** 2 (1982) 538-545 [doi:10.1016/0021-8693(82)90055-2]

[[!redirects lex total categories]]
[[!redirects lex-total categories]]
[[!redirects lex-total category]]
[[!redirects Lex total category]]
[[!redirects Lex total categories]]


lexicographic order

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## Idea

The _lexicographic order_ is a generalization of the order in which words are listed in a dictionary, according to the order of letters where the spelling of two words first differs.

## Definition

+-- {: .num_defn}
###### Definition
Let $\{L_i\}_{i \in I}$ be a [[well-ordered set|well-ordered]] family of [[strict total order|strictly totally ordered sets]]. The **lexicographic order** on the [[product]] of sets $L = \prod_{i \in I} L_i$ is the [[strict total order]] defined as follows: if $x, y \in L$ and $x \neq y$, then $x \lt y$ iff $x_i \lt y_i$ where $i$ is the least element in the set $\{j \in I: x_j \neq y_j\}$.
=--

While this notion is most often seen for strict total orders, it can be applied also toward more general [[relations]]. For example, one might apply the construction to sets equipped with a transitive relation $\lt$, dropping the [[strict total order|trichotomy assumption]].

Often this notion is extended to subsets of $\prod_{i \in I} L_i$ as well. For instance, the [[free monoid]] $S^\ast$ on a linearly ordered set $S$ can be embedded in a countable power

$$i \colon S^\ast \hookrightarrow (1 + S)^\mathbb{N} = \prod_{n \in \mathbb{N}} (1 + S)$$

where $1 + S$ is the result of freely adjoining a bottom element $e$ to $S$, and for each finite list $(s_1, \ldots, s_k)$ we have

$$i(s_1, \ldots, s_k) = (s_1, s_2, \ldots, s_k, e, e, e, \ldots).$$

Then the lexicographic order on $S^\ast$ is the one inherited from its embedding into the lexicographically ordered set $(1 + S)^\mathbb{N}$.

+-- {: .num_remark #antidictionary}
###### Remark
The decision to freely adjoin a _bottom_ element $e$ is of course purely a convention, based on the ordinary dictionary convention that the Scrabble word AAH should come after AA. Alternatively, we could equally well deem that $e$ is a freely adjoined top element, so that AA comes after AAH; this might be called the "anti-dictionary" convention.
=--

### Corecursive definition

if $L$ is linearly ordered and the underlying set $C = L^\mathbb{N}$ is regarded as the [[terminal coalgebra]] for the functor $L \times - \colon Set \to Set$, with coalgebra structure $\langle h, t \rangle \colon C \to L \times C$, then the lexicographic order on $C$ may be defined [[corecursion|corecursively]]:

* $c \lt c'$ if $h(c) \lt h(c')$ or ($h(c) = h(c')$ and $t(c) \lt t(c')$).

For the special case $L = \mathbb{N}$, the terminal coalgebra $\mathbb{N}^\mathbb{N}$ with this lexicographic order is order-isomorphic to the real interval $[0, \infty)$. This isomorphism is described more precisely [here](/nlab/show/continued+fraction#PavPratt).

[[!redirects lexicographic ordering]]
[[!redirects dictionary order]]
[[!redirects dictionary ordering]]


LF

[[!redirects Elf]]

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## Idea

**LF** ([Pfenning 91](#Pfenning91), [Harper-Honsell-Plotkin 93](#HarperHonsellPlotkin93)) is a certain [[logical framework]] based on [[dependent type theory|dependent]] [[intuitionistic type theory]]. Variants include the _Edinburgh logical framework_, abbreviated "Elf", and _linear LF_, abbreviated LLF ([Pfenning 96](Pfenning96)), the latter capturing also [[dependent linear type theory]].

## Related concepts

* [[Automath]]

* [[Twelf]]

## References

The logical framework LF originates around

* {#Pfenning91} [[Frank Pfenning]], _Logic Programming in the LF Logical Framework_ (1991) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.40.3910))

* {#HarperHonsellPlotkin93} [[Robert Harper]], F. Honsell, G. Plotkin, _A framework for defining logics_, Journal for the association for computing machinery 40(1):143-184 (1993)

For Elf see

* _[Elf Homepage](http://www.cs.cmu.edu/~fp/elf.html)_

Extension to a framework for [[dependent linear type theory]] called LLF is discussed in

* {#Pfenning96} Iliano Cervesato, [[Frank Pfenning]], _A Linear Logical Framework_, 1996, ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.1152))

* {#WCFW03} Kevin Watkins, Iliano Cervesato, [[Frank Pfenning]], David Walker, _A concurrent logical framework I: Judgments and properties_, CMU technical report CMU-CS-02-101, revised May 2003 ([web](http://www.cs.cmu.edu/~fp/papers/CMU-CS-02-101.pdf))

category: software

[[!redirects Edinburgh Logical Framework]]
[[!redirects Edinburgh logical framework]]
[[!redirects ELF]]

[[!redirects LF]]


LHC

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## Description

An [[experiment]] in [[particle physics]] measuring the details of the [[standard model of particle physics]]. Detected the [[Higgs boson]].

Subexperiments include: [[LHCb]], ...

## Properties

Resolution about $10^{-20}m \simeq 10$ [[TeV]] and currently reached $\simeq 7 TeV$  (e.g. [Mazumdar, slide 7](#Mazumdar))

## Related entries

* [[LIGO]], [[Super-Kamiokande]], [[EDGES]], [[RHIC]]

* [[LHCb]], [[Belle experiment]]

* [[standard model of particle physics]]

* [[Higgs boson]]

* [[flavour anomaly]]

## References

* Wikipedia, _[CERN](http://en.wikipedia.org/wiki/CERN)_

* Kajari Mazumdar, _Essentials of the LHC_ ([pdf](http://www.tifr.res.in/~mazumdar/talks/glbrg2.pdf))
 {#Mazumdar}

* [[John Ellis]], _Summary of the Nobel Symposium on LHC Results_ ([arXiv:1309.3549](http://arxiv.org/abs/1309.3549))

List of [[hadrons]] discovered at the LHC:

* {#Koppenburg21} [[Patrick Koppenburg]] for the [[LHCb collaboration]]: *List of hadrons observed at the LHC* (2021) [[pdf](https://cds.cern.ch/record/2749030/files/FIGURE-2021-001.pdf), [webpage](https://www.nikhef.nl/~pkoppenb/particles.html), [cds:2749030](https://cds.cern.ch/record/2749030)]

The [[LHCb]] experiment on [[B-meson]] decays (see also at _[[flavour anomaly]]_):

* [lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/Summary_all.html](http://lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/Summary_all.html)

Outlook on results such as [[flavour anomalies]], search for [[leptoquarks]] etc.

* [[Salvatore Rappoccio]], _The experimental status of direct searches for exotic physics beyond the standard model at the Large Hadron Collider_, Reviews in Physics 4 (2019) 100027 ([arXiv:1810.10579](https://arxiv.org/abs/1810.10579))

[[!redirects Large Hadron Collider]]

[[!redirects LHC experiment]]

[[!redirects ATLAS collaboration]]
[[!redirects CMS collaboration]]


LHCb experiment

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## Idea

The _LHCb experiment_ is one detector/[[experiment]] within the [[Large Hadron Collider]] ("LHC") with focus on the [[physics]] of [[B-mesons]] (similar to the [[Belle experiment]]).

After the discovery of the [[Higgs field]] at the two major other detectors of [[LHC]], the LHCb experiment currently stands out as seeing (together with the [[Belle experiment]] and the [[BaBar experiment]]) hints for physics beyond the [[standard model of particle physics]] with considerable [[statistical significance]]: the _[[flavour anomalies]]_ (see there for more).

## Related concepts

* related [[experiments]]

  * [[Belle experiment]]

  * [[BaBar experiment]]

* observations

  * [[flavour anomaly]]

  * [[V_cb puzzle]]

  * [[pentaquark]]

  * [[XYZ particle]]

* [[precision experiment]]

## Selected publications

### On flavour anomalies

On [[flavour anomalies]]:

* {#Koppenburg07} [[Patrick Koppenburg]], _Selection of $B_u \to \ell \ell K$ at LHCb and Sensitivity to $R_K$_, 2007 ([cern:1027442](cds.cern.ch/record/1027442), [pdf](cds.cern.ch/record/1027442/files/lhcb-2007-034.pdf))

* [[LHCb collaboration]], _Differential branching fractions and isospin asymmetries of $B \to K^{(\ast)}\mu^+ \mu^-$ decays_, JHEP 06 (2014) 133 ([arXiv:1403.8044](https://arxiv.org/abs/1403.8044))

* [[LHCb collaboration]], _Test of lepton universality using $B^+\to K^+ \ell^+ \ell^-$ decays_, Phys. Rev. Lett. 113, 151601 (2014) ([arXiv:1406.6482](https://arxiv.org/abs/1406.6482))

* [[LHCb collaboration]], _Angular analysis and differential branching fraction of the decay $B^0_s \to \phi \mu^+ \mu^-$_, JHEP09(2015)179 ([arXivL1506.08777](https://arxiv.org/abs/1506.08777))

* [[LHCb collaboration]], _Differential branching fraction and angular analysis of $\Lambda^0_b \to \Lambda \mu^+ \mu^-$ decays_, JHEP 06 (2015) 115; JHEP 09 (2018) 145 ([arXiv:1503.07138](https://arxiv.org/abs/1503.07138))

* [[LHCb collaboration]], _Measurement of the ratio of branching fractions of the decays $\Lambda^0_b \to \psi(2S)\Lambda$ and $\Lambda^0_b \to J/\psi\Lambda$_ ([arXiv:1902.02092](https://arxiv.org/abs/1902.02092))

* [[LHCb collaboration]], _Test of lepton universality using $B^0\to K^{\ast 0} \ell^+ \ell^-$ decays_, JHEP 08 (2017) 055 ([arXiv:1705.05802](https://arxiv.org/abs/1705.05802))

* [[LHCb collaboration]], _Measurement of the ratio of the $B^0 \to D^{\ast -} \tau^+ \vu_\tau$ and $B^0 \to D^{\ast - \mu^+ \nu_\mu}$ branching fractions using three-prong τ-lepton decays_ ([arXiv:1708.08856](https://arxiv.org/abs/1708.08856))

* {#Cartelle18} Paula Alvarez Cartelle on behalf of the [[LHCb collaboration]], _New results on flavor anomalies at LHCb_, April 2018 ([pdf](https://cds.cern.ch/record/2311960/files/FlavourAnomaliesLHCbAlvarez.pdf))

* {#Dey18} Biplab Dey on behalf of the [[LHCb collaboration]], _Flavor anomalies at LHCb_, May 2018 (pdf)

* {#CPCharm19} [[LHCb collaboration]],  _Observation of CP violation in charm decays_, March 2019 ([pdf](http://cds.cern.ch/record/2668357/files/LHCb-PAPER-2019-006.pdf), [arXiv:1903.08726](https://arxiv.org/abs/1903.08726))

  exposition: _[LHCb sees a new flavour of matter–antimatter asymmetry](https://home.cern/news/press-release/physics/lhcb-sees-new-flavour-matter-antimatter-asymmetry)_, March 2019

* {#Betti19} Federico Betti on behalf of [[LHCb collaboration]], _Observation of CP violation in charm decays at LHCb_, Moriond 2019 ([pdf](http://moriond.in2p3.fr/2019/EW/slides/5_Thursday/1_morning/4_betti_federico_v4.pdf))

* [[LHCb collaboration]], _Search for lepton-universality violation in $B^+ \to K^+ \ell^+ \ell^-$ decays_ ([cern:2668514](https://cds.cern.ch/record/2668514))

* [[LHCb collaboration]], _Search for lepton-universality violation in $B^+ \to K^+ \ell^+ \ell^-$ decays_ ([cern:2668514](https://cds.cern.ch/record/2668514))

* {#LHCb19} [[LHCb collaboration]], _Lepton flavour universality in charged-current $B$ decays_ ([arXiv:1907.01500](https://arxiv.org/abs/1907.01500))

* [[LHCb collaboration]], _Test of lepton universality with $\Lambda^0_b\to p K^-\ell^+\ell^-$ decays_ ([arXiv:1912.08139](https://arxiv.org/abs/1912.08139))

  also:

  CERN courier, _[Debut for baryons in flavour puzzle](https://cerncourier.com/a/debut-for-baryons-in-flavour-puzzle/)_, 20 Dec. 2019

  CERN news, _[LHCb explores the beauty of lepton universality](https://home.cern/news/news/physics/lhcb-explores-beauty-lepton-universality)_, 15 Jan 2020

  (first measurement of lepton universality for $\Lambda^0_b$ [[Lambda baryons]], finding the [[flavour anomaly]] at (only) 1 [[standard deviation|sigma]] but pointing in the same direction as seen in other channels)

* [[LHCb collaboration]], _Measurement of CP-averaged observables in the $B^0 \to K^{\ast 0} \mu^+ \mu^-$ decay_ ([cds:2712641](http://cds.cern.ch/record/2712641), [arXiv:2003.04831](https://arxiv.org/abs/2003.04831))

  exposition:

  {#Smith20} Eluned Smith on behalf of the [[LHCb collaboration]], _Update of the $B^0 \to K^{\ast 0} \mu^+ \mu^-$ angular analysis at LHCb_, 2020 ([[SmithatLHCbOnFlavourAnomalies2020.pdf:file]])

  {#Lopes20} Ana Lopes, _[New LHCb analysis still sees previous intriguing results](https://home.cern/news/news/physics/new-lhcb-analysis-still-sees-previous-intriguing-results)_, CERN News 2020

  {#CERNCourir200311} CERN Courier, _[Anomalies persist in flavour-changing B decays](https://cerncourier.com/a/anomalies-persist-in-flavour-changing-b-decays/)_, March 13 2020

* [[LHCb collaboration]], _Measurement of CP-averaged observables in the $B^0\to K^{\ast 0} \mu^+ \mu^-$ decay_ ([spire:1784890](https://inspirehep.net/literature/1784890), [arXiv:2003.04831](https://arxiv.org/abs/2003.04831))

* [[LHCb collaboration]] *Tests of lepton universality using $B^0 \to K^0_S l^+ l^-$− and $B^+ \to K^\ast \ell^+ \ell^-$ decays* ([arXiv:2110.09501](https://arxiv.org/abs/2110.09501))

* Gregory Max Ciezarek on behalf of the [[LHCb collaboration]]: *First joint measurement of $R(D^\ast)$ and $R(D^0)$ at LHCb*, LHC seminar (2022) [[indico:1187939](https://indico.cern.ch/event/1187939), slides: [pdf](https://indico.cern.ch/event/1187939/attachments/2530158/4353109/dtaunu_CERNSeminar.pdf)]

survey:

* [[Niels Tuning]], _Recent results from LHCb_, Nikhef 2020 ([nikhef event:2253](https://indico.nikhef.nl/event/2253), [pdf](https://indico.nikhef.nl/event/2253/material/slides/0.pdf), [[TuningFlavourAnomaly20.pdf:file]])

  > In recent years a number of results in flavour physics have drawn some attention due to tensions with respect to Standard Model predictions. An overview of these results will be shown, together with a few recent results from this Spring, both on the flavour anomalies and 'classical' flavour physics

Claim that the apparent $R_K$-anomalies were due to mis-identification of hadronic backgrounds:

* {#LHCB22} [[LHCb collaboration]], *Test of lepton universality in $b \to s \ell^+ \ell^-$ decays LHCb collaboration* [[arXiv:2212.09152](https://arxiv.org/abs/2212.09152)]

  [[LHCb collaboration]], *Measurement of lepton universality parameters in $B^+ \to K^+ \ell^+ \ell^-$ and $B^0 \to K^{\ast 0} \ell^+ \ell^-$ decays* [[arXiv:2212.09153](https://arxiv.org/abs/2212.09153)]

  Davide Castelvecchi, *[Hint of crack in standard model vanishes in LHC data](https://www.nature.com/articles/d41586-022-04545-z)*, Nature News (20 Dec 2022)

### On tetraquarks

On [[tetraquarks]]:

fully charmed:

* [[LHCb collaboration]], *Observation of structure in the $J/\psi$-pair mass spectrum*, Science Bulletin 65 (2020) 1983 ([arXiv:2006.16957](https://arxiv.org/abs/2006.16957), [doi:10.1016/j.scib.2020.08.032](https://doi.org/10.1016/j.scib.2020.08.032), [CERN news](https://phys.org/news/2020-07-cern-physicists-discovery-unique-particle.html))

doubly charmed:

* [[LHCb collaboration]], *Observation of an exotic narrow doubly charmed tetraquark* ([arXiv:2109.01038](https://arxiv.org/abs/2109.01038))

* [[LHCb collaboration]], *Study of the doubly charmed tetraquark $T^+_{c c}$* ([arXiv:2109.01056](https://arxiv.org/abs/2109.01056))

## References

* [LHCb public page](http://lhcb-public.web.cern.ch/lhcb-public/)

* [LHCb collaboration](http://lhcb.web.cern.ch/lhcb/)

* [LHCb public results](http://lhcbproject.web.cern.ch/lhcbproject/Publications/LHCbProjectPublic/Summary_all.html)

* {#LHCbCollaboration18} [[LHCb collaboration]], _Physics case for an LHCb Upgrade II - Opportunities in flavour physics, and beyond, in the HL-LHC era_ ([arXiv:1808.08865](https://arxiv.org/abs/1808.08865))

See also

* Wikipedia, _[LHCb experiment](https://en.wikipedia.org/wiki/LHCb_experiment)_

[[!redirects LHCb]]

[[!redirects LHCb collaboration]]
[[!redirects LHCb collaborations]]

[[!redirects Large Hadron Collider beauty]]


Li Zhou

* [GoogleScholar page](https://scholar.google.com/citations?user=oEZ7iPkAAAAJ&hl=zh-CN)

## Selected writings

Introducing the [[quantum programming language]] *[[CoqQ]]*:

* [[Li Zhou]], [[Gilles Barthe]], Pierre-Yves Strub, Junyi Liu, [[Mingsheng Ying]], *CoqQ: Foundational Verification of Quantum Programs* [[arXiv:2207.11350](https://arxiv.org/abs/2207.11350)]

  > (in [[Coq]])

Understanding the [[linear logic]]-sector of [[bunched logic]] as a kind of [[quantum logic]]:

* [[Li Zhou]], [[Gilles Barthe]], [[Justin Hsu]], [[Mingsheng Ying]], [[Nengkun Yu]], *A Quantum Interpretation of Bunched Logic for Quantum Separation Logic*, LICS '21: Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science  75 (2021) 1–14 [[arXiv:2102.00329](https://arxiv.org/abs/2102.00329), [doi:10.1109/LICS52264.2021.9470673](https://doi.org/10.1109/LICS52264.2021.9470673)]

category: people


Liam McAllister

* [personal page](https://physics.cornell.edu/liam-mcallister)

* [Research group page](https://liammcallistergroup.com/)

## Selected writings

On [[moduli stabilization]] in [[heterotic string theory]]:

* [[Sergei Gukov]], [[Shamit Kachru]], Xiao Liu, [[Liam McAllister]], _Heterotic Moduli Stabilization with Fractional Chern-Simons Invariants_,  Phys.Rev.D **69** (2004) 086008 [[arXiv:hep-th/0310159](http://arxiv.org/abs/hep-th/0310159)]

On [[cosmic inflation]] in [[string theory]]:

* [[Daniel Baumann]], [[Liam McAllister]], _Inflation and String Theory_, Cambridge University Press (2015) [[arXiv:1404.2601](https://arxiv.org/abs/1404.2601), [doi:10.1017/CBO9781316105733]( https://doi.org/10.1017/CBO9781316105733)]

On realistic values of the [[cosmological constant]] in [[string theory]]:

* [[Liam McAllister]], _Small Cosmological Constants in String Theory_ [[indico:4940826](https://indico.cern.ch/event/1085701/contributions/4940826), [slides](https://indico.cern.ch/event/1085701/contributions/4940826/attachments/2481781/4260542/Liam%20McAllister%20Monday-Research-McAllister.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=1e5171ec-597a-4c34-bfb7-58d243b39ece)]

On [[moduli stabilization]] in [[string theory]]:

* [[Liam McAllister]], [[Fernando Quevedo]], *Moduli Stabilization in String Theory*, in: *[[Handbook of Quantum Gravity]]* [[arXiv:2310.20559](https://arxiv.org/abs/2310.20559)]

On [[KK-compactification]] in [[string theory]]:

* [[Michael R. Douglas]], [[Liam McAllister]], *Compactification of Superstring Theory*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]* [[arXiv:2310.20118](https://arxiv.org/abs/2310.20118)]

category: people


Liang Fu

* [personal page](https://physics.mit.edu/faculty/liang-fu/)

## Selected writings

On [[topological insulators]]:

* [[Liang Fu]], [[Charles Kane]], [[Eugene Mele]], *Topological Insulators in Three Dimensions*, Phys. Rev. Lett. **98** (2007) 106803 $[$[doi:10.1103/PhysRevLett.98.106803](https://doi.org/10.1103/PhysRevLett.98.106803)$]$

On realizing [[Majorana zero modes]] via [[topological insulators]]:

* {#FuKane08} [[Liang Fu]], [[Charles Kane]], *Superconducting Proximity Effect and Majorana Fermions at the Surface of a Topological Insulator*, Phys. Rev. Lett. **100** (2008) 096407 [[doi:10.1103/PhysRevLett.100.096407](https://doi.org/10.1103/PhysRevLett.100.096407)]

On [[topological crystalline insulators]]:

* [[Liang Fu]], *Topological Crystalline Insulators*, Phys. Rev. Lett. **106** (2011) 106802 ([doi:10.1103/PhysRevLett.106.106802](https://doi.org/10.1103/PhysRevLett.106.106802))

* [[Yoichi Ando]] and [[Liang Fu]], *Topological Crystalline Insulators and Topological Superconductors: From Concepts to Materials*, Annual Review of Condensed Matter Physics, **6** (2015) 361-381 $[$[doi:10.1146/annurev-conmatphys-031214-014501](https://doi.org/10.1146/annurev-conmatphys-031214-014501)$]$

* Timothy H Hsieh, Hsin Lin, Junwei Liu, Wenhui Duan, Arun Bansil, [[Liang Fu]], *Topological crystalline insulators in the SnTe material class*,
Nature Communications **3** (2012) 982 $[$[doi:10.1038/ncomms1969](https://doi.org/10.1038/ncomms1969)$]$

Further on [[symmetry protected topological phases]] under [[crystallographic symmetry]]:

* Hao Song, Sheng-Jie Huang, [[Liang Fu]], Michael Hermele, *Topological phases protected by point group symmetry*, Phys. Rev. X **7** (2017) 011020 $[$[arXiv:1604.08151](https://arxiv.org/abs/1604.08151), [doi:10.1103/PhysRevX.7.011020](https://doi.org/10.1103/PhysRevX.7.011020)$]$

category: people


Liang Kong

* [page at SIQSE](https://siqse.sustech.edu.cn/Index/staff_detail/mid/58/id/345)

* [page at SUST](https://faculty.sustech.edu.cn/kongl/en/)

* [Blog](https://kongliang.wordpress.com/)

* [Publication list](http://en.scientificcommons.org/liang_kong)

## Selected writings

On [[full field algebra]] for [[2d conformal field theory]]:

* {#HuangKong05} [[Yi-Zhi Huang]], [[Liang Kong]], _Full field algebras_, Commun. Math. Phys. **272** (2007) 345-396 [[arXiv:0511328](http://arxiv.org/abs/math/0511328), [doi:10.1007/s00220-007-0224-4](https://doi.org/10.1007/s00220-007-0224-4)]

* {#Kong06} [[Liang Kong]], _Full field algebras, operads and tensor categories_, Adv. Math.213:271-340, 2007 [[arXiv:0603065](http://arxiv.org/abs/math/0603065), [doi:10.1016/j.aim.2006.12.007](https://doi.org/10.1016/j.aim.2006.12.007)]

* {#Kong11} [[Liang Kong]], *Conformal field theory and a new geometry*, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), *[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Proceedings of Symposia in Pure Mathematics **83**, AMS (2011) [[arXiv:1107.3649](http://arxiv.org/abs/1107.3649), [ISBN:978-0-8218-5195-1](http://bookstore.ams.org/PSPUM-83)]

On [[enriched monoidal categories]]:

* [[Liang Kong]], Wei Yuan, Zhi-Hao Zhang, Hao Zheng, *Enriched monoidal categories I: centers* [[arXiv:2104.03121](https://arxiv.org/abs/2104.03121)]

On [[topological order]], [[braided fusion categories]] and the [[holographic principle]]:

* [[Liang Kong]], *Topological Wick Rotation and Holographic Dualities*, talk at [[CQTS]] (Oct 2022) [[[Kong-TalkAtCQTS-20221019.pdf:file]]]

## Related entries

* [[conformal field theory]]

* [[vertex operator algebra]]

category: people


Liang Ze Wong

* [personal page](http://sheaves.github.io/me/)

* [GoogleScholar page](https://scholar.google.com/citations?user=mkhDZ0xlXDIC)

## Selected writings

On the [[enriched Grothendieck construction]]:

* {#BeardsleyWong18} [[Jonathan Beardsley]], [[Liang Ze Wong]], _The Enriched Grothendieck Construction_, Advances in Mathematics **344** (2019) 234-261 [[arXiv:1804.03829](https://arxiv.org/abs/1804.03829), [doi:10.1016/j.aim.2018.12.009](https://doi.org/10.1016/j.aim.2018.12.009)]

* {#BeardsleyWong19} [[Jonathan Beardsley]], [[Liang Ze Wong]], *The operadic nerve, relative nerve and the Grothendieck construction*, Theory and Applications of Categories, **34** 13 (2019) 349-374 [[tac:34-13](http://www.tac.mta.ca/tac/volumes/34/13/34-13abs.html)]

On the [[Grothendieck construction]] in [[enriched category theory|enriched-]], [[internal category|internal-]] and [[(infinity,1)-category theory|$\infty$-category theory]] (the *[[enriched Grothendieck construction|enriched-]] and [[(infinity,1)-Grothendieck construction|$\infty$-Grothendieck construction]]):

* [[Liang Ze Wong]], *The Grothendieck Construction in Enriched, Internal and ∞-Category Theory*, PhD thesis, Univ. Washington (2019) [[pdf](http://sheaves.github.io/slides/thesis.pdf), [[Wong-GrothendieckConstruction.pdf:file]]]

category: people


liar paradox

This page is false.

See also [[paradox]] and _[[Russell's paradox]]_.

## References

In

* [[Bertrand Russell]], _[[Principia Mathematica]]_, 1910

a [[type theory]] of "[[ramified types]]" is introduced to prevent liar-type paradoxes from the [[foundations of mathematics]].

[[!redirects liar paradox]]
[[!redirects liar's paradox]]
[[!redirects liar's paradox]]
[[!redirects liar\' paradox]]


Lichnerowicz formula

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## Idea

An explicit expression in [[analysis]] for the square of a [[Dirac operator]] on a [[Riemannian manifold]].

The crucial point of the formula is that it shows a scalar summand (i.e. 0th order derivative) proportional to the [[scalar curvature]].

## Related concepts

* [[Stolz conjecture]]

## References

* Wikipedia, _[Lichnerowicz formula](http://en.wikipedia.org/wiki/Lichnerowicz_formula)_

[[!redirects Lichnerowicz theorem]]


Lichtenbaum-Quillen conjecture

The Quillen-Lichtenbaum conjecture is a conjecture of [[Daniel Quillen]] relating [[etale cohomology]] and [[algebraic K-theory]].
It is implied by the [[Bloch-Kato conjecture]], which was proved by [[Markus Rost]] and [[Vladimir Voevodsky]].

## Related concepts

* [[etale cohomology]]
* [[algebraic K-theory]]
* [[Bloch-Kato conjecture]]

## References

* Wikipedia, _[Quillen-Lichtenbaum conjecture](http://en.wikipedia.org/wiki/Quillen–Lichtenbaum_conjecture)_

[[!redirects Quillen-Lichtenbaum conjecture]]


Lie 2-algebra

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## Idea

A Lie 2-algebra is to a [[Lie 2-group]] as a [[Lie algebra]] is to a [[Lie group]].  Thus, it is a [[vertical categorification]] of a [[Lie algebra]].

## Definition

### Semistrict case

A ("semistrict") Lie 2-algebra $\mathfrak{g}$ is an [[L-∞-algebra]] with generators concentrated in the lowest two degrees.

This means that it is

* a pair of [[vector space]]s $\mathfrak{g}_0, \mathfrak{g}_1$

* equipped with [[linear functions]] as follows:

  a unary bracket $[-]$ encoding a _[[differential]]_

  $$
    \delta : \mathfrak{g}_1 \to \mathfrak{g}_0
    \,
  $$

  and a binary bracket $[-,-]$, whose component on elements in degree 0
  is a [[Lie bracket]]

  $$
    [-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_0
  $$

  and whose component on elements in degree 0 and degree 1 is a _weak [[action]]_

  $$
    \alpha(-,-) : \mathfrak{g}_0 \otimes \mathfrak{g}_1 \to
     \mathfrak{g}_1
     \,;
  $$

  and a trinary bracket

  $$
    [-,-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_1
  $$

  called the _Jacobiator_;

* such that

  * $[-,-]$ and $[-,-,-]$ are skew-symmetric in their arguments, as indicated;

  * the [[differential]] respects the brackets: for all $x \in \mathfrak{g}_0$ and $h \in \mathfrak{g}_1$ we have

    $$
      \delta [x,h] = [x, \delta h]
    $$

    hence

    $$
      \delta \alpha(x,h) = [x, \delta h]
      \,;
    $$

  * the [[Jacobi identity]] of $[-,-]$ holds up to the image under $\delta$ of the _Jacobiator_ $[-,-,-]$: for all $x,y,z \in \mathfrak{g}_0$ we have

    $$
      [x,[y,z]] + [y,[z,x]] + [z,[x,y]]
      =
      \delta [x,y,z]
    $$

  * as does the [[action]] property:

    $$
      \alpha(x,[y,h]) - \alpha(y,[x,h])
      =
      \alpha([x,y],h)
      +
      [x,y,\delta h]
    $$

  * the Jacobiator is [[coherence law|coherent]]:

    $$
      [[w,x,y], z] + [[w,y,z],x] + [[w,y],x,z]
      +
      [[x,z],w,y]
      =
      [[w,x,z], y] + [[x,y,z], w]
      +
      [[w,x],y,z]
      +
      [[w,z], x,y]
      +
      [[x,y], w,z]
      +
      [[y,z],w,x]
      \,.
    $$

The Jacobiator identity equivalently expresses the [[commuting diagram|commutativity]] of the following [[diagram]] in the given [[2-vector space]] (analogous to the [[pentagon identity]])

<img src="https://ncatlab.org/nlab/files/JacobiatorIdentity.jpg" width="560">

> (graphics grabbed from [Baez-Crans 04, p. 19](#BaezCrans04))

### Strict case

If the trinary bracket $[-,-,-]$ in a Lie 2-algebra is trivial, one speaks of a [[strict Lie 2-algebra]]. Strict Lie 2-algebras are equivalently [[differential crossed module]]s (see there for details).

## Examples

* [[derivation Lie 2-algebra]]

* [[string Lie 2-algebra]]

* [[4d supergravity Lie 2-algebra]]

## Related concepts

* [[Lie algebra]], [[Lie group]]

* **Lie 2-algebra**, [[Lie 2-group]]

  * [[membrane matrix model]]

* [[Lie 3-algebra]], [[Lie 3-group]]

* [[L-∞ algebra]], [[smooth ∞-group]]

* [[Lie algebroid]], [[Lie groupoid]]

* [[L-∞ algebroid]], [[smooth ∞-groupoid]]

## References

* {#BaezCrans04} [[John Baez]], [[Alissa Crans]], _Higher-Dimensional Algebra VI: Lie 2-Algebras_ Theory and Applications of Categories, Vol. 12, (2004) No. 15, pp 492-528. ([TAC:12-15](http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html))

* [[Daniel Berwick-Evans]], [[Eugene Lerman]], _Lie 2-algebras of vector fields_, [arxiv/1609.03944](http://arxiv.org/abs/1609.03944)

On [[weak Lie 2-algebras]]:

* {#Roytenberg07} [[Dmitry Roytenberg]], _On weak Lie 2-algebras_, AIP Conference Proceedings 956, 180 (2007) ([arXiv:0712.3461](http://arxiv.org/abs/0712.3461), [doi:10.1063/1.2820967]( https://doi.org/10.1063/1.2820967))

[[!redirects Lie 2-algebras]]


Lie 2-group

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## Idea

The notion of _Lie 2-group_ is the generalization of the notion of [[Lie group]] as [[group]]s are generalized to [[2-groups]]:

it is a [[smooth 2-group]] that happens to have a model given by a [[Lie groupoid]] equipped with the structure of a [[group object]] (in general only up to [[homotopy]]).

One general way to make the notion precise is as a special case of an [[smooth ∞-groupoid]], namely a [[1-truncated]] [[∞-group]] object in [[∞-stack]]s over the [[site]] [[CartSp]]/[[SmthMfd]], possibly with some representability condition:

these are [[stack]]s on the [[site]] of [[smooth manifold]]s (representable by [[Lie groupoid]]s and) equipped with group structure:  "group stacks" or "gr-stacks".

Special cases of this have simpler definitions. For instance a [[crossed module]] [[internal category|internal]] to [[Diff]] is a model for a strict and comparatively tame Lie 2-group.

Analogous to how the [[infinitesimal object|infinitesimal]] version of a [[Lie group]] is a [[Lie algebra]], the infinitesimal version of a Lie 2-group is a [[Lie 2-algebra]].

## Examples

* Every ordinary [[Lie group]] $G$ is in particular a Lie 2-group with $G$ as its space of [[object]]s and only [[identity morphism]]s.

* Every [[discrete infinity-groupoid|discrete]] [[2-group]] is a Lie 2-group equipped with [[discrete space|discrete smooth structure]];

* For $A$ an [[abelian group|abelian]] [[Lie group]] there is a Lie 2-group denoted $\mathbf{B}A$ or $(A \to 1)$, which has a single [[object]] and $A$ as its space of [[morphism]].

  For $A = U(1)$ the [[circle group]], the Lie 2-group $\mathbf{B}U(1)$ is the _[[circle n-group|circle 2-group]]_ .

* Every [[crossed module]] of Lie groups $(H \to G )$, $G \to Aut(H)$ gives an example of a [[strict 2-group|strict]] Lie 2-group, using the general relation between crossed modules and [[strict 2-group]]s.

  For instance the crossed module $Spin(n) \to O(n)$. Or $(U(1) \to 1)$.
  Etc.

* The [[string 2-group]] $String(G)$
  has various different but equivalent incarnations
  as a Lie 2-group. One is given by the [[crossed module]]
  $\hat \Omega_* G \to P_* G$ internal to [[Frechet manifold|Frechet]]
  Lie groups.

* For $X$ a [[Lie groupoid]], its [[automorphism infinity-group]] is a smooth 2-group.

## Constructions and Applications

### Delooping 2-groupoids
 {#DeloopingLie2Groupoid}

By the discussion at [[looping and delooping]], every Lie 2-group $G$ induces a [[delooping]] [[Lie 2-groupoid]] $\mathbf{B}G$: this has a single [[object]], the space of [[morphism]]s is $G_0$, the space of [[2-morphism]]s is $G_1$ and the [[horizontal composition]] is given by the group product.

### Principal 2-bundles

For $X$ a [[smooth manifold]] (or itself a [[Lie groupoid]] such as an [[orbifold]], or generally any [[smooth ∞-groupoid]]), morphisms

$$
  g : X \to \mathbf{B}G
$$

of [[smooth ∞-groupoid]]s from $X$ to the [delooping Lie 2-groupoid](#DeloopingLie2Groupoid) $\mathbf{B}G$ classify smooth $G$-[[principal 2-bundle]]s over $X$.

If $G = AUT(H)$ is the [[automorphism 2-group]] of a [[Lie group]] $H$ then these are equivalently smooth $H$-[[gerbe]]s over $X$.

Notice that a morphism of smooth $\infty$-groupoids $X \to \mathbf{B}G$ is presented by an [[infinity-anafunctor|2-anafunctor]] of 2-groupoid valued presheaves, given by a [[span]]

$$
  \array{
     C(U_i) &\stackrel{g}{\to}& \mathbf{B}G
     \\
     \downarrow^{\simeq}
     \\
     X
  }
 \,,
$$

where $C(U_i)$ is the [[Cech nerve]] 2-groupoid of some [[covering]]. The top morphism here encodes degree-1 [[nonabelian cohomology|nonabelian]] [[Cech cohomology|Cech]] [[hypercohomology]] with coefficients in $G$.

## Related concepts

* [[group]], [[Lie group]], [[Lie groupoid]]

* [[2-group]], **Lie 2-group**, [[Lie 2-groupoid]]

  * [[Poisson Lie 2-group]]

* [[∞-group]], [[smooth ∞-group]], [[smooth ∞-groupoid]]

## References

A first exposition is in the lecture notes

* Alissa Crans, _A survey of higher Lie theory_ ([pdf](http://myweb.lmu.edu/acrans/SurveyHigherLieTheory.pdf))

A general review of Lie 2-groups, as well as a discussion of the example of the [[string 2-group]] is in

* [[John Baez]], Alissa Crans, [[Urs Schreiber]], [[Danny Stevenson]], _From Loop Groups to 2-Groups_ ([arXiv:math/0504123](http://arxiv.org/abs/math/0504123))

Discussion in a more comprehensive context is in

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

with an introduction in section 1.3.1 and a general abstract discussion in 3.3.2.

On the [[cohomology]] of Lie 2-groups:

* [[Grégory Ginot]], [[Ping Xu]], _Cohomology of Lie 2-groups_ ([pdf](http://people.math.jussieu.fr/~ginot/papers/CohLie2gp.pdf))

* [[Christoph Wockel]],  _Categorified central extensions, étale Lie 2-groups and Lieʼs Third Theorem for locally exponential Lie algebras_ ([web](http://www.sciencedirect.com/science/article/pii/S0001870811002313))

* [[Chris Schommer-Pries]], _Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group_ ([arXiv](http://arxiv.org/abs/0911.2483))

[[!redirects Lie 2-groups]]


Lie 2-groupoid

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## Definition

A **Lie 2-groupoid** is a 2-[[truncated]] [[∞-Lie groupoid]].

## Examples

* Every [[Lie groupoid]] is a special case of a Lie 2-groupoid.

* For $A$ an [[abelian group|abelian]] [[Lie group]], its double [[delooping]] is a Lie 2-groupoid $\mathbf{B}^2 A$.

* More generally for $G$ a [[Lie 2-group]], its [[delooping]] $\mathbf{B}G$ is a one-object lie 2-groupoid

* For $X$ a smooth manifold, the [[path n-groupoid|path 2-groupoid]] $\mathbf{P}_2(X)$ is a Lie 2-groupoid (a 2-groupoid internal to [[diffeological space]])s.

## Related concepts

* [[double Lie algebroid]], [[Lie algebroid-groupoid]], [[double Lie groupoid]]

* [[higher differential geometry]]

  * [[smooth infinity-groupoid]]

[[!redirects Lie 2-groupoids]]


Lie 3-algebra

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## Definition

A **Lie 3-algebra** or **$3$-Lie algebra** is a 3-[[truncated]] [[L-∞-algebra]]. Under [[Lie integration]] is the [[infinitesimal space|infinitesimal]] approximation to a [[Lie 3-group]].

## Models by 2-crossed modules

A semistrict model for Lie 3-algebras, analogous to that of 3-groupoids that are [[Gray groupoid]]s is given by [[differential 2-crossed module]]s.

## Examples

* [[11d supergravity Lie 3-algebra]]

* [[4d supergravity Lie 3-algebra]]

## References

Lie 3-algebras coming from [[differential 2-crossed module]]s and [[parallel transport]] for [[3-groupoid of Lie 3-algebra valued forms|Lie 3-algebra valued forms]] is discussed in

* [[João Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_ ([arXiv:0907.2566](http://arxiv.org/abs/0907.2566))

On [[weak Lie 3-algebras]]:

* [[Malte Dehling]], *On weak Lie 3-algebras* [[arXiv:1710.11104](https://arxiv.org/abs/1710.11104)]

[[!redirects Lie 3-algebras]]


Lie 3-group

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## Definition

A _Lie 3-group_ of _3-Lie group_ is a 3-[[truncated]] [[∞-group]] in [[?LieGrpd]].

## From Lie 2-crossed module

A smooth [[2-crossed module]] is a model for semistrict Lie 3-groups, Lie [[Gray group]]s.

## Related concepts

* [[Lie 2-group]]

* [[smooth infinity-group]]

* [[higher differential geometry]]

## References

Lie 3-groups coming from Lie [[2-crossed module]]s and morphisms from a [[path 3-groupoid]] into their [[delooping]] [[3-groupoid]] is discussed in

* [[João Faria Martins]], [[Roger Picken]], _The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module_ ([arXiv:0907.2566](http://arxiv.org/abs/0907.2566))

[[!redirects Lie 3-group]]

[[!redirects Lie 3-groups]]