gauge transformation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### Local gauge transformations

In [[physics]] the term local **gauge transformation** or **gauge equivalence** means essentially [[isomorphism]] or rather [[equivalence in an (infinity,1)-category]]: the configuration [[space]] of a physical theory is typically a [[groupoid]] (an [[orbifold]]) and a _gauge transformation between configurations_ is a [[morphism]] in this groupoid.

More specifically, in physical theories called [[gauge theory]] -- such as describing [[electromagnetism]] and [[Yang-Mills field]]s -- the configuration space is a space of $G$-[[principal bundle]]s (over [[spacetime]] $X$) with [[connection on a bundle|connection]] over a [[Lie group]] $G$ called the **[[gauge group]]**. A _gauge transformation_ in the strict and original sense of the word is a morphism in the groupoid $G Bund_\nabla(X)$ of these bundles with connection.

In this context the connection $\nabla$ itself -- usually thought of in terms of its local [[Lie-algebra valued 1-form]]s $(A_i \in \Omega^1(U_i \subset X, \mathfrak{g}))$ -- is the _[[gauge field]]_ . The equation

$$
  A'_i = h_i^{-1} A_i h_i + h_i^{-1} d h_i \;\;\;\; ; \;\; h_i \in C^\infty(U_i,G)
$$

that characterizes morphisms $A_i \stackrel{g}{\to} A'_i$ in the [[groupoid of Lie-algebra valued forms]] is historically the hallmark of the "gauge principle" and is often what is meant specifically by _gauge transformation_ .

But from there on the terminology generalizes to almost all physical theories. For one this is because the configuration space of many theories may be thought of as spaces of bundles with connections, notably also for [[gravity]].

Moreover, many formal physical theories such as [[Chern-Simons theory]], [[supergravity]], etc. are described by [[higher category theory|higher categorical]] generalizations of bundles with connections: [[principal ∞-bundle]]s with [[connection on a principal ∞-bundle]] such as the [[Chern-Simons circle 3-bundle]]. Their configuration spaces form not just groupoids but [[∞-groupoids]]. The [[higher morphisms]] in these are called **[[higher gauge transformations]]**.

For instance the configuration space of the [[Kalb-Ramond field]] is the [[2-groupoid]]  of [[circle n-bundle with connection|circle 2-bundles with connection]] over [[spacetime]]. An object in there is locally given by a 2-form $B_i \in \Omega^2(U_i \subset X)$. A 1-[[morphism]] in there is a **first order gauge transformation** $B_i \stackrel{\lambda}{\to} B'_i$ characterized by the equation $B_i' = B_i + d_{dR} \lambda_i$. A [[2-morphism]] in there is a **second order gauge transformation** $\lambda_i \stackrel{g_i}{\Rightarrow} \lambda'_i$ characterized by $\lambda'_i = \lambda_i + d_{dR} g_i$.

The [[Lie algebroid]] of the groupoid of configurations and gauge transformations is known in physics in terms of its dual [[Chevalley-Eilenberg algebra]] called the [[BRST-complex]]. The degree 1 generators in this [[dg-algebra]] are hence the functions on _[[infinitesimal object|infinitesimal]]_ gauge transformations. (A discussion of such infinitesimal transformations is here.) These graded functions on infintesimal gauge transformations are called **[[ghost fields]]** or **ghosts** for short, in the physics literature.

If the space of configurations is not just a groupoid in ordinary spaces but a groupoid in _derived_ spaces such as [[derived smooth manifold]]s, then the CE-algebra of the corresponding [[derived ∞-Lie algebroid]] is called the [[BV-BRST complex]].

### Global gauge transformations

See [[global gauge group]].

## Details

### For local connection forms

The formulas for the local gauge transformations and higher gauge transformations for [[connections on bundles]], [[connections on 2-bundles]] and [[connections on 3-bundles]] are discussed, respectively, at

* [[groupoid of Lie-algebra valued 1-forms]]

* [[2-groupoid of Lie 2-algebra valued forms]]

* [[3-groupoid of Lie 3-algebra valued forms]].

The fully general description for [[connections on ∞-bundles]] is at

* [[∞-groupoid of ∞-Lie-algebra valued forms]].

## Examples

### In electromagnetism

(...)

### In Yang-Mills theory

(...)

### In gravity and topological field theories

* [[general covariance]]

## Related concepts

* [[gauge]]

* [[gauge parameter]]

* [[gauge group]], [[gauge fixing]]

* [[large gauge transformation]]

* [[higher gauge transformation]]

* [[gauge invariance]]

* [[group averaging]]

* [[Haag–Łopuszański–Sohnius theorem]] on symmetries of an [[S-matrix]]

* [[spontaneously broken symmetry]]

* [[principle of equivalence]]

* [[general covariance]]

* [[generalized global symmetry]]

[[!include gauge field - table]]

[[!redirects gauge transformations]]

[[!redirects gauge symmetry]]
[[!redirects gauge symmetries]]

[[!redirects global gauge symmetry]]
[[!redirects local gauge symmetry]]
[[!redirects global gauge symmetries]]
[[!redirects local gauge symmetries]]

[[!redirects global symmetry]]
[[!redirects local symmetry]]
[[!redirects global symmetries]]
[[!redirects local symmetries]]

[[!redirects infinitesimal gauge transformation]]
[[!redirects infinitesimal gauge transformations]]

[[!redirects infinitesimal gauge symmetry]]
[[!redirects infinitesimal gauge symmetries]]

[[!redirects gauge equivalence]]
[[!redirects gauge equivalences]]


gauge-Higgs unification

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[particle physics]], _gauge-Higgs unification_ ([Manton 79](#Manton79), [Hosotani 83](#Hosotani83)) also known as the _Hosotani mechanism_ (see [Hosotani 12](#Hosotani12)) refers to hypothetical [[model (in theoretical physics)|models]] of [[particle physics]] where the [[Higgs field]] on 4d [[spacetimes]] arises as a component of a [[gauge field]] in higher dimensions, either after [[Kaluza-Klein compactification]] or by passage to a 4d [[asymptotic boundary]] in a [[Randall-Sundrum model]] or similar.

> graphics grabbed from [Lim 09](#Lim09)

Gauge-Higgs unification is naturally combined with [[GUT|grand unification]] of gauge groups, whence one then also speaks of _gauge-Higgs grand unification_ (e.g. [Hosotani-Yamatsu 15](#HosotaniYamatsu15)). Also the [[Connes-Lott models]] realize gauge-Higgs unification, in a [[non-commutative geometry|non-commutative geometric]] [[KK-compactification]].

Attractive aspects of gauge-Higgs unification include the following:

1. a potential for providing a mechanism behind the Higgs-sector in the [[standard model of particle physics]] which could possibly solve the [[naturalness]]/[[hierarchy problem]] (e.g. [Kaul 08](#Kaul08), [Csáki-Tanedo 13](#CsakiTanedo13)).

   As such, gauge-Higgs unification is a possible alternative to [[MSSM]]-models.

1. a mechanism for ruling out [[proton decay]] in [[GUT]]-models, see [below](#NoProtonDecay).

## Properties

### No proton-decay
 {#NoProtonDecay}

Gauge-Higgs grand unification is claimed to predict stability of the [[proton]], hence no [[proton decay]] ([Hosotani-Yamatsu 15](#HosotaniYamatsu15), [Furui-Hosotani-Yamatsu 16, Sec. 2.6](#FuruiHosotaniYamatsu16) [Hosotani 17, Section 6](#Hosotani17)).

## References

### General

The mechanism is due to

* {#Manton79} [[Nicholas Manton]], _A new six-dimensional approach to the Weinberg-Salam model_, Nuclear Physics B Volume 158, Issue 1, 8 October 1979, Pages 141-153 (doi:10.1016/0550-3213(79)90192-5)

* {#Hosotani83} [[Yutaka Hosotani]], _Dynamical Mass Generation by Compact Extra Dimensions_, Phys. Lett. 126B (1983) 309-313 ([spire:188768](http://inspirehep.net/record/188768), doi:10.1016/0370-2693(83)90170-3)

* {#Hosotani89} [[Yutaka Hosotani]], _Dynamics of non-integrable phases and gauge symmetry breaking_, Annals of Physics Volume 190, Issue 2, March 1989, Pages 233-253 (doi:10.1016/0003-4916(89)90015-8)

and was revived with:

* H. Hatanaka, T. Inami, C.S. Lim, _The Gauge Hierarchy Problem and Higher Dimensional Gauge Theories_, Mod. Phys. Lett. A13 (1998) 2601-2612 ([arXiv:hep-th/9805067](https://arxiv.org/abs/hep-th/9805067))

Review:

* {#Hosotani12} [[Yutaka Hosotani]], _Gauge-Higgs Unification Approach_, AIP Conference Proceedings 1467, 208 (2012) ([arXiv:1206.0552](https://arxiv.org/abs/1206.0552))

* {#CsakiTanedo13} Csaba Csáki, Philip Tanedo, _Beyond the Standard Model_, lectures at [2013 European School of High-Energy Physics](http://cds.cern.ch/record/1522578), 2013  ([doi:10.5170/CERN-2015-004.169](http://dx.doi.org/10.5170/CERN-2015-004.169))

Deatiled [[phenomenology]]:

* Naoyuki Haba, Kazunori Takenaga, Toshifumi Yamashita, _Higgs mass in the gauge-Higgs unification_, Phys. Lett. B615 (2005) 247-256 ([arXiv:hep-ph/0411250](https://arxiv.org/abs/hep-ph/0411250))

* {#Lim09} C. S. Lim, _Precision Tests and CP Violation in Gauge-Higgs Unification_, 2009 ([slides pdf](https://www.iop.vast.ac.vn/theor/conferences/icfp09/files/Lim))

* Jason Carson, Nobuchika Okada, _125 GeV Higgs boson mass from 5D gauge-Higgs unification_, Progress of Theoretical and Experimental Physics, Volume 2018, Issue 3, March 2018, 033B03 ([doi:10.1093/ptep/pty018](https://doi.org/10.1093/ptep/pty018))

Discussion in view of [[naturalness]]/[[hierarchy problem]]:

* {#Kaul08} Romesh K. Kaul, _Naturalness and Electro-weak Symmetry Breaking_ ([arXiv:0803.0381](https://arxiv.org/abs/0803.0381))

### In $Spin(11)$-GUT ("$SO(11)$"):

Discussion in [[Spin(11)]]-[[GUT]] ("SO(11)-GUT"):

* {#HosotaniYamatsu15} [[Yutaka Hosotani]], Naoki Yamatsu, _Gauge–Higgs grand unification_, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 11, November 2015 ([arXiv:1504.03817](https://arxiv.org/abs/1504.03817), [doi:10.1093/ptep/ptw116](https://doi.org/10.1093/ptep/ptw116))

* {#FuruiHosotaniYamatsu16} Atsushi Furui, [[Yutaka Hosotani]], Naoki Yamatsu, _Toward Realistic Gauge-Higgs Grand Unification_, Progress of Theoretical and Experimental Physics, Volume 2016, Issue 9, September 2016, 093B01 ([arXiv:1606.07222](https://arxiv.org/abs/1606.07222))

* {#Hosotoni16} [[Yutaka Hosotani]], _Gauge-Higgs EW and Grand Unification_, International Journal of Modern Physics AVol. 31, No. 20n21, 1630031 (2016)  ([arXiv:1606.08108](https://arxiv.org/abs/1606.08108))

* {#Hosotani17} [[Yutaka Hosotani]], _New dimensions from gauge-Higgs unification_ ([arXiv:1702.08161](https://arxiv.org/abs/1702.08161))

* {#HosotaniYamatsu17} [[Yutaka Hosotani]], Naoki Yamatsu, _Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification_ ([arXiv:1710.04811](https://arxiv.org/abs/1710.04811))

### In $Spin(12)$-GUT ("$SO(12)$"):

Discussion in [[Spin(11)]]-[[GUT]] ("SO(11)-GUT"):

* S. Rajpoot and P. Sithikong, _Implications of the $SO(12)$ gauge symmetry for grand unification_, Phys. Rev. D 23, 1649 (1981) ([doi:10.1103/PhysRevD.23.1649](https://doi.org/10.1103/PhysRevD.23.1649))

* {#NomuraSato08} Takaaki Nomura, Joe Sato, _Standard(-like) Model from an $SO(12)$ Grand Unified Theory in six-dimensions with $S^2$ extra-space_, Nucl.Phys.B811:109-122, 2009 ([arXiv:0810.0898](https://arxiv.org/abs/0810.0898))

* {#Nomura09} Takaaki Nomura, _Physics beyond the standard model with $S^2$ extra-space_, 2009 ([pdf](http://krishna.th.phy.saitama-u.ac.jp/joe/doctor/nomura-doctor.pdf), [[NomuraSO12GUT.pdf:file]])

* {#ChiangNomuraSato11} Cheng-Wei Chiang, Takaaki Nomura, Joe Sato, _Gauge-Higgs unification models in six dimensions with $S^2/\mathbb{Z}_2$ extra space and GUT gauge symmetry_ ([arXiv:1109.5835](https://arxiv.org/abs/1109.5835))

[[!redirects Hosotani mechanism]]
[[!redirects gauge-Higgs grand unification]]


gauged linear sigma-model

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[2d (2,0)-supersymmetric sigma-model]]

* [[gauged WZW model]]

## References

* Huijun Fan, Tyler Jarvis, Yongbin Ruan, _A Mathematical Theory of the Gauged Linear Sigma Model_ ([arXiv:1506.02109](https://arxiv.org/abs/1506.02109))

* Huijun Fan, Tyler Jarvis, [[Yongbin Ruan]], _A Mathematical Theory of the Gauged Linear Sigma Model_ ([arXiv:1506.02109](https://arxiv.org/abs/1506.02109))

Review:

* _Introduction to the gauged linear sigma model_ ([pdf](http://www2.yukawa.kyoto-u.ac.jp/~tetsuji.kimura/NOTEs/2004/GLSM.pdf))

* [[Jock McOrist]], _The Revival of $(0,2)$ Linear Sigma Models_, Int. J. Mod. Phys. A26:1-41, 2011 ([arXiv:1010.4667](http://arxiv.org/abs/1010.4667))

* [[Eric Sharpe]], *A survey of recent developments in GLSMs* [[arxiv:2401.11637](https://arxiv.org/abs/2401.11637)]

Discussion for [[target spaces]] that are [[gerbes]]/[[stacks]] includes

* [[Tony Pantev]], [[Eric Sharpe]], _GLSM's for gerbes (and other toric stacks)_ ([arXiv:hep-th/0502053](http://arxiv.org/abs/hep-th/0502053))

[[!redirects gauged linear sigma-models]]

[[!redirects GLSM]]
[[!redirects GLSMs]]


gauged sigma-model

> under construction. For the moment see _[[sigma-model]]_.

## Related concepts

* [[gauged WZW model]]

## References

* _Quasi-Topological Gauged Sigma Models, The Geometric Langlands Program, And Knots_ ([arXiv:1111.0691](http://arxiv.org/abs/1111.0691))

* _Properties of gauged Sigma models_ ([pdf](http://sq2007.seenet-mtp.info/lectures/alikaya.pdf))

(...)

[[!redirects gauged field theory]]


gauged supergravity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Theories of [[supergravity]] in [[dimension]] $l \lt 11$ always contain a global [[symmetry]] called [[R-symmetry]] (a remnant of the full [[11-dimensional supergravity]] $Spin(10,1)$-symmetry after  [[KK-compactification]]). In some cases this is promoted to a local symmetry, such that there is a [[gauge field]] ([[connection on a bundle]]) with coefficients in that group. These are called _gauged supergravity_ theories.

## Properties

### From dimensional reduction

In many cases, gauged supergravity theories are obtained by [[KK-compactification|dimensional reduction]] from [[11-dimensional supergravity]] or [[type II supergravity]], which themselves do not contain a gauge field but higher degree fields ("fluxes"), the [[supergravity C-field]] and the [[B-field]] respectively. These induce gauged supergravities (e.g. [Samtleben 08, figure 1](#Samtleben08)). The [[gauge groups]] are the [[U-duality]] groups of the compactification (e.g. [Samtleben 08, table 1](#Samtleben08)).

### From double and exceptional field theory

[[Exceptional field theory]] (but also [[double field theory]]) reduces to [[gauged supergravity]] under [[generalized Scherk-Schwarz reduction]], i.e. a [[KK-compactification|dimensional reduction]] combined with [[U-duality]] (or just [[T-duality]]) twists.

\begin{imagefromfile}
        "file_name": "gaugedsugraexft.png",
        "width": 600,
        "unit": "px",
        "caption": "Picture grabbed from [Musaev 13](#Musaev13)"
\end{imagefromfile}

## Examples

Some examples are discussed at

* [[7-dimensional supergravity]]

* [[4-dimensional supergravity]]

* [[3-dimensional supergravity]]

## Related concepts

* [[tensor hierarchy]]

* [[embedding tensor]]

* [[double field theory]], [[exceptional field theory]]

## References

Review includes

* Martin Weidner, _Gauged Supergravities in Various Spacetime Dimensions_, Fortsch.Phys.55:843-945,2007 ([arXiv:hep-th/0702084](http://arxiv.org/abs/hep-th/0702084))

* {#Samtleben08} [[Henning Samtleben]], _Lectures on Gauged Supergravity and Flux Compactifications_ ([arXiv:0808.4076](http://arxiv.org/abs/0808.4076))

* Mario Trigiante, _Gauged Supergravities_, Physics Reports Volume 680, 31 March 2017, Pages 1-175 ([arXiv:1609.09745](https://arxiv.org/abs/1609.09745), [doi:10.1016/j.physrep.2017.03.001](https://doi.org/10.1016/j.physrep.2017.03.001))

Discussion of the origin in [[KK-compactification]] of [[11-dimensional supergravity]]/[[M-theory]] is in

* {#Roest04} [[Diederik Roest]], _M-theory and Gauged Supergravities_, Fortsch.Phys.53:119-230,2005 ([arXiv:hep-th/0408175](http://arxiv.org/abs/hep-th/0408175))

Maximally gauged [[4d supergravity]] was first discussed in

* [[Bernard  de Wit]], [[Hermann Nicolai]], _$N = 8$ supergravity_, Nucl. Phys. B208 (1982) 323

and gauged [[D=5 supergravity]] in

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Gauged $N = 8$ Supergravity in Five Dimensions_, Phys. Lett. 154B, (1985) 268 ([spire:207663](http://inspirehep.net/record/207663), doi:10.1016/0370-2693(85)90361-2)

* [[Murat Günaydin]], [[L. J. Romans]] and [[Nicholas Warner]], _Compact and Non–Compact Gauged Supergravity Theories in Five Dimensions_, Nucl. Phys. B272 (1986) 598 ([spire:219727](http://inspirehep.net/record/219727), doi:10.1016/0550-3213(86)90237-3)

*  M. Pernici, K. Pilch, [[Peter van Nieuwenhuizen]], _Gauged $N=8$ $D=5$ Supergravity_, Nucl.Phys. B259 (1985) 460 ([spire](https://inspirehep.net/record/16067?ln=en))

Discussion in the context of [[flux compactification]] of [[type II superstring theory]] includes

* [[Mariana Graña]], section 5.3 of _Flux compactifications in string theory: a comprehensive review_, Phys.Rept. 423 (2006) 91-158 ([arXiv:hep-th/0509003](http://arxiv.org/abs/hep-th/0509003))

Discussion in the context of the [[D'Auria-Fré formulation of supergravity]] is in

* {#Fre05} [[Pietro Fré]], _M-theory FDA, Twisted Tori and Chevalley Cohomology_, Nucl.Phys.B742:86-123, 2006 ([arXiv:hep-th/0510068](http://arxiv.org/abs/hep-th/0510068))

Discussion related to [[orbifold]] singularities includes

* Richard Corrado, Murat Gunaydin, Nicholas P. Warner, Marco Zagermann, _Orbifolds and Flows from Gauged Supergravity_, Phys.Rev.D65:125024,2002 ([arXiv:hep-th/0203057](http://arxiv.org/abs/hep-th/0203057))

Review of the relation with [[double field theory]] and [[exceptional field theory]]:

* {#Musaev13} [[Edvard Musaev]], _U-dualities in Type II string theories and M-theory_, _Symmetry_ 2019, _11_(8), 993 ([arXiv:1311.3331](https://arxiv.org/abs/arXiv:1311.3331), [doi:10.3390/sym11080993](https://doi.org/10.3390/sym11080993))

[[!redirects gauged supergravities]]


gauged WZW model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Wess-Zumino-Witten theory
+--{: .hide}
[[!include infinity-Wess-Zumino-Witten theory - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _gauged WZW model_ is a [[field (physics)|field]] [[theory (physics)]] which combines the [[WZW model]] with [[gauge theory]]:
given a ([[simple Lie group|simple]]) [[Lie group]] $G$ and a [[subgroup]] $H \hookrightarrow G \times G$, the corresponding gauged WZW model is a 2-dimensional [[prequantum field theory]] on some [[worldvolume]] $\Sigma_2$ whose [[field (physics)|fields]] are pairs consisting of a smooth function $\Sigma_2 \to G$ and a [[Lie algebra valued 1-form]] $A \in \Omega^1(\Sigma_2, \mathfrak{h})$, with values in the [[Lie algebra]] of $H$.

Where the [[Lagrangian]]/[[action functional]] of the ordinary [[WZW model]] is the sum/product of a standard [[kinetic action]] and the [[surface holonomy]] of a [[circle 2-bundle with connection]] whose [[curvature]] 3-form is the canonical 3-form $\langle \theta \wedge [\theta \wedge \theta]\rangle \in \Omega^3(G)^G$, so the action functional of the gauged WZW model is that obtained by refining this circle 2-bundle to the $H$-[[equivariant cohomology|equivariant]] [[differential cohomology]] of $G$, with [[curvature]] 3-form in [[equivariant de Rham cohomology]].

## Definition
 {#Definition}

The [[Chevalley-Eilenberg algebra]] of the [[Lie algebra]] $\mathfrak{g}$ is naturally identified with the sub-algebra of [[left invariant differential forms]] on $G$:

$$
  CE(\mathfrak{g}) \simeq \Omega^\bullet(G)^G
  \,.
$$

The ordinary [[WZW model]] is given by the basic [[circle 2-bundle with connection]] on $G$ whose [[curvature]] 3-form is

$$
  H = \langle \theta \wedge [\theta \wedge \theta]\rangle
  \in \Omega^3(G)^G
  \,.
$$

Now for $H \hookrightarrow G$ a [[subgroup]], write

$$
  CE(\mathfrak{g}//\mathfrak{h})
 \coloneqq
  \Omega^\bullet(G, \mathfrak{h}^\ast[1])^G
$$

for the corresponding dg-algebra of (say) the [[Cartan model]] for [[equivariant de Rham cohomology]] on $G$. There is a canonical projection

$$
  CE(\mathfrak{g}//\mathfrak{h})
  \to
  CE(\mathfrak{g})
  \,.
$$

A curvature 3-form for the **gauged WZW model** is a 3-cocycle

$$
  \tilde H \in CE^3(\mathfrak{g}//\mathfrak{h})
$$

in this [[equivariant de Rham cohomology]] which lifts $H \coloneqq \langle \theta \wedge [\theta \wedge \theta]\rangle$ through this projection.

 One finds ([Witten 92, appendix](#Witten92)) that in terms of the degree-2 generators $\{F^a\}$ of the [[Cartan model]] (see there) with respect to some [[basis]] $\{t_a\}$ of $\mathfrak{g}$, these lifts are of the form ([Witten 92, (A.14)](#Witten92))

{#WZWCurvatureInCartanModel}
$$
  \tilde H = H + \lambda_a \wedge F^a
$$

where $\lambda_a \in \Omega^1(G)$ is given by (in [[matrix Lie algebra]] notation)

$$
  \lambda_a
  =
  \left\langle
    t_a^l \cdot (d g)g^{-1}
    +
    t_a^r \cdot g^{-1} d g
  \right\rangle
$$

and exist precisely if ([Witten 92, (A.16)](#Witten92)) for all pairs of basis elements

$$
  \langle
    t_a^l \cdot t_b^l
    -
    t_a^r \cdot t_b^r
  \rangle
  = 0
  \,.
$$

This condition had originally been seen as a [[anomaly cancellation]]-condition of the gauged WZW model. A systematic discussion of these obstructions in [[equivariant de Rham cohomology]] is in ([Figueroa-O'Farrill-Stanciu 94](#FigueroaOFarrillStanciu94)).

Now by [[schreiber:∞-Wess-Zumino-Witten theory]], the corresponding WZW model has as target the [[smooth groupoid]] $\tilde G//H$ such that maps into it are locally a map $g$ into $G$ together with 1-form potentials $A^a$ for the $F^a$, and the WZW term is locally a 2-form built from $d g$ and $A^a$ such that its curvature 3-form is $\tilde H$. This is the **gauged WZW model** ([Witten 92, (A.16)](#Witten92)).

## Properties

### Partition function in (equivariant) elliptic cohomology
 {#PartitionFunctionInEllipticCohomology}

The [[partition function]] of the gauged WZW model as an [[elliptic genus]] is considered in ([Henningsonn 94, (8)](#Henningson94)). When done properly this should give elements in [[equivariant elliptic cohomology]], hence an [[equivariant elliptic genus]].

## Related concepts

* [[coset WZW model]]

* [[gauged linear sigma model]]

## References

[[!include WZW term of QCD chiral perturbation theory -- references]]

### General gauged WZW models

The original articles are

* [[Edward Witten]], _Nonabelian bosonization in two dimensions_, Commun. Math. Phys. 92 (1984) 455 ([euclid:cmp/1103940923](https://projecteuclid.org/euclid.cmp/1103940923))

* [[Krzysztof Gawedzki]], A. Kupiainen, _G/H conformal field theory from gauged WZW model_ Phys. Lett. 215B, 119 (1988);

* [[Krzysztof Gawedzki]], A. Kupiainen, _Coset construction from functional integrals_, Nucl. Phys. B 320 (FS), 649 (1989)

* [[Krzysztof Gawedzki]], in _From Functional Integration, Geometry and Strings_, ed. by Z. Haba and J. Sobczyk (Birkhaeuser, 1989).

The (curvature of the)gauged WZW term was recognized/described as a  [[cocycle]] in [[equivariant de Rham cohomology]] is in the appendix of

* {#Witten92} [[Edward Witten]], _On holomorphic factorization of WZW and coset models_,  Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. ([EUCLID](http://projecteuclid.org/euclid.cmp/1104249222))

This is expanded on in

* {#FigueroaOFarrillStanciu94} [[José Figueroa-O'Farrill]], S Stanciu, _Gauged Wess-Zumino terms and Equivariant Cohomology_, Phys.Lett. B341 (1994) 153-159 ([arXiv:hep-th/9407196](http://arxiv.org/abs/hep-th/9407196))

* [[José de Azcárraga]],  J. C. Perez Bueno, _On the general structure of gauged Wess-Zumino-Witten terms_ ([arXiv:hep-th/9802192](http://arxiv.org/abs/hep-th/9802192))

A quick review of this class of 3-cocycles in equivariant de Rham cohomology is also in section 4.1 of

* Hugo Garcia-Compean, Pablo Paniagua, [[Bernardo Uribe]], _Equivariant extensions of differential forms for non-compact Lie groups_ ([arXiv:1304.3226](http://arxiv.org/abs/1304.3226))

which further generalizes the discussion to non-compact Lie groups.

On [[boundary conditions]] ([[BCFT]]/[[D-branes]]) for the [[gauged WZW model]] via [[parafermions]]:

* [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], *Geometrical interpretation of D-branes in gauged WZW models*, JHEP 0107:046 (2001) $[$[arXiv:hep-th/0105038](https://arxiv.org/abs/hep-th/0105038), [doi:10.1088/1126-6708/2001/07/046](https://doi.org/10.1088/1126-6708/2001/07/046)$]$

See also:

* [[Edward Witten]], _The $N$ matrix model and gauged WZW models_, Nuclear Physics B Volume 371, Issues 1–2, 2 March 1992, Pages 191–245

* Stephen-wei Chung, S.-H. Henry Tye, _Chiral Gauged WZW Theories and Coset Models in Conformal Field Theory_, Phys. Rev. D47:4546-4566,1993 ([arXiv:hep-th/9202002](http://arxiv.org/abs/hep-th/9202002))

* Konstadinos Sfetsos, _Gauged WZW models and Non-abelian duality_,  	Phys.Rev. D50 (1994) 2784-2798 ([arXiv:hep-th/9402031](http://arxiv.org/abs/hep-th/9402031))

* [[Elias Kiritsis]], _Duality in gauged WZW models_ ([pdf](http://hep.physics.uoc.gr/~kiritsis/papers2/dual.pdf))

The [[partition function]]/[[elliptic genus]] of the SU(2)/U(1) gauged WZW model is considered in

* {#Henningson94} [[Måns Henningson]], _N=2 gauged WZW models and the elliptic genus_, Nucl.Phys. B413 (1994) 73-83 ([arXiv:hep-th/9307040](http://arxiv.org/abs/hep-th/9307040))

Emphasis of the special case of abelian gauging in Section 2 of

* {#Forste04} [[Stefan Förste]],  _Deformations of WZW models_, Class. Quant. Grav. 21 (2004) S1517-1522 ([arXiv:hep-th/0312202](https://arxiv.org/abs/hep-th/0312202))

[[!redirects gauged WZW models]]

[[!redirects gauged Wess-Zumino-Witten model]]
[[!redirects gauged Wess-Zumino-Witten models]]

[[!redirects gauged WZW-model]]
[[!redirects gauged WZW-models]]


gaugino

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] the term _gaugino_ denotes a field that is a [[superpartner]] of a [[gauge boson]]. This appears in [[super Yang-Mills theory]]. For instance the partner of a [[gluon]] in the [[MSSM]] is called a [[gluino]], and so on.

In terms of [[Chern-Weil theory]]/[[differential cohomology]] we have that

* [[gauge bosons]] are (the [[quanta]] of) [[connection on a bundle|connections]] on [[principal bundle]] or more generally the even components of [[superconnections]];

* gauginos are the odd components of superconnections.

## Experiment

Experimental exclusion bounds of gluino [[rest masses]] due to the [[LHC]] [[experiment]] excludes gluinos/[[squarks]] of mass below about 3-4 [[TeV]] ([Particle Data Group Review 17, figure 113.2 and 113.9](#ParticleDataGroup17)):

[[LHCSuperpartnerMassExclusion2017.png:file]]

## Related concepts

* in [[10d super Yang-Mills theory]]: see [there](D%3D10+super+Yang-Mills+theory#EquationsOfMotionFromTheBianchiIdentity)

* [[sfermions]]

  * [[squarks]]

* [[split supersymmetry]]

* [[gaugino condensation]]

## References

Theoretical discussion includes

* F. Gieres, _Geometry of Supersymmetric Gauge Theories: Including an Introduction to BRS Differential Algebras and Anomalies_  Springer Lecture Notes in Physics 302 (1998)

Experimental exclusion bounds are discussed iin

* {#ParticleDataGroup17} [[Particle Data Group]] [Review 2017](http://pdg.lbl.gov/), 113. _Supersymmetry , part II (experiment)_ ([pdf](http://pdg.lbl.gov/2017/reviews/rpp2017-rev-susy-2-experiment.pdf))

See also

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

[[!redirects gauginos]]


gaugino condensation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

[[condensation]] of [[gauginos]] (e.g. [[gluinos]])

## Related concepts

* [[gravitino condensation]]

* [[supersymmetry breaking]]

## References

### General

See also

* Wikipedia, _[Gaugino condensation](https://en.wikipedia.org/wiki/Gaugino_condensation)_

### In heterotic M-theory

Discussion in [[heterotic string theory|heterotic]] [[M-theory]] aka [[Horava-Witten theory]], where the two heterotic [[MO9-branes]] provide a natural mechanism for [[supersymmetry breaking]] on one (the physical brane) by gaugino condensation on the other (the "dark sector" brane):

lecture notes:

* {#Nilles04} [[Hans-Peter Nilles]], _Gaugino Condensation and SUSY Breakdown_, Lectures at Cargese School of Physics and Cosmology, Cargese, France, August 2003 ([arXiv:hep-th/0402022](https://arxiv.org/abs/hep-th/0402022))

original articles:

* I. Antoniadis, M. Quiros, _On the M-theory description of gaugino condensation_, Phys.Lett. B416 (1998) 327-333 ([arXiv:hep-th/9707208](https://arxiv.org/abs/hep-th/9707208))

* I. Antoniadis, M. Quiros, _Supersymmetry breaking in M-theory and gaugino condensation_, Nucl.Phys. B505 (1997) 109-122 ([arXiv:hep-th/9705037](https://arxiv.org/abs/hep-th/9705037))

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Gaugino condensation in M theory on $S^1/Z_2$_, Phys. Rev. D 57, 7529 (1998) ([arXiv:hep-th/9711197](https://arxiv.org/abs/hep-th/9711197), [doi:10.1103/PhysRevD.57.7529](https://doi.org/10.1103/PhysRevD.57.7529))

* [[André Lukas]], [[Burt Ovrut]], [[Daniel Waldram]], _Five-Branes and Supersymmetry Breaking in M-Theory_, JHEP 9904:009, 1999 ([arXiv:hep-th/9901017](https://arxiv.org/abs/hep-th/9901017))

specifically for [[M-theory on S1/G_HW times H/G_ADE]]:

* Zygmunt Lalak, Steven Thomas, _Gaugino Condensation, Moduli Potentials and Supersymmetry Breaking in M-Theory Models_, Nuclear Physics B
Volume 515, Issues 1–2, 30 March 1998, Pages 55-72
Nuclear Physics B ([hep-th/9707223](https://arxiv.org/abs/hep-th/9707223), doi:10.1016/S0550-3213(97)00784-0)

[[!redirects gaugino condensations]]

[[!redirects gaugino condensate]]
[[!redirects gaugino condensates]]


gauginos > history

see _[[gaugino]]_


gaunt category

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

# Contents
* table of contents
{: toc}

## Idea

A [[category]] is called **gaunt** if all its [[isomorphisms]] are in fact [[identities]]. This is a property of [[strict categories]]; that is, it is not [[principle of invariance|invariant]] under [[equivalence of categories]]. See below for some related concepts that are invariant.

Every gaunt category is [[skeletal category|skeletal]], but not conversely.  Every gaunt category is also [[univalent category|univalent]]; the converse holds in [[set-level foundations]] but not in the [[higher foundations]] where univalent categories are usually considered.  More generally, gaunt categories are precisely the categories that are both [[strict category|strict]] and univalent.

## Definitions

### In set-level foundations

\begin{definition}
A **gaunt category** or **stiff category** is a [[skeletal category]] $C$ whose [[core]] is [[thin category|thin]].

\end{definition}

Expanding out, this becomes

\begin{definition}
A **gaunt category** or **stiff category** is a category $C$ such that the set of isomorphisms $A \cong B$ between every two objects $A \in Ob(C)$ and $B \in Ob(C)$ of $C$ is a [[subsingleton]], and there is an isomorphism $f:A \cong B$ if and only if $A = B$.

\end{definition}

### In type theory

In a [[dependent type theory]] with some notion of [[identity type]] or [[path type]], there is another definition of gaunt category available.

\begin{definition}
A **gaunt category** or **stiff category** is a strict univalent category.
\end{definition}

Expanded out, this becomes

\begin{definition}
A **gaunt category** or **stiff category** is a category whose type of objects is 0-truncated and which additionally satisfies a condition called *Rezk completeness*: the canonical function
$$idtoiso : (a=b) \to (a \cong b)$$
from the identity type between two objects $a$ and $b$ to the type of isomorphisms between $a$ and $b$ is an equivalence of types.
\end{definition}

In [[set-level type theory]], the type theoretic definition and the set-level definitions are equivalent to each other.  For in this case, every [[identity type]] is a [[proposition]]/[[subsingleton]].  Thus, by the Rezk completeness condition, the set of isomorphisms is a subsingleton, and two objects which are isomorphic to each other are also equal to each other.

## Examples

* Every [[poset]] is a gaunt category.
* The [[walking parallel pair]] is a gaunt category which is not a [[poset]].
* The [[free category]] on a [[directed graph]] is a gaunt category.
* The [[category]] of [[Archimedean ordered fields]] and [[monotonic]] [[field homomorphisms]] is a gaunt category.
* The [[delooping]] of a group is (in set-level foundations) a [[skeletal category]] that is not gaunt (unless the group is [[trivial group|trivial]]).
* The [[walking isomorphism]], as usually defined, is not gaunt.  (In [[homotopy type theory]], one can Rezk-complete the walking isomorphism, obtaining a univalent category that is gaunt and indeed equivalent to the [[terminal category]].)
* The [[simplex category]] and more generally any [[Reedy category]] is gaunt.

## Properties

### Relation to other properties of categories

Gaunt categories are necessarily [[skeletal]]; a skeletal category is gaunt iff every [[automorphism]] is an [[identity morphism]]. Consequently a [[thin]] gaunt category is skeletal, and since a thin skeletal category is a [[poset category]] a thin gaunt category is also a poset category.

Note that a gaunt category need not be thin, since we may have parallel non-isomorphisms which are not equal. Similarly, a thin category need not be gaunt since we may have isomorphisms that aren't the identity.

The [[core]] of a gaunt category is a [[set]].

### Relation to complete Segal spaces

The [[nerve]] [[simplicial set]] of a [[category]], regarded as a [[simplicial object]] in [[homotopy types]] under the inclusion $Set \hookrightarrow \infty Grpd$, is a _[[complete Segal space]]_ precisely if the category is gaunt. More discussion of this is at _[Segal space -- Examples -- In Set](Segal%20space#InSetByNervesOfCategories)_.

### In higher foundations

As noted above, in [[higher foundations]] the gaunt categories are precisely the [[strict category|strict]] and [[univalent categories]]. Thus, every property which is satisfied of strict categories and univalent categories apply to gaunt categories.

In addition, we have

\begin{theorem}
The [[Rezk completion]] of a core-thin category is a gaunt category.
\end{theorem}

## Related invariant concepts

To make sense of the definition of a gaunt category, we need to use equality of objects: For every isomorphism $f : a \simeq b$, there is an equality $p : a = b$, relative to which $f$ equals the identity at $a$. Replacing the equality $p$ by an isomorphism $g : a \simeq b$, the resulting condition holds for all categories.
This echoes how one might understand the definition in [[univalent foundations]]: the univalence condition for a [[univalent category]] is another way of saying that every isomorphism is an identity (and uniquely so).

Alternatively, we could avoid the equality on objects by requiring only that every [[automorphism]] $f : a \cong a$ be equal to the identity at $a$. This amounts to requiring that the [[core]] be a [[thin category]], i.e., that parallel isomorphisms are equal.  A category has this property if and only if it is *equivalent* to a gaunt one.

{#Incidentally} Incidentally, we may view both [[strict categories]] and categories up to [[equivalence of categories|equivalence]] as embedded in the type of [[flagged categories]]. Recall that a [[flagged category]] consists of a category $C$, a [[groupoid]] $X$, and an [[essentially surjective functor]] $p \colon X \to C$. In this way, we can view categories as those [[flagged categories]] where $p$ is an [[equivalence of groupoids|equivalence]] onto the [[core]] of $C$, and [[strict categories]] as those flagged categories where $X$ is a [[set]] (up to homotopy: an "[[h-set]]"): see at *[[category with an atlas]]*. The [[intersection]] of categories-up-to-equivalence and [[strict categories]] within the type of flagged categories is then exactly this type of core-thin categories.  (This is the semantics in [[Gpd]] of the fact that in [[homotopy type theory]], the gaunt categories are precisely the [[strict category|strict]] and [[univalent categories]].)

## See also

* [[strict category]]

* [[univalent category]]

* [[skeletal category]]

* [[thin category]]

* [[symmetric proset]]

* [[core]]

## References

The term "gaunt category" was apparently introduced in

* {#BarwickSchommerPries} [[Clark Barwick]], [[Chris Schommer-Pries]], _On the Unicity of the Homotopy Theory of Higher Categories_ ([arXiv:1112.0040](http://arxiv.org/abs/1112.0040), [slides](http://prezi.com/w0ykkhh5mxak/the-uniqueness-of-the-homotopy-theory-of-higher-categories/))

in the context of a discussion of [[(infinity,n)-categories]].

In the [[Elephant]], gaunt categories are briefly mentioned under the name "stiff categories", in the paragraph preceding B1.3.11 (about [[split fibration|splitting]] of [[Grothendieck fibrations]]).

[[!redirects gaunt category]]
[[!redirects gaunt categories]]
[[!redirects stiff category]]
[[!redirects stiff categories]]

[[!redirects strict saturated category]]
[[!redirects strict univalent category]]
[[!redirects strict saturated categories]]
[[!redirects strict univalent categories]]


Gauss decomposition

#Contents#
* table of contents
{:toc}

## Idea

Let $k$ be a [[field]] and $g\in M_n(k)$ $n\times n$ square matrix whose entries are in $k$. One would like to decompose this matrix as $g = UDL$ where $L$ is the lower triangular unidiagonal matrix, $U$ upper triangular unidiagonal and $D$ a diagonal matrix. Unidiagonal means with only units on a diagonal. One can also try to decompose $g = UA$ where $A$ is lower triangular and $U$ upper triangular unidiagonal. Not every matrix however has such a decomposition: one tries to find the solution via [[Gauss elimination procedure]], and the conditions for the solution involve non-vanishing of certain "principal" minors of the matrix. Then the solution for entries of $U$ or $L$ involves ratios of (determinants of) minors of the same size, while the solution for $A$ the ratio of (determinants of) minors of sizes different by one. These  decompositions of a matrix $G$ are called the Gauss decompositions.

If $g\in GL_n(k)$ then there exist a permutation $n\times n$ matrix $w$ such that the problem $G = wUDL$ has a solution. The subset of matrices $G\in GL_n(k)$ for which such a decomposition exists (and then it is automatically unique and given by universal formulas) is a Zariski open subset in $GL_n(k)$, one of the "Gauss cells"; the decomposition of matrices is for each $w$ also called Gauss decomposition. Then $w=1$ is said to be the main cell. The group $GL_n(k)$ is __covered__ by $n!$ open subsets, this is the Gauss of global decomposition of $GL_n$; similarly one can do for various subgroups of $GL_n$ by inducing the decomposition from $GL_n$. In the case of $SL_n(\mathbb{C})$, and some other cases, the matrices corresponding to $A$ in the decomposition for $w=1$ are forming the corresponding (lower) Borel subgroup.

This should be distinguished from Bruhat decomposition where one wants $g=UwA$ instead of $g=wUA$. Except for the case when $w=1$ when the two decompositions coincide, the matrices which decompose for given $w$ make a subset of higher codimension, hence nonprincipal (that is $w\neq 1$) Bruhat cells are not open. Furthermore, while in Gauss case  the cells make a cover of $GL_n$, in Bruhat case they make a partition of $GL_n$ into disjoint subsets of elements.

If $B$ is a subgroup of lower triangular matrices, then for the fixed $w$, the entries of $U$ as a function of $G$ in the decomposition $g=wUA$ are the coordinates on the patch in $GL_n/B$ evaluated at the coset of $g$. The decomposition therefore for fixed $w$ corresponds to the trivialization of the principal $B$-bundle $GL_n\to GL_n/B$ over the open Gauss cell corresponding to $w$.

Gauss decomposition can be generalized to the matrices with noncommutative entries and there are explicit formulas involving [[quasideterminants]], and in the case of quantum linear groups in terms of quantum minors.

Remark. The upper conventions are more from mathematical physics literature. The lowest weight representations have often positive energy, what is normal requirement there and the lower triangular matrices are used to act on the lowest weight vector. In mathematical community, one prefers to talk about upper triangular matrices forming Borel subgroup and about the decomposition $LDU$ and $LA$ where $A$ is upper triangular and act on heighest weight representations instead.

## Related concepts

See also [[quantum Gauss decomposition]].

## References

Named after [[Carl Friedrich Gauß]].

See also

* [[eom]], _[Gauss decomposition](https://www.encyclopediaofmath.org/index.php/Gauss_decomposition)_


Gauss law

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

The *Gauss law* is, originally, one of the [[equations of motion]] of [[electromagnetism]], specifically of its [[electric field]]-sector, relating the [[divergence]] of the [[electric field]] to the [[electric charge]] density at each point of space. But the name has come to be commonly used also more generally for the corresponding equation of [[Yang-Mills theory]] and also for [[higher gauge theories]], all of which are closely similar equations when formulated via [[differential forms]] in [[exterior calculus]].

In the [[Hamiltonian mechanics|Hamiltonian]] formulation of these theories, the Gauss law appears as a [[first class constraint]] on the [[canonical phase space]] whose [[Hamiltonian vector field]] is that which generates the [[gauge transformations]].

In form analogous to the Gauss law is a dual statement for the [[magnetic field]], accordingly also known as the *magnetic Gauss law* and as such also part of [[Maxwell's equations]]. However, in traditional [[Hamiltonian mechanics|Hamiltonian]] [[phase space]]-formulation of [[electromagnetism]], where the magnetic but not the electric field is assumed to have a [[gauge potential]], the symmetry between these two laws is broken (the magnetic Gauss law is not a phase space constraint but a differential [[Bianchi identity]] implied by the existence of a gauge potential).

## Statement

### In electromagnetism
 {#InElectromagnetism}

In its original form (named after [Gauß 1877](#Gauß1877)), the *Gauß law* of [[electromagnetism]] asserts that the total [[flux]] of the [[electric field]] through any [[closed manifold|closed]] [[surface]] (in physical space) is proportional to the total [[electric charge]] enclosed by that surface, otherwise irrespective of the charge distribution. In its differential form, the Gauss law is part of what later became [[Maxwell's equations]] of [[electromagnetism]].

In terms of modern [[differential geometry]], with

* $X^3$ denoting the [[dimension of a manifold|3-dimensional]] [[smooth manifold]] representing physical space (in this context often but not necessarily assumed to be the [[Euclidean space]] $\mathbb{R}^3$),

* $E \,\in\, \Omega^2_{dR}(\mathbb{R}^4)$ denoting the [[differential 2-form]] which representes the electric field's [[flux density]]

  (beware that this 2-form is the spatial [[Hodge dual]] of the electric [[vector field]] 1-form used elsewhere),

* $\Sigma^3 \hookrightarrow X^3$ denoting an [[orientation|orientable]] 3-[[dimension of a manifold|dimensional]] [[submanifold]] [[manifold with boundary|with boundary]],

* $\partial \Sigma^3 = \Sigma^2 \hookrightarrow X^3$ denoting its [[boundary]], a [[closed manifold|closed]] [[orientation|orientable]] 2-[[dimension of a manifold|dimensional]] [[submanifold]],

* $Q \in \mathbb{R}$ denoting the **total [[electric charge]] on $\Sigma^3$**,

* $\int_{\Sigma^2} E$ denoting the [[integration of differential forms|integral]] of $E$ over $\Sigma$, hence the **total electric [[flux]] through $\Sigma^2$**,

the Gauss law in its original **integral form** states that the last two quantities are proportional to each other:

\[
  \label{IntegralFormOfEMGaussLaw}
  \textstyle{\int_{\Sigma^2}} E
  \;\;\propto\;\;
  Q
  \,,
\]

with the constant of proportionality depending on conventions but typically being the inverse of the [[permittivity of the vacuum]].

By the [[Stokes theorem]] and with

* $\rho \,\in\, \Omega^3_{dR}(X^3)$ denoting a [[differential 3-form]] representing the electric charge density, i.e. such that

* $\int_{\Sigma^3} \rho \,=\, Q$,

* $\mathrm{d} E$ denoting the [[de Rham differential]] of the electric flux density, representing its [[divergence]],

the integral law (eq:IntegralFormOfEMGaussLaw) is equivalent to the following **differential form** of the Gauss law:

\[
  \label{DifferentialFormOfEMGaussLaw}
  \mathrm{d} E \,=\, \rho
  \,.
\]

It is in this differential form that the Gauss law is part of the modern form of [[Maxwell's equations]] (see their formulation in [[exterior calculus]] [here](Maxwell's+equations#InTermsOfFaradayTensor)).

### In Yang-Mills theory
 {#InYangMillsTheory}

The differential form of the Gauss law (eq:DifferentialFormOfEMGaussLaw) generalizes immediately to [[Yang-Mills theory]] with [[gauge group|gauge]] [[Lie algebra]] $mathfrak{g}$ by enhancing the [[de Rham differential]] to the corresponding [[covariant derivative|covariant differential]]  with respect to the given [[gauge potential]].

(...)

### In higher gauge theory
 {#InHigherGaugeTheory}

The generalization of Gauss laws to [[higher gauge theory]] may transparently be understood, in broad generality, by casting the corresponding [higher Maxwell-type](higher+gauge+field#HigherGaugeTheoryOfMaxwellType) [[equations of motion]] into "[[duality-symmetric gauge theory|duality-symmetric]]" form (we follow [SS23](#SS23)):

Consider:

* $D = 1 + d \in \mathbb{N}_{\geq 2}$ the [[spacetime]] dimension,

* $X^D$ a $D$-dimensional [[spacetime]] [[smooth manifold|manifold]],

* with [[Hodge star]]-operator on [[differential forms]]

  $$
    \star \,\colon\,
    \Omega^p_{dR}(X^D) \to \Omega^{D-p}_{dR}(X^D)
  $$

  satisfying (see [there](Hodge+star+operator#eq:HodgeSquareOnRiemannian))

  $$
    \star \, \star = - (-1)^{p(D-p)}
    \,,
  $$

* $I \,\in\, Set$ an [[index set]] of flux species

  (subsuming both electric fluxes and magnetic fluxes, both now allowed of further different species),

* $\big( deg_i \,\in\, \mathbb{N}_{\geq 1} \big)_{i \in I}$ an [[indexed set]] of degrees,

* $\vec F \,\equiv\,\big( F^{(i)} \,\in\, \Omega^{deg_i}_{dR}(X^D) \big)_{i \in I}$ an [[indexed set]] of [[flux densities]]

* $\vec P = \big( P^{(i)}  \big)_{i \in I}$ an [[indexed set]] of graded-symmetric [[polynomials]] in $I$ [[variables]],

* $\vec \mu$ an [[invertible matrix|invertible]] $I \times I$-[[matrix]]

then:

\begin{definition}\label{HigherMaxwellInDualitySymmetricForm}
The corresponding **higher Maxwell-type equations** on these [[differential forms]]/[[flux densities]] are, in [[duality-symmetric higher gauge theory|duality-symmetric form]]:

1. the system of [[Bianchi identity]] [[differential equations]]

   \[
     \label{HigherBianchiIdentities}
     \mathrm{d}\,\vec F
     \,=\,
     \vec P\big(
       \vec F
     \big)
   \]

1. the self-[[Hodge duality]] condition

   \[
     \label{HigherSelfDuality}
     \star \, \vec F
     \,=\,
     \vec \mu\big(\vec F\big)
     \,.
   \]

\end{definition}

To extract a Gauss law from such a system of equations, consider the case that the [[spacetime]] $X^D$ is [[globally hyperbolic spacetime|globally hyperbolic]] with associated (by [this Prop.](Cauchy+surface#FoliationOfGloballyHyperbolicSpacetimes)) smooth [[foliation]] by [[spacelike]] [[Cauchy surfaces]] $X^d$ exhibited by a [[diffeomorphism]]:

$$
  X^D
  \,\simeq\,
  \mathbb{R} \times X^d
  \,.
$$

This identifies the [[de Rham differential]] with the [[total differential]] of a [[double complex]] of [[differential forms]], bigraded by spatial and by temporal degree:

$$
  \mathrm{d}
  \,=\,
  \mathrm{d}_t + \mathrm{d}_s
  \,,
  \;\;\;\;\;
  \mathrm{d}_t \circ \mathrm{d}_t \,=\, 0
  \,,
  \;\;\;\;\;
  \mathrm{d}_s \circ \mathrm{d}_s \,=\, 0
  \,,
  \;\;\;\;\;
  \mathrm{d}_s \circ \mathrm{d}_t \,=\,
  -
  \mathrm{d}_t \circ \mathrm{d}_s
  \,.
$$

Denoting by

$$
  \Omega^p_{dR}\big(
    X^D
  \big)_{\iota_{\partial_t}}
  \hookrightarrow
  \Omega^p_{dR}\big(
    X^D
  \big)_{\iota_{\partial_t}}
$$

the [[kernel]] of the [[tensor contraction|contraction]] with the corresponding [[timelike]] [[vector field]] $\partial_t$, every [[flux density]] now has a unique decomposition into a summand that has none and a summand that has one factor of $\mathrm{d}t$:

$$
  \vec F
  \;=\;
  \vec B
  \,+\,
  \star
  \vec E
  \,,
  \;\;\;\;
  \text{for}
  \;
  \left\{
  \,
  \begin{array}{l}
    \vec B \,\equiv\,
    \big(
      B^{(i)}
        \in
     \Omega^{deg_i}_{dR}(X^D)_{\iota_{\partial_t} = 0}
    \big)
    \\
    \vec E \,\equiv\,
    \big(
      E^{(i)}
        \in
     \Omega^{deg_i}_{dR}(X^D)_{\iota_{\partial_t} = 0}
    \big)
    \,.
  \end{array}
  \right.
$$

In terms of these components, the higher [[Bianchi identities]] (eq:HigherBianchiIdentities) are equivalent to the following system of [[differential equations]]:

* **higher Gauss laws**

  \[
    \label{HigherGaussLaw}
    \mathrm{d}_s
    \,
    \vec B
    \;=\;
    \vec P\big( \vec B \big)
  \]

* **higher Faraday-Ampère laws**

  \[
    \label{HigherFaradayAmpereLaw}
    \mathrm{d}_t
    \,
    \vec B
    \;=\;
    -
    \mathrm{d}_s \,
    \star \vec E
    \,+\,
    \mathrm{d}_s
    \,
    \big(\star E^{(i)}\big)
    \wedge
    \frac{\delta}{\delta B^{(i)}}
    \vec P\big(
      \vec B
    \big)
    \,.
  \]

Hence in the [[duality-symmetric gauge theory|duality-symmetric formulation]] of [[higher gauge theory]], the Gauss law is just the spatial component of the [[Bianchi identity]].

Moreover, one finds that the higher Gauss law (eq:HigherGaussLaw) is a [[first class constraint]] in that it is preserved by the time evolution presribed by the higher Faraday-Ampère law (eq:HigherGaussLaw), so that the solution space of the [higher Maxwell equations](higher+gauge+field#HigherGaugeTheoryOfMaxwellType) is identified with the space of flux densities on any [[Cauchy surface]]
$$
  \iota
  \,\colon\,
  X^d \hookrightarrow X^D
$$
that satisfy (just) the higher Gauss law:

In examples relevant in practice, the [[functor|functorial]] assignment of these canonical solution spaces is [[representable functor|representable]] by the [[formal duality|formal dual]] of a [[dgc-algebra]] which embodies the [[Bianchi identities]] as abstract dg-algebraic relations:

\[
  \label{CharacteristicCEAlgebra}
  CE(\mathfrak{a})
  \;\coloneqq\;
  \mathbb{R}\Big[
    \vec b
    \,\equiv\,
    \big(
      b^{(i)}_{deg_i}
    \big)
  \Big]
  \Big/
  \Big(
    \mathrm{d} \vec b \,=\, \vec P(\vec b)
  \Big)
  \,,
\]

as

$$
  \Big\{
  \vec B
  \,\in\,
  \big(
    B^{(i)}
    \,\in\,
    \Omega^{deg_i}_{dR}(-)
  \big)_{i \in I}
  \,\Big\vert\,
  \mathrm{d}\, \vec B \,=\, \vec P\big(\vec B\big)
  \Big\}
  \;\;\;
  \simeq
  \;\;\;
  Hom_{dgAlg}\big(
    CE(\mathfrak{a})
    ,\,
    \Omega^{\bullet}_{dR}(-)
  \big)
  \,.
$$

This is equivalently the [[Chevalley-Eilenberg algebra]] of a characteristic [[L-infinity algebra|$L_\infty$-algebra]] $\mathfrak{a}$ (by the discussion [there](L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra)), making the right hand side the [[flat L-infinity-algebra valued differential forms|flat $L_\infty$-algebra valued differential forms]]:

$$
  SolSpace
  \;\simeq\;
  \Omega_{dR}\big(
    X^d
    ;\,
    \mathfrak{a}
  \big)_{flat}
  \,,
$$

with the higher Gauss law being exactly the flatness condition.

\begin{example}
**(ordinary Gauss law as special case of higher Gauss law)**
\linebreak
  The ordinary case ([above](#InElectromagnetism)) of the Gauss law in [[vacuum]] [[electromagnetism]] (ie. with vanishing [[electric charge]] density) is obtained from Def. \ref{HigherMaxwellInDualitySymmetricForm} by setting

* $D \equiv 4$

* $I \equiv \{el, mag\}$

* $\vec P \equiv 0$

* $\vec \mu \equiv \left[\array{ 0 & -1 \\ 1 & 0 }\right]$.

This gives the electric and magnetic Gauss laws as

* $\mathrm{d} B^{(el)} \,=\, 0$

* $\mathrm{d} B^{(mag)} \,=\, 0$

where self-duality constraint identifies the elctric flux density equivalently as $B^{(el)} \,=\, E^{(mag)}$.
\end{example}

\begin{example}
**([[chiral boson]] in 2d)**
\linebreak
(...)
\end{example}

\begin{example}
**([[RR-fields]] in [[D=10 supergravity]])**
\linebreak
(...)
\end{example}

\begin{example}
**([[supergravity C-field|C-field]] in [[D=11 supergravity]])**
\linebreak
(...)
\end{example}

## Related concepts

* [[constrained mechanics]]

* [[first class constraint]]

* [[uncertainty of fluxes]]

## References

### General

Named after:

* {#Gauß1877} [[Carl F. Gauß]], *Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata*, Commentationes Societatis Regiae Scientiarum Göttingensis Recentiores. Comm. Class. Math. **2** (1813) 1–24 [[PPN35283028X_0002_2NS](http://resolver.sub.uni-goettingen.de/purl?PPN35283028X_0002_2NS)]

  reprinted in: *Carl Friedrich Gauss -- Werke* **5**, Springer (1877) 2-22 [[doi:10.1007/978-3-642-49319-5_1](https://doi.org/10.1007/978-3-642-49319-5_1)]

For more see any of the references at *[[Maxwell's equations]]*, such as:

* [[Friedrich W. Hehl]], [[Yuri N. Obukhov]], I.1 & B1.45 in: *Foundations of Classical Electrodynamics -- Charge, Flux, and Metric*, Progress in Mathematical Physics **33**, Springer (2003) [[doi:10.1007/978-1-4612-0051-2](https://doi.org/10.1007/978-1-4612-0051-2)]

See also:

* Wikipedia, *[Gauss's law](https://en.wikipedia.org/wiki/Gauss%27s_law)*

### As a phase space constraint

Discussion of the Gauss law as a [[first-class constraint|first-class]] [[constrained mechanics|constraint]] on [[phase space]]:

In [[Maxwell theory]]:

* {#BlaschkeGieres21} [[Daniel N. Blaschke]], [[François Gieres]], equation (5.8) in: *On the canonical formulation of gauge field theories and Poincaré transformations*, Nuclear Physics B **965** (2021) 115366 [[doi:10.1016/j.nuclphysb.2021.115366](https://doi.org/10.1016/j.nuclphysb.2021.115366), [arXiv:2004.14406](https://arxiv.org/abs/2004.14406)]

* [[Marc Henneaux]], [[Claudio Teitelboim]], p. 456 in: *[[Quantization of Gauge Systems]]*, Princeton University Press 1992 [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

In [[Yang-Mills theory]]:

* {#FriedmanPapastamatiou83} [[John L. Friedman]], Nicholas J. Papastamatiou, equation (3.9) in: *On the canonical quantization of Yang-Mills theories*, Nuclear Physics B **219** 1 (1983) 125-142 \[doi:10.1016/0550-3213(83)90431-5\]

In [[higher gauge theory]]:

* {#SS23} [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Flux Quantization on Phase Space]]* [[arXiv:2312.12517](https://arxiv.org/abs/2312.12517)]

### In relation to locality

The Gauss law in relation to non-[[local field theory|locality]] of charged states in [[quantum field theory]]:

* [[Giovanni Morchio]], [[Franco Strocchi]], *Localization and symmetries*, J. Phys. A **40** (2007) 3173-3188 [[arXiv:math-ph/0607015](https://arxiv.org/abs/math-ph/0607015), [doi:10.1088/1751-8113/40/12/S17](https://doi.org/10.1088/1751-8113/40/12/S17)]

* [[Detlev Buchholz]], Fabio Ciolli, [[Giuseppe Ruzzi]], Ezio Vasselli, *Gauss's law, the manifestations of gauge fields, and their impact on local observables* [[arXiv:2212.11009](https://arxiv.org/abs/2212.11009)]

see also in relation to [[infraparticles]]:

* {#Buchholz86} [[Detlev Buchholz]], _Gauss’ law and the infraparticle problem_, Phys.  Lett.  B **174** (1986) 331-334 \[doi:10.1016/0370-2693(86)91110-X\]

[[!redirects Gauss laws]]

[[!redirects Gauss's law]]
[[!redirects Gauss's laws]]

[[!redirects Gauss' law]]
[[!redirects Gauss' laws]]

[[!redirects Gauß law]]
[[!redirects Gauß laws]]

[[!redirects magnetic Gauss law]]
[[!redirects magnetic Gauss laws]]

[[!redirects magnetic Gauss' law]]
[[!redirects magnetic Gauss' laws]]

[[!redirects magnetic Gauß law]]
[[!redirects magnetic Gauß laws]]

[[!redirects electric Gauss law]]
[[!redirects electric Gauss laws]]

[[!redirects electric Gauss' law]]
[[!redirects electric Gauss' laws]]

[[!redirects electric Gauß law]]
[[!redirects electric Gauß laws]]


Gauss lemma


Gauss multiplication formula

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

#Contents#
* table of contents
{:toc}

## Statement

The following is known as the *[[Carl Gauss|Gauss]] multiplication formula* or the *[[Legendre]] relation*:

\begin{prop}\label{GaussMultiplicationFormula}

For

* a [[positive number|positive]] [[integer]] $N \in \mathbb{N}_+$,

* any $z \in \mathbb{R} \setminus  \{0, -1/N, -2/N, \cdots\} $,

the [[Gamma function]] $\Gamma(-)$ satisfies

\[
  \underoverset
    {j = 0}
    {N-1}
    {\prod}
  \Gamma
  \left(
    z + \tfrac{j}{N}
  \right)
  \;=\;
  (2 \pi)^{ \tfrac{1}{2}(N-1) }
  \cdot
  N^{ (\tfrac{1}{2} - N z) }
  \cdot
  \Gamma( N z )
  \,.
\]

\end{prop}

\begin{example}
In the special case of $N = 2$ this is known as the *duplication formula*:

$$
  \Gamma(z)
  \cdot
  \Gamma
  \big(
    z + 1/2
  \big)
  \;=\;
  (2\pi)^{1/2}
  \cdot
  2^{1/2 - 2z}
  \cdot
  \Gamma(2 z)
  \,.
$$
\end{example}

\begin{remark}
  A related expression gives a product of [[Barnes G-functions]] (see [there](Barnes+G-function#MultiplicationFormulaForGammaFunctionInTermsOfGFunction)):
$$
  \underoverset
    {j = 1}
    {N}
    {\prod}
  \Gamma(j/2)
  \;=\;
  \frac
    {
      G(N/2+ 1)
      \cdot
      G(N/2 + 1/2)
    }
    { G(1/2) }
  \,.
$$
\end{remark}

## References

* J. Sándor and L. Tóth, *A remark on the gamma function*, Elem. Math. 44 (3), pp. 73–76 (1989) ([dml:141455](https://eudml.org/doc/141455))

See also:

* Wikipedia, *[Multiplication theorem](https://en.wikipedia.org/wiki/Multiplication_theorem)*

* ProofWiki, *[Gauss Multiplication Formula](https://proofwiki.org/wiki/Gauss_Multiplication_Formula)*

* DLMF, *[Gamma function §5.5(iii) Multiplication](https://dlmf.nist.gov/5.5#iii)*

* Wolfram MathWorld, *[Gauss multiplication formula](https://mathworld.wolfram.com/GaussMultiplicationFormula.html)*

* Wolfram MathWorld, *[Legendre Duplication formula](https://mathworld.wolfram.com/LegendreDuplicationFormula.html)*

[[!redirects Legendre relation]]

[[!redirects multiplication theorem]]

[[!redirects duplication formula]]
[[!redirects Legendre duplication formula]]


Gauss-Bonnet theorem

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

#Contents#
* table of contents
{:toc}

## Idea

The _Chern-Gauss-Bonnet theorem_ gives a formula that computes the [[Euler characteristic]] of an even-[[dimension]]al [[smooth manifold]] as the [[integration]] of a [[curvature characteristic form]] of the [[Levi-Civita connection]] on its [[tangent bundle]].

For [[surface]]s the theorem simplifies and in this simpler version is the older _Gauss-Bonnet theorem_ .

## Statement

### For surfaces

(...)

### For smooth manifolds

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

Let $X$ be a [[compact space|compact]] [[smooth manifold]] of even [[dimension]] $dim X = 2k$. Write $\chi(X)$ for its [[Euler characteristic]].

For $\nabla$ any [[Levi-Civita connection]] on its [[tangent bundle]], write $F_\nabla$ for its [[curvature]] [[differential form|2-form]], valued in the [[orthogonal Lie algebra]] $\mathfrak{so}(2k)$ and $Pf(F_\nabla)$ for its [[Pfaffian]] $2k$-form.

Then

$$
  \chi(X) =
  \left(
    \frac{-1}{2 \pi}
  \right)^{k}
  \int_X Pf(F_\nabla)
  \,.
$$

=--

### For orbifolds

There is a generalization for $X$ an [[orbifold]] due to ([Satake](#Satake)).

## Related concepts

* [[Euler characteristic]]

* [[Euler form]]

* [[Poincaré–Hopf theorem]]

## References

Named after [[Carl Friedrich Gauss]].

The Chern-Gauss-Bonnet theorem goes back to

* [[ Shiing-shen Chern]], _A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds_ , Annals, 45 (1944), 747-752.

A classical textbook reference is chapter X of volume II of

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_ Academic Press (1973)

Discussion is for instance in

* Denis Bell, _The Gauss-Bonnet theorem for vector bundles_ ([pdf](http://www.unf.edu/~dbell/GBT.pdf))

Expositions include

* [[Liviu Nicolaescu]], _The many faces of the Gauss-Bonnet theorem_ ([pdf](http://www.nd.edu/~lnicolae/GradStudSemFall2003.pdf))

* Denis Bell, _The Gauss-Bonnet theorem_ ([pdf](http://www.unf.edu/~dbell/GBVB.pdf))

* [[Chenchang Zhu]], _The Gauss-Bonnet theorem and its applications_ ([pdf](http://math.berkeley.edu/~alanw/240papers00/zhu.pdf))

* [[Johannes Ebert]], _[MO comment](http://mathoverflow.net/questions/53302/is-there-a-chern-gauss-bonnet-theorem-for-orbifolds/53351#53351)_

The generalization to [[orbifold]]s is considered in

* {#Satake} [[Ichiro Satake]], _The Gauss-Bonnet Theorem for V-manifolds_ J. Math. Soc. Japan Volume 9, Number 4 (1957), 464-492.  ([ProjectEclid](http://projecteuclid.org/euclid.jmsj/1261153826))

Generalization to [[orbifolds]]:

* Christopher Seaton, _Two Gauss–Bonnet and Poincaré–Hopf theorems for orbifolds with boundary_, Differential Geometry and its Applications
Volume 26, Issue 1, February 2008, Pages 42-51 ([doi:10.1016/j.difgeo.2007.11.002](https://doi.org/10.1016/j.difgeo.2007.11.002))

For an approach via 0|2-dimensional [[supersymmetric Euclidean field theory]], see

* [[Daniel Berwick-Evans]], _The Chern-Gauss-Bonnet Theorem via supersymmetric Euclidean field theories_, Communications in Mathematical Physics, Volume 335 (2015), ([arXiv:1310.5383](https://arxiv.org/abs/1310.5383)).

[[!redirects Chern-Gauss-Bonnet theorem]]
[[!redirects Gauss-Bonnet density]]


Gauss-Manin connection

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

#Contents#
* table of contents
{:toc}

## Definition

A *Gauss-Manin connection* is a canonical [[flat connection]] on [[fiber bundles]] of [[relative cohomology|relative]] [[ordinary cohomology|ordinary]] (but possibly [[twisted cohomology|twisted]]) ([[cohomology groups|co]]-)[[homology groups]], hence a way of transforming (co-)homology classes as a *parameter* varies.

## Details

(...)

## For fiber bundles
 {#ForFiberBundles}

The existence of a flat connection on bundles of fiberwise-cohomology groups is easy to understand in the case that the [[fibers]] form a [[locally trivial bundle|locally trivial]] [[fiber bundle]]: In this case the fiberwise cohomology of the local trivialization of this fiber bundle is a local trivialization of the bundle of its fiberwise cohomology groups. The Gauss-Manin connection in this case is made explicit for instance in [Voisin 2002, above Def. 9.13](#Voisin02) ([[Voisin-DefinitionOfGaussManinConn.pdf:file]]), see also [Etingof, Frenkel & Kirillov 1998, §7.5](#EtingofFrenkelKirillov98) ([[EtingofEtAl-DefOfGaussManin.pdf:file]]).

Simple as this may appear, at least in the case of fiberwise [[twisted cohomology]] this case subsumes (for forgetful fibrations of [[configuration spaces of points]] in the plane) the profound example of solutions to the [[Knizhnik-Zamolodchikov equation]], identified as such via the  [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) -- this is highlighted as such in [Etingof, Frenkel & Kirillov 1998, §7.5](#EtingofFrenkelKirillov98), reproduced as Example \ref{ExampleOfHypergeometricKZEquations} below.

In the following we give (following [SS22](#SS22)) a more abstract/formal ([[model category|model]] [[category theory|category theoretic]]/[[homotopy theory|homotopy theoretic]]) argument for the existence and construction of the Gauss-Manin connection for [[twisted cohomology]] on a class of [[fiber bundles]] which subsume the case that gives the [[KZ-equation]], namely projections of [[configuration spaces of points]] (Ex. \ref{ForgetfulMapBetweenConfigurationSpaces} below).

This following more abstract argument has the advantage that it applies at once to all [[generalized cohomology]]-theories for which [[classifying spaces]] exist ([[Whitehead-generalized cohomology theories]] and more general [[non-abelian cohomology]]-theories, such as unstable [[Cohomotopy]]), and that its structure is readily formulated in general [[(infinity,1)-topos|$(\infty,1)$-toposes]] and in [[homotopy type theory]]. In fact, in that formal language the following construction of the Gauss-Manin connection for twisted cohomology on fibers of a fiber bundle is so simple that it becomes essentially a tautology.

### For non-abelian cohomology
 {#ForNonAbelianCohomology}

The following is a (non-standard) general-abstract argument for the existence and construction of the structure of local systems on fiberwise cohomology groups, which applies in the generality of [[Whitehead-generalized cohomology theories]], and in fact of [[non-abelian cohomology|non-abelian cohomology theories]].

The simple idea -- using that all these notions of cohomology have [[classifying spaces]] -- is that the fiberwise 0-truncation of the *[fiberwise mapping space](parameterized+homotopy+theory#FiberwiseMappingSpaces)* to that classifying space provides at once the [[covering space]] which exhibits the local system.

\linebreak

We consider the following situation, see also (eq:OverviewUntwisted) and (eq:NotationOverview).

Let

1. $B \in kHaus \hookrightarrow kTop$ be

   1. a [[compactly generated topological space|compactly generated]] [[Hausdorff space]],

   1. [[locally path-connected topological space|locally path-connected]] and [[semi-locally simply-connected topological space|semi-locally simply connected]],

1. $(X \xrightarrow{p_X} B) \in kTop_{/B}$ be

   1. a [[compactly generated topological space]] (i.e. a [[k-space]]),

   1. in the [[slice category]]  over $B$, via a [[Hurewicz fibration]]

      \[
        \label{AssumingpXIsFibration}
        p_X \,\in\, hFib
        \,.
      \]

   1. which is [[fiber bundle|locally trivial]] over a [[numerable cover]] $\big\{ U_j \xhookrightarrow{\;\iota_j\;} B\big\}_{j \in J}$ for some [[typical fiber]] $X_0$:

      \[
        \label{LocalTrivialityOfFibration}
        \underset{j \in J}{\forall}\;\;\;\;
        \iota_j^\ast (X, p_X) \,\simeq\, p_B^\ast X_0
      \]

   1. what has the [[mathematical structure|structure]] of a [[CW-complex]]:

      \[
        \label{FiberIsCWComplex}
        X_0 \,\in\, CWCpl \to kTop
        \,.
      \]

For $b \in B$ we write $X_b$ for the [[fiber]] of $p_X$ over $b$:

\[
  \label{FiberOfTheFibration}
  \array{
    X_b &\xrightarrow{\phantom{----}}& X
    \\
    \big\downarrow
    &{}_{{}^{(pb)}}&
    \big\downarrow{}^{\mathrlap{p_X}}
    \\
    \{b\}
    &\underset{\phantom{----}}{\hookrightarrow}&
    B
  }
\]

\begin{example}\label{ForgetfulMapBetweenConfigurationSpaces}
**(forgetful map between configuration spaces of points)**
\linebreak
  For $n,N \,\in\, \mathbb{N}$, the map of [[configuration spaces of ordered points]] in the [[plane]]
$$
  X
    \;=\;
  \underset{
    \{ 1, \cdots, n+N \}
  }{Conf}
  \big(
    \mathbb{R}^2
  \big)
  \xrightarrow{\;\;\;}
  \underset{
    \{1, \cdots, N\}
  }{Conf}
  \big(
    \mathbb{R}^2
  \big)
  \;=\;
  B
  \,
$$
(which forgets the position of the first $n$ among $n + N$ points)
satisfies the above assumption, by [this Prop.](configuration+space+of+points#ForgettingPointsIsAFibration) and [this Prop.](configuration+space+of+points#CnfSpaceOfPointsInPlaneIsEMSpace).
\end{example}

\begin{definition}\label{NonAbelianCohomology}
**(non-abelian cohomology sets)**
\linebreak
For any $A \,\in\, kTop$, and for $X_b$ a [[CW-complex]], we write
\[
  \label{NonAbelianCohomologySets}
  A^0(X_b)
    \;\coloneqq\;
  H^0(X_b;\, A)
    \;\coloneqq\;
  \pi_0
  Map
  \big(
    X_b
    ,\,
    A
  \big)
\]
for the *[[non-abelian cohomology]]* of $X_b$ with [[coefficients]] in $A$.
\end{definition}
\begin{example}
**(ordinary complex cohomology)**
\linebreak
  For $A \,=\, K(n,\mathbb{C})$ an [[Eilenberg-MacLane space]] (here of the additive [[abelian group]] of [[complex numbers]]), Def. \ref{NonAbelianCohomology} yields [[ordinary cohomology|ordinary]] [[complex cohomology]]
$$
  H^n(X_b; \mathbb{C})
  \,\simeq\,
  \pi_0 Map\big(X_b, \, K(\mathbb{C},n)\big)
  \,.
$$
\end{example}
\begin{example}
**(Cohomotopy)**
\linebreak
  For $A \,=\, S^n$ the [[n-sphere|$n$-sphere]], Def. \ref{NonAbelianCohomology} yields [[Cohomotopy]]
$$
  \pi^n(X_b)
  \,\simeq\,
  \pi_0 Map\big(X_b, \, S^n\big)
  \,.
$$
\end{example}

\begin{proposition}\label{FiberwiseNonAbelianCohomologySetsFormLocalSystem}
**(fiberwise non-abelian cohomology sets form local system)**
\linebreak
  For any [[k-space]] $A \,\in\, kTop$, the [[non-abelian cohomology]]-sets (eq:NonAbelianCohomologySets) of the [[fibers]] $X_b$ (eq:FiberOfTheFibration) with [[coefficients]] in $A$
constitute a [[local system]] over $B$, in that they arrange into a [[covering space]] over $B$ ([[fundamental theorem of covering spaces|equivalently]], they are the values of a [[functor]] from the [[fundamental groupoid]] of $B$ to [[Sets]]).
\[\label{OverviewUntwisted}\]
\begin{tikzcd}[
  sep=small
]
    &
    H^n(\mathrm{X}_{b_2})
    \ar[
      ddr,
      "{\sim}"{sloped},
      shorten=-2pt
    ]
    \\[-27pt]
    &&&
    \mathrm{X}
    \ar[
      dddd,
      "{p_{\mathrm{X}}}"
    ]
    \ar[
      white,
      dr,
      "{\color{white}\tau}"{swap, xshift=1pt}
    ]
    \ar[
      rr,
      dashed
    ]
    &&
    K(n,\mathbb{C})
    \\[-12pt]
    H^n(\mathrm{X}_{b_1})
    \ar[
      uur,
      shorten=-2pt,
      "{
        \sim
      }"{sloped}
    ]
    \ar[
      rr,
      "{\sim}"{swap}
    ]
    &&
    H^n(\mathrm{X}_{b_3})
    &&
    \phantom{B G}
    \ar[
      white,
      dddd,
      "{\color{white}
        p_{B G}
      }"
    ]
    \\
    \\
    &
    \{b_2\}
    \ar[
      ddr,
      "{
        [\gamma_{23}]
      }"
    ]
    \\[-27pt]
    &&&
    \mathrm{B}
    \ar[
      white,
      dr,
      "{\color{white}p_{\mathrm{B}}}"{swap}
    ]
    \\[-12pt]
    \{b_1\}
    \ar[
      uur,
      "{
        [\gamma_{12}]
      }"
    ]
    \ar[
      rr,
      "{
        [\gamma_{23} \circ \gamma_{12}]
      }"{swap}
    ]
    &&
    \{b_3\}
    &
    &
    \ast
\end{tikzcd}

Moreover, if $\big\{U_j \xhookrightarrow{\iota_j} B\big\}_{j \in J}$ is an [[open cover]] over which $X \to B$ [[local trivialization|locally trivializes]] with [[typical fiber]] $X_0$, then the covering space of cohomology sets [[local trivialization|locally trivializes]] over the same cover, with [[typical fiber]] $H^0(X_0; A)$.
\end{proposition}
\begin{proof}
  Write $p_B^\ast A \,\in\, kTop_{/B}$ for the [[base change]] of $A$ to the [[slice category|slice]] over $B$, i.e. for the trivial fibration
\[
  \label{ATrivialFibration}
  p_B^\ast A
  \;\;\;\coloneqq\;\;\;
  B \times A \xrightarrow{\; pr_B \;} B
  \,.
\]
This is clearly a [[Hurewicz fibration]]. Since also $p_X$ is such by assumption (eq:AssumingpXIsFibration), it follows (by [this Prop.](parameterized+homotopy+theory#FiberwiseMappingSpacePreservesHFibrations)) that the [fiberwise mapping space](parameterized+homotopy+theory#FiberwiseMappingSpaces) between the two ([this Def.](parameterized+homotopy+theory#FiberwiseMappingSpace)) is also a [[Hurewicz fibration]], hence its [[fibers]] are [[homotopy fibers]] and as such coincide with the ordinary [[compact-open topology|mapping space]] between the fibers (by [this Example](parameterized+homotopy+theory#FiberOfFiberwiseMappingSpace)):
\[
  \label{HomotopyFiberOfFiberwiseMappingSpace}
  \array{
    Map
    \big(
      X_b
      ,\,
      A
    \big)
    &
    \xrightarrow{\phantom{--}}
    \;\;\;\;\;
    &
    \phantom{---}
    \mathclap{
    Map
    \big(
      (X,p_X)
      ,\,
      p_B^\ast A
    \big)
    }
    \phantom{---}
    \\
    \big\downarrow
    &{}_{{}^{(pb)}}&
    \big\downarrow{}^{ \mathrlap{\in Fib}}
    \\
    \{b\}
    &\xrightarrow{\phantom{-----}}&
    B
    \mathrlap{\,.}
  }
\]
Hence we may think of this diagram equivalently as exhibiting the [[homotopy pullback]] (in the [[classical model structure on topological spaces]]) of the mapping fibration along $\{b\} \to B$.

But since homotopy pullback preserves fiberwise 0-truncation (by [this Prop.](n-connected/n-truncated+factorization+system#HomotopyPullbackPreservesNImageFactorization)) it follows that the fiberwise 0-truncation of the fiberwise mapping space fibration has as [[homotopy fibers]] the [[non-abelian cohomology]]-sets (eq:NonAbelianCohomologySets):

$$
  \array{
    \mathllap{
      H^0(X_b;\, A)
      \;=\;
    }
    \pi_0
    Map
    \big(
      X_b
      ,\,
      A
    \big)
    &
    \xrightarrow{\phantom{--}}
    \;\;\;\;\;\;\;
    &
    \phantom{----}
    \mathclap{
    \pi_{0/B}
    \,
    Map
    \big(
      (X,p_X)
      ,\,
      p_B^\ast A
    \big)
    }
    \phantom{----}
    \\
    \big\downarrow
    &{}_{{}^{(pb)}}&
    \big\downarrow{}^{ \mathrlap{\in Fib} }
    \\
    \{b\}
    &\xrightarrow{\phantom{-----}}&
    B
    \mathrlap{\,.}
  }
$$
Here by [[fibrant replacement]] we may assume that the right vertical map is again a [[Serre fibration]], as indicated, in which case it is a [[covering space]]-projection whose fibers are the desired cohomology groups.
This proves the first statement.

The second statement follows by the same argument, after using over any patch $U_j \xhookrightarrow{\iota_j} B$ of the given open cover the following [[natural isomorphism|natural]] identification:
$$
  \begin{array}{ll}
    \iota_j^\ast
    Map
    \big(
      (X, p_X)
      ,\,
      p_B^\ast A
    \big)
    \\
    \;\simeq\;
    Map
    \big(
      \iota_j^\ast
      (X, p_X)
      ,\,
      \iota_j^\ast
      p_B^\ast A
    \big)
    &
    \text{by (eq:PullbackOfFiberwiseMappingSpaces)}
    \\
    \;\simeq\;
    Map
    \big(
      p_{U_j}^\ast X_0
      ,\,
      p_{U_j}^\ast A
    \big)
    &
    \text{by (eq:LocalTrivialityOfFibration)}
    \\
    \;\simeq\;
    p_{U_j}^\ast
    Map
    \big(
      X_0
      ,\,
      A
    \big)
    &
    \text{by (eq:PullbackOfFiberwiseMappingSpaces)}
  \end{array}
$$
\end{proof}
Here we have used that [[pullback]] is a [[closed functor]] with respect to [[fiberwise mapping spaces]] (by [this Prop.](parameterized+homotopy+theory#PullbackOfFiberwiseMappingSpace)):
\[
  \label{PullbackOfFiberwiseMappingSpaces}
  f^\ast
  Map
  \big(
    (X,p_X)
    ,\,
    (Y,p_Y)
  \big)
  \;\simeq\;
  Map\big( f^\ast(X,p_X),\, f^\ast(Y,p_Y) \big)
  \,.
\]

### For twisted non-abelian cohomology
 {#ForTwistedNonabelianCohomology}

We generalize the [above](#ForNonAbelianCohomology) discussion to [[twisted cohomology|twisted]] [[non-abelian cohomology]] (Def. \ref{TwistedNonAbelianCohomology} below).

The argument is essentially the same as in the previous Prop. \ref{FiberwiseNonAbelianCohomologySetsFormLocalSystem}, only that now:

1. the base spaces starts out being the [[product space]] $B \times B G$ of the previous base space $B$ with the [[classifying space]] $B G$ for the twists,

1. before passing to homotopy classes of fiberwise maps we form the right [[base change]] $(B \times p_{B G})_\ast$ ([[dependent product]]) along this classifying space.

To see that the previous argument generalizes to this case one needs to

1. observe that this right base change is right Quillen (Lem. \ref{RightBaseChangeAlongProjectionIsQuillen} below),

1. use a [[Beck-Chevalley relation]] (eq:BeckChevalleyRelationForProjection) to see that it is compatible with pullback along $\{b\} \to B$.

\linebreak

For $G$ a [[discrete group]], we write $B G \,\coloneqq\, \vert N (G \rightrightarrows \ast)\vert$ for the [[topological realization]] of the [[nerve]] of the [[delooping groupoid]] of $G$, hence for the usual [[classifying space]] (which, by discreteness of $G$, is an [[Eilenberg-MacLane space]] $K(G,1)$) in its usual realization as a [[CW-complex]]:
\[
  \label{BGBeingCWCOmplex}
  B G \,\in\, CWCpx \hookrightarrow kTop^{cof}
  \,.
\]

For $A \,\in\, G Act(kTop)$ a [[topological G-space|topological $G$-space]] (a space equipped with a [[continuous map]] $G$-[[group action|action]]), its *[[Borel construction]]* is -- regarded as the $A$-[[fiber bundle]] [[associated bundle|associated]] to the [[universal principal bundle|universal $G$-principal bundle]] -- an object in the [[slice category]] over the [[classifying space]] $B G$ (eq:BGBeingCWCOmplex)

\[
  \label{BorelConstructionAsObjectInSliceOverBG}
  \big(
    A \times_G E G
    ,\,
    p_{A \times_G E G }
  \big)
  \;\;
  \in
  \;
  kTop_{/B G}
\]

\begin{lemma}\label{RightBaseChangeAlongProjectionIsQuillen}
  For $B \in kTop$, the right [[base change]] along the [[projection]] $pr_B \,\colon\, B \times B G \to B $
  $$
    kTop_{B \times B G}
     \underoverset
       {
         \underset{ (pr_B)^\ast }{\longrightarrow}
       }
       {
         \overset{ (pr_B)^\ast }{\longleftarrow}
       }
       {\bot_{{}_{\mathrlap{Qu}}}}
    kTop_{B}
  $$
  is a [[Quillen adjunction]] between the respective [[slice model categories]] of the [[classical model structure on topological spaces]].
\end{lemma}
\begin{proof}
  Since $pr_B \,\colon\, B \times B G \to B$ is just [[projection]] out of a [[Cartesian product]], the [[pullback]] $(pr_B)^\ast$ acts by forming the [[product topological space]] with the space $B G$. But since $B G$ is [[cofibrant object|cofibrant]] (eq:BGBeingCWCOmplex) this operation preserves [[cofibrations]] and [[acyclic cofibrations]] of $kTop_{Qu}$ (by [this Prop.](classical+model+structure+on+topological+spaces#HomProductAdjunctionForCofibrantObjectInTopCGIsQuillen)) and hence also those of its [[slice model structure]].
\end{proof}

\begin{definition}\label{TwistedNonAbelianCohomology}
**(twisted non-abelian cohomology, as in [[schreiber:The Character Map in Twisted Non-Abelian Cohomology|FSS20, §2.2]])**
\linebreak
For

* $G$ a [[discrete group]],

* $A$ a [[G-space|$G$-space]],

* $\tau_b \,\colon \,X_b \xrightarrow{\;\;} B G$ a map to the [[classifying space]] (eq:BGBeingCWCOmplex),

the *$\tau_b$-[[twisted cohomology|twisted]] [[non-abelian cohomology]]* of $X_b$ with [[coefficients]] in $A$ is the [[connected components]] of the [[space of sections]] of the [[pullback bundle]] of $A \times_G E G$ along $\tau_b$:
\[
  \label{TwistedGeneralizedCohomologySets}
  \begin{aligned}
  A^\tau(X_b)
  \;\coloneqq\;
  H^\tau(X_b;\, A)
  &
  \;\coloneqq\;
  \pi_0
  \,
  \Gamma_{X_b}
  \big(
    \tau_b^\ast
    (A \times_G E G)
  \big)
  \\
  &
  \;\simeq\;
  \pi_0
  \bigg(
  (p_{B G})_\ast
  Map
  \Big(
    (X_b, \tau_b)
    ,\,
    \big(
      A \times_G E G
      ,\,
      p_{A \times_G E G}
    \big)
  \Big)
  \bigg)
  \,,
  \end{aligned}
\]
where the [[codomain]] on the right is the [[Borel construction]]  (eq:BorelConstructionAsObjectInSliceOverBG) on $A$.
\end{definition}

\begin{remark}\label{SpaceOfSectionsByRightBaseChangeOfFiberwiseMapping}
To see that $(p_{B G})_\ast Map\big( (-,-),\, (-,-) \big)$ in the last line of (eq:TwistedGeneralizedCohomologySets) is really the same as the [[space of sections]], use the [[Yoneda lemma]] (for $kTop^{op}$) on the following sequence of [[natural isomorphisms]]:
$$
  \begin{array}{ll}
    kTop
    \Big(
      U
      ,\,
      (p_{B G})_\ast
      Map
      \big(
        (X_b, \tau_b)
        ,\,
        (A \times_G E G, p_{A \times_G E G})
      \big)
    \Big)
    \\
    \;\simeq\;
    kTop_{/ B G}
    \Big(
      (p_{B G})^\ast
      U
      ,\,
      Map
      \big(
        (X_b, \tau_b)
        ,\,
        (A \times_G E G, p_{A \times_G E G})
      \big)
    \Big)
    \\
    \;\simeq\;
    kTop_{/ B G}
    \Big(
      \big(
        (p_{B G})^\ast U
      \big)
      \times
      (X_b, \tau_b)
      ,\,
      (A \times_G E G, p_{A \times_G E G})
    \Big)
    \\
    \;\simeq\;
    kTop_{/ B G}
    \Big(
      (\tau_b)_!
      (U \times X_b, pr_{X_b})
      ,\,
      (A \times_G E G, p_{A \times_G E G})
    \Big)
    &
    \text{by (eq:APullback)}
    \\
    \;\simeq\;
    kTop_{/ X_b}
    \Big(
      (U \times X_b, pr_{X_b})
      ,\,
      (\tau_b)^\ast
      (A \times_G E G, p_{A \times_G E G})
    \Big)
    \\
    \;\simeq\;
    kTop
    \big(
      U \times X_b
      ,\,
      (\tau_b)^\ast (A \times_G E G)
    \big)
    \underset{
      kTop
      \big(
        U \times X_b
        ,\,
        X_b
      \big)
    }{\times}
    \big\{
      pr_{X_b}
    \big\}
    \\
    \;\simeq\;
    kTop
    \Big(
    U
    ,\,
    Map
    \big(
      X_b
      ,\,
      (\tau_b)^\ast (A \times_G E G)
    \big)
    \underset{
      Map
      \big(
        X_b
        ,\,
        X_b
      \big)
    }{\times}
    \big\{
      \mathrm{id}_{X_b}
    \big\}
    \Big)
    \\
    \;\simeq\;
    kTop
    \Big(
      U
      ,\,
      \Gamma_X
      \big(
        (\tau_b)^\ast
        ( A \times_G E G )
      \big)
    \Big)
  \end{array}
$$
Here each step is the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of various [[adjoint functors]], except the marked one which is immediate from this [[pullback]] square:
\[
  \label{APullback}
  \array{
    U \times X_b
    &\overset{pr_{X_b}}{\longrightarrow}&
    X_b
    \\
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow\mathrlap{{}^{\tau_b}}
    \\
    U \times B G
    &\underset{pr_{B G}}{\longrightarrow}&
    B G
    \mathrlap{\,.}
  }
\]
and remembering that left [[base change]] $(\tau_b)_!$ is given simply by post-[[composition]] with $\tau_b$.
\end{remark}

Now we may state and prove the twisted-cohomology generalization of Prop. \ref{FiberwiseNonAbelianCohomologySetsFormLocalSystem}:

\begin{proposition}\label{FiberwiseTwistedNonAbelianCohomologySetsFormLocalSystem}
**(fiberwise twisted non-abelian cohomology sets form local system)**
\linebreak
  With $(X, p_X)$ a [[fiber bundle]] as [above](#ForNonAbelianCohomology), and
  given a *global twist* on its total space
  $$
    \tau \,\colon\, X \xrightarrow{\;\;} B G
    \,,
  $$
  the $\tau_b$-twisted $A$-cohomology sets
  (Def. \ref{TwistedNonAbelianCohomology})
of $X_b$,
for any $b \in B$, arrange into a [[covering space]] over $B$, generalizing the un-twisted statement from Prop. \ref{FiberwiseNonAbelianCohomologySetsFormLocalSystem}.
\end{proposition}
\[\label{NotationOverview}\]
\begin{tikzcd}[
  sep=small
]
    &
    A^\tau(\mathrm{X}_{b_2})
    \ar[
      ddr,
      "{\sim}"{sloped, pos=.35},
      shorten=-2pt
    ]
    \\[-27pt]
    &&&
    \mathrm{X}
    \ar[
      dddd,
      "{p_{\mathrm{X}}}"
    ]
    \ar[
      dr,
      "{\tau}"{swap, xshift=1pt}
    ]
    \ar[
      rr,
      dashed
    ]
    &&
    \mathrm{A} \times_G E G
    \ar[
      dl,
      shorten=-2pt,
      "{
        p_{\mathrm{A} \times_G E G}
      }"
    ]
    \\[-12pt]
    A^\tau(\mathrm{X}_{b_1})
    \ar[
      uur,
      shorten=-2pt,
      "{
        \sim
      }"{sloped, pos=.6}
    ]
    \ar[
      rr,
      "{\sim}"{swap}
    ]
    &&
    A^\tau(\mathrm{X}_{b_3})
    &&
    B G
    \ar[
      dddd,
      "{
        p_{B G}
      }"
    ]
    \\
    \\
    &
    \{b_2\}
    \ar[
      ddr,
      "{
        [\gamma_{23}]
      }"
    ]
    \\[-27pt]
    &&&
    \mathrm{B}
    \ar[
      dr,
      "{p_{\mathrm{B}}}"{swap}
    ]
    \\[-12pt]
    \{b_1\}
    \ar[
      uur,
      "{
        [\gamma_{12}]
      }"
    ]
    \ar[
      rr,
      "{
        [\gamma_{23} \circ \gamma_{12}]
      }"{swap}
    ]
    &&
    \{b_3\}
    &
    &
    \ast
\end{tikzcd}
\begin{proof}
In the evident generalization of (eq:HomotopyFiberOfFiberwiseMappingSpace), we have the following [[homotopy pullback]] diagram:
$$
  \array{
    \phantom{---}
    \mathclap{
    Map
    \Big(
      \big(
        (X_b, \tau_b)
      \big)
      ,\,
      \big(
        A \times_G E G, p_{A \times_G E G}
      \big)
    \Big)
    }
    \phantom{-------}
    &
    \!\!\!\!\!\!\!\!\!\!\!\!
    \xrightarrow{\phantom{---}}
    \;\;\;\;\;\;\;\;\;\;
    &
    \phantom{-----}
    \mathclap{
    Map
    \Big(
      \big(
        X, (p_X, \tau)
      \big)
      ,\,
      p_B^\ast
      \big(
        A \times_G E G, p_{A \times_G E G}
      \big)
    \Big)
    }
    \phantom{-----}
    \\
    \big\downarrow
    &{}_{{}^{(pb)}}&
    \big\downarrow{}^{ \mathrlap{\in Fib}}
    \\
    \{b\} \times B G
    &\xrightarrow{\phantom{-----}}&
    B \times B G
    \mathrlap{\,.}
  }
$$
Now by the [[Beck-Chevalley relation]]
\[
  \label{BeckChevalleyRelationForProjection}
  \big(
    b_{(-)}
  \big)^\ast
    \circ
  \big(
    B
    \times
    p_{B G}
  \big)_\ast
  \;\;
  \simeq
  \;\;
  \big(
    p_{B G}
  \big)_\ast
  \circ
  \big(
    b_{(-)} \times B G
  \big)^\ast

\]
\begin{tikzcd}[
    column sep={between origins, 36pt}
  ]
    &
    B G
    \ar[
      dr,
      "{
        b_{(-)}
        \times
        B G
      }"
    ]
    \ar[
      dl,
      "{
        \mathrm{p}_{B G}
      }"{swap}
    ]
    \\
    \ast
    \ar[
      dr,
      "{
        b_{(-)}
      }"{swap}
    ]
    &&
    B
    \times
    B G
    \ar[
      dl,
      "{
        B
        \times
        p_{B G}
      }"
    ]
    \\
    &
    B
  \end{tikzcd}
this gives the following pullback diagram:
$$
  \array{
    \phantom{---}
    \mathclap{
    (p_{B G})_\ast
    Map
    \Big(
      \big(
        (X_b, \tau_b)
      \big)
      ,\,
      \big(
        A \times_G E G, p_{A \times_G E G}
      \big)
    \Big)
    }
    \phantom{---------}
    &
    \!\!\!\!\!\!\!\!\!\!\!\!
    \xrightarrow{\phantom{---}}
    \;\;\;\;\;\;\;\;\;\;
    &
    \phantom{--------}
    \mathclap{
    (B \times p_{B G})_\ast
    Map
    \Big(
      \big(
        X, (p_X, \tau)
      \big)
      ,\,
      p_B^\ast
      \big(
        A \times_G E G, p_{A \times_G E G}
      \big)
    \Big)
    }
    \phantom{-----}
    \\
    \big\downarrow
    &{}_{{}^{(pb)}}&
    \big\downarrow{}^{ \mathrlap{\in Fib}}
    \\
    \{b\}
    &\xrightarrow{\phantom{----------}}&
    B
    \mathrlap{\,,}
  }
$$
where the vertical map on the right is still a fibration, by Lem. \ref{RightBaseChangeAlongProjectionIsQuillen}.

From here we conclude as in the proof of Prop. \ref{FiberwiseNonAbelianCohomologySetsFormLocalSystem} that for all $b \in B$ we have:
$$
  \array{
    \phantom{---}
    \mathclap{
    \mathllap{
      H^{\tau_b}(X_b;\, A)
      \,=\,
    }
    \pi_0
    (p_{B G})_\ast
    Map
    \Big(
      \big(
        (X_b, \tau_b)
      \big)
      ,\,
      \big(
        A \times_G E G, p_{A \times_G E G}
      \big)
    \Big)
    }
    \phantom{---------}
    &
    \!\!\!\!\!\!\!\!\!\!\!\!
    \xrightarrow{\phantom{---}}
    \;\;\;\;\;\;\;\;\;\;
    &
    \phantom{---------}
    \mathclap{
    \pi_{0/B}
    (B \times p_{B G})_\ast
    Map
    \Big(
      \big(
        X, (p_X, \tau)
      \big)
      ,\,
      p_B^\ast
      \big(
        A \times_G E G, p_{A \times_G E G}
      \big)
    \Big)
    }
    \phantom{-----}
    \\
    \big\downarrow
    &{}_{{}^{(pb)}}&
    \big\downarrow{}^{ \mathrlap{\in Fib}}
    \\
    \{b\}
    &\xrightarrow{\phantom{----------}}&
    B
    \mathrlap{\,,}
  }
$$
so that the fibration on the right is the covering space which exhibits the claimed local system.
\end{proof}

\begin{example}\label{ExampleOfHypergeometricKZEquations}
**(Hypergeometric Knizhnik-Zamolodchikov equations)**
\linebreak
  Let

1. $p_X \colon X \to B$ be the forgetful map of [[configuration spaces of points]] from Ex. \ref{ForgetfulMapBetweenConfigurationSpaces}

   $$
     Conf_{n + N}(\mathbb{R}^2)
     \xrightarrow{\;\;}
     Conf_{N}(\mathbb{R}^2)
     \,,
   $$

1. $A \,=\, K(\mathbb{C},n)$ the [[Eilenberg-MacLane space]] classifying [[ordinary cohomology|ordinary]] [[complex cohomology]];

1. $G \,\coloneqq\, \mathbb{Z}/k \,\subset\, U(1)$ be a [[cyclic group]], acting by multiplication by [[roots of unity]] on the [[complex numbers]] $\mathbb{C}$ and hence on $K(\mathbb{C},n)$.

Then Prop. \ref{FiberwiseTwistedNonAbelianCohomologySetsFormLocalSystem} constructs the Gauss-Manin connection
claimed in [Etingof, Frenkel & Kirillov 1998, §7.5](#EtingofFrenkelKirillov98).
\end{example}

## Examples

* [[Picard-Fuchs equation]]

* [[Knizhnik-Zamolodchikov connection]]

## Related concepts

* [[Ehresmann theorem]]

* [[flat connection]]

* [[flat vector bundle]]

* [[local system]]

## References

Original articles:

* {#Manin58} [[Yuri Manin]], _Algebraic curves over fields with differentiation_, Izv. Akad. Nauk SSSR Ser. Mat. **22** 6 (1958) 737-756 $[$[mathnet:izv3998](http://mi.mathnet.ru/izv3998), [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=3998&volume=22&year=1958&issue=6&fpage=737&what=fullt&option_lang=eng)$]$ (in Russian)

* [[Nicholas M. Katz]], *On the differential equations satisfied by period matrices*, Publications Mathématiques de l'IHÉS, Tome **35** (1968) 71-106 $[$[numdamPMIHES_1968__35__71_0](http://www.numdam.org/item/PMIHES_1968__35__71_0)$]$

* {#Grothendieck66} [[Alexander Grothendieck]], *On the de Rham cohomology of algebraic varieties*, Publications Mathématiques de l'IHÉS, Tome **29** (1966) 95-103 $[$[numdam:PMIHES_1966__29__95_0](http://www.numdam.org/item/?id=PMIHES_1966__29__95_0)$]$

* [[Nicholas M. Katz]], [[Tadao Oda]], *On the differentiation of De Rham cohomology classes with respect to parameters*, J. Math. Kyoto Univ. **8** 2 (1968) 199-213 $[$[doi:10.1215/kjm/1250524135](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-8/issue-2/On-the-differentiation-of-De-Rham-cohomology-classes-with-respect/10.1215/kjm/1250524135.full)$]$

* [[Egbert Brieskorn]], *Die Monodromie der isolierten Singularitäten von Hyperflächen*, Masucripta Math. **2** (1970) 103-161  $[$[doi:10.1007/BF01155695](https://doi.org/10.1007/BF01155695), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/brieskorn7.pdf)$]$

* [[Pierre Deligne]], *Equations différentielles à points singuliers réguliers*, Lecture Notes  Math. **163**, Springer (1970) $[$[publications.ias:355](https://publications.ias.edu/node/355)$]$

* [[Phillip A. Griffiths]], *Periods of integrals on algebraic manifolds: Summary of main results and discussion of open problems*, Bull. Amer. Math. Soc. **76** (1970) 228-296 $[$[doi:10.1090/S0002-9904-1970-12444-2](https://doi.org/10.1090/S0002-9904-1970-12444-2)$]$

Textbook accounts:

* [[Pierre Deligne]], *Equations différentielles à points singuliers réguliers*, Lecture Notes  Math. **163**, Springer (1970) $[$[publications.ias:355](https://publications.ias.edu/node/355)$]$

* {#Kulikov98} [[Valentine S. Kulikov]], Part I of: *Mixed Hodge Structures and Singularities*, Cambridge University Press (1998) $[$[doi:10.1017/CBO9780511758928](https://doi.org/10.1017/CBO9780511758928)$]$

* [[Alexandru Dimca]], Theorem 2.5.14 in: *Sheaves in Topology*, Universitext, Springer (2004) $[$[doi:10.1007/978-3-642-18868-8](https://doi.org/10.1007/978-3-642-18868-8)$]$

and with focus on the special case of [[surjective submersions]] of [[smooth manifolds]]:

* {#Voisin02} [[Claire Voisin]] (translated by [[Leila Schneps]]), Section 9.2.1 (Def. 9.13) of: _[[Hodge theory and Complex algebraic geometry]] I_, Cambridge Stud. in Adv. Math. **76** (2002) [[doi:10.1017/CBO9780511615344](https://doi.org/10.1017/CBO9780511615344), def: [[Voisin-DefinitionOfGaussManinConn.pdf:file]]]

Lecture notes:

* {#Bloch16} [[Spencer Bloch]], *Gauss-Manin connections* (2016) [[pdf](http://www.math.uchicago.edu/~bloch/Beijing_lectures_hypergeometric_161028.pdf), [[Bloch-GaussManinConnections.pdf:file]]]

Gauss-Manin connections over [[configuration spaces of points]]:

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, I Eigenvalues*, Compositio Math. **136** (2003) 299-316 $[$[arXiv:math/0105063](https://arxiv.org/abs/math/0105063), [doi:10.1023/A:1023262022279](https://doi.org/10.1023/A:1023262022279)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, II Nonresonant weights*, Amer. J. Math. **127** (2005) 569-594 $[$[arXiv:math/0207114](https://arxiv.org/abs/math/0207114), [jstor:40067930](https://www.jstor.org/stable/40067930)$]$

* [[Daniel C. Cohen]], [[Peter Orlik]], *Gauss-Manin connections for arrangements, III Formal connections*, Trans. Amer. Math. Soc. **357** (2005) 3031-3050 $[$[arXiv:math/0307210](https://arxiv.org/abs/math/0307210), [doi:10.1090/S0002-9947-04-03621-9](https://doi.org/10.1090/S0002-9947-04-03621-9)$]$

and review in the context of [hypergeometric solutions](BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) to the [[Knizhnik-Zamolodchikov equation]]:

* {#EtingofFrenkelKirillov98} [[Pavel Etingof]], [[Igor Frenkel]], [[Alexander Kirillov]], Section 7.5 in: *Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations*, Mathematical surveys and monographs **58**, American Mathematical Society (1998) [[ISBN:978-1-4704-1285-2](https://bookstore.ams.org/surv-58), [review pdf](http://www.ams.org/journals/bull/2000-37-02/S0273-0979-00-00853-3/S0273-0979-00-00853-3.pdf), def: [[EtingofEtAl-DefOfGaussManin.pdf:file]]]

The discussion [above](#ForFiberBundles) follows:

* {#SS22} *[Gauss-Manin- and Knizhnik-Zamolodchikov connections via abstract homotopy theory](/schreiber/show/Topological+Quantum+Computation+in+TED-K#GMConAbs)*, Chapter 4 of: [[David J. Myers]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Topological Quantum Gates in Homotopy Type Theory]]*, Comm. Math. Phys. (2024) [[arXiv:2303.02382](https://arxiv.org/abs/2303.02382)]

See also:

* [[eom]], *[Gauss-Manin connection](https://encyclopediaofmath.org/wiki/Gauss-Manin_connection)*

* Wikipedia, *[Gauss-Manin connection](http://en.wikipedia.org/wiki/Gauss%E2%80%93Manin_connection)*

* Allan Yashinski, _Smooth deformations and the Gauss-Manin connection_ ([arxiv/1410.0715](http://arxiv.org/abs/1410.0715))

Discussion in [[cyclic homology]]:

* [[Boris Tsygan]], _On the Gauss-Manin connection in cyclic homology_, Methods Funct. Anal. Topology __13__ (2007), no. 1, 8394.

* [[Victor Ginzburg]], [[Travis Schedler]], [[Boris Tsygan]], _Free products, cyclic homology, and the Gauss–Manin connection_, Advances in Mathematics __231__:3-4 (2012) 2352--2389 [doi](https://doi.org/10.1016/j.aim.2012.06.016)

* [[Ezra Getzler]], _Cartan homotopy formulas and the Gauss/Manin connection in cyclic homology_, in: “Quantum deformations of algebras and their representations,” Israel Math. Conf. Proc. 7 (1993) 65--78 [pdf](https://sites.northwestern.edu/getzler/files/2019/08/barilan.pdf)

In [[noncommutative geometry]]:

* [[Vasily Dolgushev]], [[Dmitry Tamarkin]], [[Boris Tsygan]], _Noncommutative calculus and the Gauss-Manin connection_ ([arxiv/0902.2202](https://arxiv.org/abs/0902.2202))

[[!redirects Gauss-Manin connections]]


Gaussian elimination

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

#Contents#
* table of contents
{:toc}

## Idea

What is called _Gaussian elimination_ is an [[algorithm]] for solving systems of [[linear equations]]. It proceeds by encoding the system as a suitable [[matrix]] and then applying [[elementary row operations]] to bring  it into [[upper triangular form]].

## In constructive mathematics

Gaussian elimination only works for [[discrete fields]] such as the [[rational numbers]] or the [[algebraic numbers]]. The algorithm does not work for other types of fields such as [[Heyting fields]], which means that it doesn't work for the [[real numbers]], however defined.

## Related concepts

* [[Gram-Schmidt process]]

## References

* Wikipedia, _[Gaussian elimination](https://en.wikipedia.org/wiki/Gaussian_elimination)_

[[!redirects Gaussian eliminations]]

[[!redirects Gauss elimination]]
[[!redirects Gauss eliminations]]

[[!redirects row reduction]]
[[!redirects row reductions]]

[[!redirects elementary row operation]]
[[!redirects elementary row operations]]

[[!redirects elementary column operation]]
[[!redirects elementary column operations]]

[[!redirects Gauss elimination procedure]]
[[!redirects Gauss elimination procedures]]


Gaussian integral

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Integration theory
+--{: .hide}
[[!include integration theory - contents]]
=--
=--
=--

## Related concepts

* [[Gaussian measure]]

* [[free field theory]]

* [[vacuum energy]]

## References

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

[[!redirects Gaussian integrals]]


Gaussian measure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

## Related concepts

* [[Gaussian integral]]

* [[free field theory]]

## References

* Wikipedia, _[Gaussian measure](https://en.wikipedia.org/wiki/Gaussian_measure)_

[[!redirects Gaussian measures]]