G2

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#### Exceptional structures
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[[!include exceptional structures -- contents]]
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#### Group Theory
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[[!include group theory - contents]]
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#### Lie theory
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[[!include infinity-Lie theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[Lie group]] $G_2$ is one (or rather: three) of the [[exceptional Lie groups]]. One way to characterize it is as the [[automorphism group]] of the [[octonions]] as a [[normed algebra]]:

$$
  G_2 = Aut(\mathbb{O})
  \,.
$$

Another way to characterize it is as the [[stabilizer subgroup]] inside the [[general linear group]] $GL(7)$ of the canonical [[differential n-form|differential 3-form]]  $\langle ,(-)\times (-) \rangle$ on the [[Cartesian space]] $\mathbb{R}^7$

$$
  G_2 \simeq Stab_{GL(7)}(\langle -, -\times -\rangle)
  \,.
$$

As such, the group $G_2$ is a higher analog of the [[symplectic group]] (which is the group that preserves a canonical 2-form on any $\mathbb{R}^{2n}$), obtained by passing from [[symplectic geometry]] to [[2-plectic geometry]].

## Definition
 {#Definition}

+-- {: .num_defn #As2PlectomorphismsOnR7}
###### Definition

On the [[Cartesian space]] $\mathbb{R}^7$ consider the _[[associative 3-form]]_, the constant [[differential n-form|differential 3-form]]  $\omega \in \Omega^3(\mathbb{R}^7)$ given on tangent vectors $u,v,w \in \mathbb{R}^7$ by

$$
  \omega(u,v,w) \coloneqq \langle u , v \times w\rangle
  \,,
$$

where

* $\langle -,-\rangle$ is the canonical [[bilinear form]]

* $(-)\times(-)$ is the [[cross product]] of vectors.

Then the group $G_2 \hookrightarrow GL(7)$ is the [[subgroup]] of the [[general linear group]] acting on $\mathbb{R}^7$ which preserves the canonical [[orientation]] and preserves this 3-form $\omega$. Equivalently, it is the subgroup preserving the orientation and the [[Hodge star operator|Hodge dual]] differential 4-form $\star \omega$.

=--

See for instance the introduction of ([Joyce](#Joyce)).

## Properties

### Orientation

The inclusion $G_2 \hookrightarrow GL(7)$ of def. \ref{As2PlectomorphismsOnR7} factors through the [[special orthogonal group]]

$$
  G_2
    \hookrightarrow
  SO(7)
    \hookrightarrow
  GL(7)
  \,.
$$

### Dimension
 {#Dimension}

The [[dimension]] of (the [[manifold]] underlying) $G_2$ is

$$
  dim(G_2) = 14
  \,.
$$

One way to see this is via [octonionic basic triples](octonion#ElementaryTriples) $(e_1, e_2, e_3) \in \mathbb{O}^3$ and the fact ([this proposition](octonion#BasicTriplesFormAutomorphism)) that these form a [[torsor]] over $G_2$, hence that the space of them has the same dimension as $G_2$:

* the space of choices for $e_1$ is the 6-[[sphere]] of imaginary unit octonions;

* given that, the space of choices for $e_2$ is a 5-sphere of imaginary unit octonions orthogonal to $e_1$;

* given that, then the space of choices for $e_3$ is the [[3-sphere]] of imaginary unit octonions orthogonal to both $e_1$ and $e_2$.

Hence

$$
  dim(G_2) = dim(S^6) + dim(S^5) + dim(S^3) = 14
  \,.
$$

(e.g. [Baez, 4.1](#Baez))

### Cohomology

The _[[Dwyer-Wilkerson space]]_ $G_3$ ([Dwyer-Wilkerson 93](Dwyer-Wilkerson+H-space#DwyerWilkerson93)) is a [[p-adic completion|2-complete]] [[H-space]], in fact a finite [[loop space]]/[[infinity-group]], such that the mod 2 [[cohomology ring]] of its [[classifying space]]/[[delooping]] is the mod 2 [[Dickson invariants]] of rank 4. As such, it is the fourth and last space in a series of [[infinity-groups]] that starts with 3 [[compact Lie groups]]:

| $n=$ | 1 | 2 | 3 | 4 |
|--|--|--|--|--|
| $DI(n)=$ | [[Z/2]] | [[SO(3)]] | [[G2]] | [[G3]] |
|          | [= Aut(C)](complex+number#AutomorphismsOfComplexNumbersIsZ2)        | [= Aut(H)](quaternion#AutomorphismsOfQUatrnionsAlgebraIsSO3)       |  [= Aut(O)](octonion#AutomorphismsOfOctonionAlgebraIsG2)  |   |

### Subgroups
 {#Subgroups}

We discuss various [[subgroups]] of $G_2$.

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

Write

* $G_2 = Aut(\mathbb{O})$, the [[automorphism group]] of the octonions as a normed alegbra,

* $Stab_{G_2}(\mathbb{H})$, the [[stabilizer subgroup]] of [[generalized the|the]] [[quaternions]] inside the octonions, i.e. of elements $\sigma\in G_2$ such that $\sigma_{|\mathbb{H}}\colon \mathbb{H}\to \mathbb{H} \hookrightarrow\mathbb{O}$;

* $Fix_{G_2}(\mathbb{H})$ for the further subgroup of elements that [[fixed point|fix]] each quaternions (the "elementwise stabilizer group"), i.e. those $\sigma$ with $\sigma_{\vert \mathbb{H}} = id_{\mathbb{H}}$.

=--

+-- {: .num_prop #ElementwiseStabilizerOfHIsSU2}
###### Proposition

The elementwise stabilizer group of the [[quaternions]] is [[SU(2)]]:

$$
  Fix_{G_2}(\mathbb{H}) \simeq SU(2)
  \,.
$$

=--

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

Consider [octonionic basic triples](octonion#ElementaryTriples) $(e_1, e_2, e_3) \in \mathbb{O}^3$ and the fact ([this proposition](octonion#BasicTriplesFormAutomorphism)) that these form a [[torsor]] over $G_2$.

The choice of $(e_1,e_2)$ is equivalently a choice of inclusion $\mathbb{H} \hookrightarrow \mathbb{O}$. Then the remaining space of choices for $e_3$ is the [[3-sphere]] (the space of unit imaginary octonions orthogonal to both $e_1$ and $e_2$). This carries a unit group structure, and by the torsor property this is the required subgroup of $SU(2)$.

=--

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

The subgroups in def. \ref{StabilizerOfQuaternions} sit in a [[short exact sequence]] of the form

$$
  \array{
    1 &=& 1
    \\
    \downarrow && \downarrow
    \\
    Fix_{G_2}(\mathbb{H}) & \simeq & SU(2)
    \\
    \downarrow && \downarrow
    \\
    Stab_{G_2}(\mathbb{H}) & \simeq & SO(4)
    \\
    \downarrow && \downarrow
    \\
    Aut(\mathbb{H}) &\simeq& SO(3)
    \\
    \downarrow && \downarrow
    \\
    1 &=& 1
  }
$$

exhibiting [[special orthogonal group|SO(4)]] as a [[group extension]] of the [[special orthogonal group]] $SO(3)$ by the [[special unitary group]] $SU(2)$.

=--

(e.g. [Ferolito, section 4](#Ferolito))

Furthermore there is a [[subgroup]] $SU(3) \hookrightarrow G_2$ whose [[intersection]] with $SO(4)$ is $U(2)$. The [[simple Lie group|simple]] part $SU(2)$ of this intersection is a [[normal subgroup]] of $SO(4)$.

(see e.g. [Miyaoka 93](#Miyaoka93))

The [[coset space]] [[G2/SU(3) is the 6-sphere]]. See there for more.

<img src="https://ncatlab.org/nlab/files/3dSubgroupsOfG2.jpg" width="700">

(from [Kramer 02](#Kramer02))

The [[Weyl group]] of $G_2$ is the [[dihedral group]] of [[order of a group|order]] 12. (see e.g. [Ishiguro, p. 3](#Ishiguro)).

$\,$

### Supgroups

+-- {: .num_prop #QuotientOfSpin7ByG2IsS7}
###### Proposition
**([[coset space]] of [[Spin(7)]] by [[G2]] is [[7-sphere]])**

Consider the canonical [[action]] of [[Spin(7)]] on the [[unit sphere]] in $\mathbb{R}^8$ (the [[7-sphere]]),

1. This action is is [[transitive action|transitive]];

1. the [[stabilizer group]] of any point on $S^7$ is [[G2]];

1. all [[G2]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other.

Hence the [[coset space]] of [[Spin(7)]] by [[G2]] is the [[7-sphere]]

$$
  Spin(7)/G_2 \;\simeq\; S^7
  \,.
$$

=--

(e.g [Varadarajan 01, Theorem 3](#Varadarajan01))

### Coset quotients

[[!include coset space structure on n-spheres -- table]]

### $G$-Structure and exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

### Relation to higher prequantum geometry

The 3-form $\omega$ from def. \ref{As2PlectomorphismsOnR7} we may regard as equipping $\mathbb{R}^7$ with [[n-plectic geometry|2-plectic structure]]. From this point of view $G_2$ is the linear subgroup of the [[2-plectomorphism group]], hence (up to the translations) the image of the [[Heisenberg group]] of $(\mathbb{R}^7, \omega)$ in the symplectomorphism group.

Or, dually, we may regard the 4-form $\star \omega$ of def. \ref{As2PlectomorphismsOnR7} as being a [[n-plectic geometry|3-plectic structure]] and $G_2$ correspondingly as the linear part in the [[3-plectomorphism group]] of $\mathbb{R}^7$.

### As zero-divisors of the sedenions

It is shown in Corollary 2.14 of [Moreno (1997)](#Moreno97) that the group of [[zero-divisor|zero-divisors]] of the [[sedenion|sedenions]] is isomorphic to $G_2$.

## Related concepts

* [[G2 manifold]], [[generalized G2-manifold]]

* [[M-theory on G2-manifolds]], [[G2-MSSM]]

* **G2**, [[F4]],

  [[E6]], [[E7]],  [[E8]], [[E9]], [[E10]], [[E11]], $\cdots$

[[!include special holonomy table]]

## References

### General

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 2 of _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000

Surveys are in

* Spiro Karigiannis, _What is... a $G_2$-manifold_ ([pdf](http://www.ams.org/notices/201104/rtx110400580p.pdf))

* [[Simon Salamon]], _A tour of exceptional geometry_, ([pdf](http://calvino.polito.it/~salamon/G/COR/tour.pdf))

* Wikipedia, _[G2](http://en.wikipedia.org/wiki/G2_%28mathematics%29)_ .

The definitions are reviewed for instance in

* {#Joyce} [[Dominic Joyce]], _Compact Riemannian 7-manifolds with holonomy $G_2$_, Journal of Differential Geometry vol 43, no 2 ([pdf](http://www.intlpress.com/JDG/archive/1996/43-2-291.pdf))

* {#Ferolito} Ferolito _The octonions and $G_2$_ ([pdf](https://www2.bc.edu/~reederma/Ferolito.pdf))

* {#Baez} [[John Baez]], section 4.1 _[G2](http://math.ucr.edu/home/baez/octonions/node14.html)_, of _The Octonions_ ([arXiv:math/0105155](http://arxiv.org/abs/math/0105155))

* Ruben Arenas, _Constructing a Matrix Representation of the Lie Group $G_2$_, 2005 ([pdf](https://www.math.hmc.edu/seniorthesis/archives/2005/rarenas/rarenas-2005-thesis.pdf))

Discussion in terms of the [[Heisenberg group]] in [[2-plectic geometry]] is in

* {#Ibort} Alberto Ibort, _Multisymplectic geometry: generic and exceptional_, _[Proceedings of the IX Fall workshop on geometry and physics](http://rsme.es/public/publi3.htm)_ ([[IbortMultisymplectic.pdf:file]])

A description of the root space decomposition of the [[Lie algebra]] $\mathfrak{g}_2$ is in

* {#Basak17} Tathagata Basak, _Root space decomposition of $\mathfrak{g}_2$ from octonions_, arXiv:[1708.02367](https://arxiv.org/abs/1708.02367)

As the group of [[zero-divisor|zero divisors]] of the [[sedenion|sedenions]]

* {#Moreno97} Guillermo Moreno. *The zero divisors of the Cayley-Dickson algebras over the real numbers*. (1997) ([doi](https://arxiv.org/abs/q-alg/9710013))

Cohomological properties are discussed in

* Younggi Choi, _Homology of the gauge group of exceptional Lie group $G_2$_, J. Korean Math. Soc. 45 (2008), No. 3, pp. 699–709

Discussion of [[subgroups]] includes

* {#Miyaoka93} [[Reiko Miyaoka]], _The linear isotropy group of $G_2/SO(4)$, the Hopf fibering and isoparametric hypersurfaces_, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. ([Euclid](http://projecteuclid.org/euclid.ojm/1200784357))

* {#Ishiguro} Kenshi Ishiguro, _Classifying spaces and a subgroup of the exceptional Lie group $G_2$_ [pdf](http://hopf.math.purdue.edu/Ishiguro/G2.pdf)

* {#Kramer02} Linus Kramer, 4.27 of _Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces_, AMS 2002

Discussion of $G_2$ as a subgroup of [[Spin(7)]]:

* {#Varadarajan01} [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 ([pdf](https://core.ac.uk/download/pdf/81114499.pdf))

### Applications in physics

Discussion of [[Yang-Mills theory]] with $G_2$ as [[gauge group]] is in

* Ernst-Michael Ilgenfritz, Axel Maas, _Topological aspects of $G_2$ Yang-Mills theory_ ([arXiv:1210.5963](http://arxiv.org/abs/1210.5963))


G2 manifold

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###Context###
#### Riemannian geometry
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[[!include Riemannian geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _$G_2$-structure_ on a [[manifold]] $X$ of [[dimension]] 7 is a choice of [[G-structure]] on $X$, for $G$ the [[exceptional Lie group]] [[G2]]. Hence it is a reduction of the [[structure group]] of the [[frame bundle]] of $X$ along the canonical (the defining) inclusion $G_2 \hookrightarrow GL(\mathbb{R}^7)$ into the [[general linear group]].

Given that $G_2$ is the [[subgroup]] of the [[general linear group]] on the [[Cartesian space]] $\mathbb{R}^7$ which preserves the [[associative 3-form]] on $\mathbb{R}^7$, a $G_2$ structure is a higher analog of an  [[almost symplectic structure]] under lifting from [[symplectic geometry]] to [[2-plectic geometry]] ([Ibort](#Ibort)).

A _$G_2$-manifold_ is a manifold equipped with $G_2$-structure that is [[integrability of G-structures|integrable to first order]], i.e. [[torsion of a G-structure|torsion-free]] (prop. \ref{CovariantlyConstantDefinite3FormMeansTorsionVanishes} below). This is equivalently a [[Riemannian manifold]] of [[dimension]] 7 with [[special holonomy]] group being the [[exceptional Lie group]] [[G2]].

$G_2$-manifolds may be understood as 7-dimensional [[analogs]] of real 6-dimensional [[Calabi-Yau manifolds]]. Accordingly the relation between [[Calabi-Yau manifolds and supersymmetry]] lifts from [[string theory]] to [[M-theory on G2-manifolds]].

## Definition

### $G_2$-structure
 {#G2Structure}

+-- {: .num_defn #G2Structure}
###### Definition

For $X$ a [[smooth manifold]] of [[dimension]] $7$ a **$G_2$-structure** on $X$ is a [[G-structure]] for $G = $ [[G2]] $\hookrightarrow GL(7)$.

=--

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

A $G_2$-structure in particular implies an [[orthogonal structure]], hence a [[Riemannian metric]].

=--

Given the definition of [[G2]] as the [[stabilizer group]] of the [[associative 3-form]] on $\mathbb{R}^7$, there is accordingly an equivalent formulation of def. \ref{G2Structure} in terms of [[differential forms]]:

+-- {: .num_defn #Definite3Forms}
###### Definition

Write $\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast $ for the [[orbit]] of the [[associative 3-form]] $\phi$ under the canonical $GL(7)$-[[action]]. Similarly for $X$ a [[smooth manifold]] of [[dimension]] 7, write

$$
  \Omega^3_+(X) \hookrightarrow \Omega^3(X)
$$

for the subset of the set of [[differential 3-forms]] on those that, as [[sections]] to the exterior power of the [[cotangent bundle]], are pointwise in $\Lambda^3_+(\mathbb{R}^7)^\ast$.

These are also called the _positive forms_ ([Joyce 00, p. 243](Joyce00)) or the _[[definite differential forms]]_ ([Bryant 05, section 3.1.1](#Bryant05)) on $X$.

=--

(e.g. [Bryant 05, definition 2](#Bryant05))

+-- {: .num_prop #G2StructureViaDefinite3Form}
###### Proposition

A $G_2$-structure on $X$, def. \ref{G2Structure}, is equivalently
a choice of [[definite 3-form]] $\sigma$ on $X$, def. \ref{Definite3Forms}.

=--

(e.g. [Joyce 00, p. 243](Joyce00), [Bryant 05, section 3.1.1](#Bryant05))

Often it is useful to exhibit prop. \ref{G2StructureViaDefinite3Form} in the following way.

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

For $X$ a [[smooth manifold]] of [[dimension]] 7, write $Fr(X) \to X$ for its [[frame bundle]]. By the discussion at _[vielbein -- in terms of basic forms on the frame bundle](vielbein#InTermsOfBasicFormsOnFrameBundle)_ there is a universal $\mathbb{R}^7$-valued differential form on the total space of the frame bundle

$$
  E_u \in \Omega^1(Fr(X), \mathbb{R}^7)
$$

(whose components we write $(E_u^a)_{a = 1}^7$) such that given an [[orthogonal structure]] $i \colon Fr_O(X)\hookrightarrow Fr(X)$ and a local section $\sigma_i \colon (U_i \subset X) \to Fr_O(X)$ of orthogonal frames, then the [[pullback of differential forms]]

$$
  E_i \coloneqq \sigma_i^\ast i^\ast E_u
$$

is the corresponding local [[vielbein]] field. Hence one obtains a universal 3-form $\phi_u \in \Omega^3(Fr(x))$ on the frame bundle by setting

$$
  \phi_u \coloneqq \phi_{a b c} E_u^a \wedge E_u^b \wedge E_u^c
$$

with $(\phi_{a b c})$ the canonical components of the [[associative 3-form]] and with summation over repeated indices understood.

By construction this is such that on a [[chart]] $(U_i \subset X)$ any [[definite 3-form]], def. \ref{Definite3Forms}, restricts to the [[pullback of differential forms|pullback]] of $\phi_u$ via a [[section]] $\sigma_i \colon U_i \to Fr(X)$ and hence is of the form

$$
  \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c
  \,.
$$

Conversely, given a 3-form $\sigma \in \Omega^3(X)$ such that on an [[atlas]] $(U_i \to X)$ over which the frame bundle trivializes it is of this form

$$
  \sigma|_{U_i} = \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c
$$

then the $GL(7)$-valued transition functions $g_{i j}$ of the given local trivialization must factor through $G_2\hookrightarrow SO(7) \hookrightarrow GL(7)$ and hence exhibit a $G_2$-structure: because we have  $\sigma|_{U_i} = \sigma|_{U_j} \;\;\; on \;U_i \cap U_j$ and hence

\[
  \label{Definite3FormsOnTwoPatches}
   \phi_{a b c} E_i^a \wedge E_i^b \wedge E_i^c
   =
   \phi_{a b c} E_j^a \wedge E_j^b \wedge E_j^c
   \;\;\;
    on \; U_i \cap U_j
  \,.
\]

But by the nature of the [universal vielbein](vielbein#InTermsOfBasicFormsOnFrameBundle), its local pullbacks are related by

$$
  E_j = g_{i j} E_i
$$

i.e.

$$
  E_j^a = (g_{i j})^a{}_b E_i^b
$$

and hence (eq:Definite3FormsOnTwoPatches) says that

$$
   \phi_{a b c}
   =
   \phi_{a' b' c'} (g_{i j})^{a'}{}_a (g_{i j})^{b'}{}_b (g_{i j})^{c'}{}_c
   \;\;\;
    on \; U_i \cap U_j
$$

which is precisely the defining condition for $g_{i j}$ to take values in $G_2$.

Viewed this way, the [[definite 3-forms]] characterizing $G_2$-structures are an example of a more general kind of differential forms obtained from a constant form on some linear model space $V$ by locally contracting with a [[vielbein]] field. For instance on a [[super-spacetime]] solving the [[equations of motion]] of  [[11-dimensional supergravity]] there is a super-4-form part of the [[field strength]] of the [[supergravity C-field]] which is constrained to be locally of the form

$$
  \Gamma_{a b \alpha \beta} E_i^a \wedge E_i^b \wedge E_i^\alpha \wedge E_i^\beta
$$

for $(E^A)= (E^a, E^\alpha) = (E^a, \Psi^\alpha)$ the [[super-vielbein]].  See at _[Green-Schwarz action functional -- Membrane in 11d SuGra Background](Green-Schwarz+action+functional#MembraneIn11dSuGraBackground)_. Indeed, by the discussion there this 4-form is required to be [[covariant derivative|covariantly constant]], which is precisely the analog of $G_2$-manifold structure as in def. \ref{G2manifold}.

=--

References that write [[definite 3-forms]] in this form locally as $\phi_{a b c}E^a \wedge E^b \wedge E^c$ include ([BGGG 01 (2.9)](#BGGG01), ...).

The following is important for the analysis:

+-- {: .num_remark #Definition3FormsGiveOpenSubset}
###### Remark

The subset $\Lambda^3_+(\mathbb{R}^7)^\ast \hookrightarrow \Lambda^3(\mathbb{R}^7)^\ast $ in def. \ref{Definite3Forms} is an [[open subset]], hence $\phi$ is a _[[stable form]]_ (e.g. [Hitchin, def. 1.1](#Hitchin)).

=--

(e.g. [Joyce 00, p. 243](#Joyce00), [Bryant 05, 2.8](#Bryant05))

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

By definition of $G_2$ as the [[stabilizer group]] of the [[associative 3-form]], the [[orbit]] it generates under the $GL_+(7)$-action is the [[coset]] $GL_+(7)/G_2$. The [[dimension]] of this as a [[smooth manifold]] is 49-14 = 35. This is however already the full dimension $\left(7 \atop 3\right) = 35$ of the space of 3-forms in 7d that the orbit sits in. Therefore (since $G_+(7)/G_2$ does not have a [[manifold with boundary|boundary]]) the orbit must be an open subset.

=--

### Closed $G_2$-structure
 {#ClosedG2Structure}

+-- {: .num_defn #ClosedG2Structure}
###### Definition

A $G_2$-structure, def. \ref{G2Structure}, is called _closed_ if the definite 3-form $\sigma$ corresponding to it via prop. \ref{G2StructureViaDefinite3Form} is a closed differential form, $\mathbf{d}\sigma = 0$.

=--

(e.g. [Bryant 05, (4.31)](#Bryant05))

+-- {: .num_prop #ClosedG2StructureByAtlas}
###### Proposition

For a closed $G_2$-structure, def. \ref{ClosedG2Structure}, on a manifold $X$ there exists an [[atlas]] by [[open subsets]]

$$\mathbb{R}^7 \underoverset{et}{f}{\leftarrow} U \underset{et}{\rightarrow} X$$

such that the globally defined 3-form $\sigma \in \Omega^3_+(X)$ is locally gauge equivalent to the canonical [[associative 3-form]] $\phi$

$$
  \sigma|_U = f^\ast \phi + \mathbf{d}\beta
$$

via a 2-form $\beta$ on $U$.

=--

(e.g. [Bryant 05, p. 21](#Bryant05))

This follows from the fact, remark \ref{Definition3FormsGiveOpenSubset}, that the definite 3-forms are an [[open subset]] inside all 3-forms: given a chart centered around any point then there is $\beta$ with $\mathbf{d}\beta$ vanishing at that point such that $\sigma|_U \simeq f^\ast \phi + \mathbf{d}\beta$ at that point. But since the $GL(7)$-action on $\phi$ is open, there is an open neighbourhood around that point where this is still the case.

+-- {: .num_remark #AtlasForClosedG2tructureInTermsOfHigherGeometry}
###### Remark

When regarding [[smooth manifolds]] in the wider context of [[higher differential geometry]], then the situation of prop. \ref{ClosedG2StructureByAtlas} corresponds to a diagram of [[formal smooth infinity-groupoids]] of the following form:

$$
  \array{
    && U
    \\
    & {}^{\mathllap{f}}\swarrow && \searrow
    \\
    \mathbb{R}^7 && \swArrow_{\mathrlap{\beta}} && X
    \\
    & {}_{\mathllap{\phi}}\searrow && \swarrow_{\mathrlap{\sigma}}
    \\
    && \flat_{dR}\mathbf{B}^3\mathbb{R}
  }
  \,,
$$

where $\flat_{dR}\mathbf{B}^3\mathbb{R}$ is the [[moduli infinity-stack|higher moduli stack]] of flat 3-forms with 2-form gauge transformations between them (and 1-form gauge transformation between these). The diagram expresses the 3-form $\sigma$ as a map to this moduli stack, which when restricted to the cover $U$ becomes gauge equivalent to the pullback of the [[associative 3-form]] $\phi$, similarly regarded as a map, to the cover, where the gauge equivalence is exhibited by a [[homotopy]]  (of maps of formal smooth $\infty$-groupoids) which is the 2-form $\beta$ on $U$.

=--

### $G_2$-holonomy / $G_2$-manifold
 {#G2Holonomy}

+-- {: .num_defn #G2manifold}
###### Definition/Proposition

A manifold $X$ equipped with a $G_2$-structure, def. \ref{G2Structure}, is called a **$G_2$-manifold** if the following equivalent conditions hold

1. we have

   1. $\mathbf{d} \omega = 0$ ([closed](#ClosedG2Structure))

   1. $\mathbf{d} \star_g \omega = 0$ (co-closed);

1. $\nabla^g \omega = 0$;

1. $(X,g)$ has [[special holonomy]] $Hol(g) \subset G_2$;

1. $Ric(g) = 0$ (vanishing [[Ricci curvature]]);

1. $R(g) = 0$ (vanishing [[scalar curvature]]);

1. $\tau = 0$ (vanishing [[torsion of a G-structure|torsion of the G2-structure]]).

Here

* $d$ is the [[de Rham differential]];

* $\omega$ is the 3-form $\omega$ corresponding to the given $G_2$-structure via prop. \ref{G2StructureViaDefinite3Form};

* $g$ is the induced [[Riemannian metric]] of remark \ref{CanonicalRiemannianMetric});

* $\star_g$ is the [[Hodge star operator]] of this metric;

* $\nabla^g$ is the [[covariant derivative]] of this metric;

=--

For the equivalence of the first items see for instance ([Joyce, p. 4](#Joyce), [Joyce 00, prop. 10.1.3](#Joyce00)). For the equivalence to the vanishing curvature invariant see also ([Bryant 05, corollary 1](#Bryant05)), and for the equivalence to the vanishing [[torsion of a G-structure]] see ([Bryant 05, prop. 2](#Bryant05)).

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

The higher [[torsion of a G-structure|torsion invariants]] of $G_2$-structures do not necessarily vanish (contrary to the case for instance of [[symplectic structure]] and [[complex structure]], see at [integrability of G-structures -- Examples](integrability+of+G-structures#Examples)). Therefore, even in view of prop. \ref{CovariantlyConstantDefinite3FormMeansTorsionVanishes}, a $G_2$-manifold, def. \ref{G2manifold}, does not, in general admit an [[atlas]] be adapted [[coordinate charts]] equal to $(\mathbb{R}^7, \phi)$.

The space of second order torsion invariants of $G_2$-structures is for instance in ([Bryant 05 (4.7)](#Bryant05)).

=--

### Variants and weakenings

There are several variants of the definition of $G_2$-manifolds,
def.\ref{G2manifold}, given by imposing other constraints on the [[torsion of a Cartan connection|torsion]].

#### With skew-symmetric torsion

Discussion for totally skew symmetric [[torsion of a Cartan connection]] includes ([Friedrich-Ivanov 01, theorem 4.7, theorem 4.8](#FriedrichIvanov01))

#### Weak $G_2$-holonomy
 {#WeakG2Holonomy}

+-- {: .num_defn #WeakG2Holonomy}
###### Definition

A 7-dimensional manifold is said to be of _weak $G_2$-holonomy_ if it carries a 3-form $\omega$ with the relation of def. \ref{G2manifold} generalized to

$$
  \mathbf{d} \omega = \lambda \star \omega
$$

and hence

$$
  \mathbf{d} \star \omega = 0
$$

for $\lambda \in \mathbb{R}$. For $\lambda = 0$ this reduces to strict $G_2$-holonomy, by \ref{G2manifold}.

=--

(See for instance ([Bilal-Derendinger-Sfetsos 02](#BilalDerendingerSfetsos), [Bilal-Metzger 03](#BilalMetzger03)).)

#### With ADE orbifold structure
 {#WithADEOrbifoldStructure}

When used as [[KK-compactification]]-fibers for [[M-theory on G2-manifolds]], then for realistic [[string phenomenology|phenomenology]] one needs to consider [[ADE orbifolds]] with "$G_2$-manifold" structure, i.e. [[G2-orbifolds]], also called _Joyce orbifolds_. Moreover, for [[F-theory]] purposes this $G_2$-orbifold is to be a fibration by a [[K3 surface]] $X_{K3}$.

For instance the [[Cartesian product]] $X_{K3} \times T^3$ admits a $G_2$-manifold structure. There is a canonical [[special orthogonal group|SO(3)]]-[[action]] on the tangent spaces of $X_{K3} \times T^3$, given on $X_{K3}$ by rotation of the [[hyper-Kähler manifold]]-structure of $X_{K_3}$ and on $T^3$ by the standard rotation. For $K_{ADE}$ a [[finite group|finite]] [[subgroup]] of $SO(3)$, hence a finite group in the  [[ADE classification]], then $(X_{K3}\times T^3)/K_{ADE}$ is a [[G2-orbifold]].
([Acharya 98, p.3](#Acharya98)). (For $K_{ADE}$ _not_ a [[cyclic group]] then this has precisely one [[parallel spinor]].)

In a local [[coordinate chart]] of $X_{K3}$ by $\mathbb{C}^2$ the orbifold $X_{K3}/K_{ADE}$ locally looks like $\mathbb{C}^2/{G_{ADE}}$, where now $G_{ADE}$ is a [[finite group|finite]] [[subgroup]] of [[special unitary group|SU(2)]]. Such local [[G2-orbifolds]] are discussed in some detail in ([Atiyah-Witten 01](#AtiyahWitten01)). Families of examples are constructed in [Reidegeld 15](#Reidegeld15).

Codimension-4 ADE singularities in $G_2$-manifolds are discussed in ([Acharya-Gukov 04, section 5.1](AcharyaGukov04), [Barrett 06](#Barrett06)).

## Properties

### Existence

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

A 7-manifold admits a $G_2$-structure, def. \ref{G2Structure}, precisely if it admits an [[orientation]] and a [[spin structure]].

=--

That orientability and spinnability is necessary follows directly from the fact that $G_2 \hookrightarrow GL(7)$ is connected and simply connected. That these conditions are already sufficient is due to ([Gray 69](#Gray69)), see also ([Bryant 05, remark 3](#Bryant05)).

### Metric structure

The canonical [[Riemannian metric]] $G_2$ manifold is [[Ricci tensor|Ricci flat]]. More generally a manifold of weak $G_2$-holonomy, def. \ref{WeakG2Holonomy}, with weakness parameter $\lambda$ is an [[Einstein manifold]] with [[cosmological constant]] $\lambda$.

### As part of the Berger classification

[[!include special holonomy table]]

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

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

### As exceptional geometry

[[!include Spin(8)-subgroups and reductions -- table]]

## Examples
 {#Examples}

### Metric cones

The [[metric cone over complex projective 3-space]] carries the [[mathematical structure|structure]] of  a [[G2-manifold]] whose [[Riemannian metric]] is [[invariant]] under the canonical [[Sp(2)]] [[action]] by left-[[matrix multiplication]] on homomogeneous coordinates in $\mathbb{H}^2 \simeq_{\mathbb{R}} \mathbb{C}^4 \to \mathbb{C}P^3$ ([Byant-Salamon 89](metric+cone+over+complex+projective+3-space#BryantSalamon89), see also [Acharya-Bryant-Salamon 20](metric+cone+over+complex+projective+3-space#AcharyaBryantSalamon20)).

### Resolution of Joyce orbifolds

[[compact topological space|compact]]$\,$[[G2-manifolds]]
by [[resolution of singularities]] in compact [[flat orbifolds]]

([Joyce 96](#Joyce96), [Joyce 00](#Joyce00))

(...)

### Twisted connected sum construction
 {#TwistedConnectedSumConstruction}

[[compact topological space|compact]] [[G2-manifolds]]
by twisted [[connected sum]]-constructions

([Kovalev 00](#Kovalev00))

\begin{center}
\begin{imagefromfile}
  "file_name": "TwistedConnectedSumG2Manifold.jpg",
  "width": 680
\end{imagefromfile}
\end{center}

> graphics grabbed from [Klemm 16](#Klemm17)

## Applications

### In supergravity

In [[string phenomenology]] [[model (in particle phyiscs)|models]] obtained from [[Kaluza-Klein mechanism|compactification]] of [[11-dimensional supergravity]]/[[M-theory on G2-manifolds]] (see for instance [Duff](#Duff)) can have attractive [[phenomenology|phenomenological]] properties, see for instance the _[[G2-MSSM]]_.

## Related concepts

* [[associative submanifold]]

* [[Hitchin functional]]

* [[M-theory on G2-manifolds]], [[G2-MSSM]]

* [[topological M-theory]], [[topological membrane]]

* [[generalized G2-manifold]]

* [[Calabi-Yau manifold]]

* [[exceptional generalized geometry]]

## References

### General

The concept goes back to

* E. Bonan, (1966), _Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)_, C. R. Acad. Sci. Paris 262: 127–129.

Non-compact $G_2$-manifolds were first  constructed in

* [[Robert Bryant]], [[Simon Salamon]],  _On the construction of some complete metrics with exceptional holonomy_, Duke Mathematical Journal 58: 829–850 (1989) ([euclid:dmj/1077307681](https://projecteuclid.org/euclid.dmj/1077307681))

[[compact topological space|Compact]] $\,G_2$-manifolds were first found in

* {#Joyce96}[[Dominic Joyce]], _Compact Riemannian 7-manifolds with holonomy $G_2$_, Journal of Differential Geometry vol 43, no 2, 1996 ([Euclid:jdg/1214458109](https://projecteuclid.org/euclid.jdg/1214458109))

* {#Joyce00} [[Dominic Joyce]], _Compact Manifolds with Special Holonomy_, Oxford Mathematical Monographs, Oxford University Press (2000)

Review in

* [[Dominic Joyce]], _Compact Riemannian manifolds with exceptional holonomy_, Surveys in Differential Geometry
Volume 6 (2001) ([doi:10.4310/SDG.2001.v6.n1.a3](https://dx.doi.org/10.4310/SDG.2001.v6.n1.a3))

The sufficiency of spin structure for $G_2$-structure is due to

* {#Gray69} A. Gray, _Vector cross products on manifolds_, Trans. Amer. Math. Soc. 141 (1969), 465–504.

and the [[compact twisted connected sum G2-manifolds]] due to

* {#Kovalev00} [[Alexei Kovalev]], _Twisted connected sums and special Riemannian holonomy_, J. Reine Angew. Math. 565 (2003) ([arXiv:math/0012189](https://arxiv.org/abs/math/0012189))

Review includes

* {#Klemm17} [[Albrecht Klemm]], _Effective Action from M-theory on twisted connected sums_, talk at Ascona Monte Verita, 6 July 2017 ([pdf](http://conf.itp.phys.ethz.ch/string17/talks/Klemm.pdf))

More compact examples are constructed in

* {#JoyceKarigiannis07} [[Dominic Joyce]], [[Spiro Karigiannis]], _A new construction of compact $G_2$-manifolds by gluing families of Eguchi-Hanson spaces_, arXiv:[1707.09325](https://arxiv.org/abs/1707.09325)

Surveys include

* [[Spiro Karigiannis]], _What is... a $G_2$-manifold_ ([pdf](http://www.ams.org/notices/201104/rtx110400580p.pdf))

* [[Spiro Karigiannis]], _$G_2$-manifolds -- Exceptional structures in geometry arising from exceptional algebra_ ([pdf](http://www.math.uwaterloo.ca/~karigian/talks/waterloo.pdf))

* {#Karigiannis14} [[Spiro Karigiannis]], _$G_2$-Conifolds: A survey_, 2014 ([pdf](http://www.math.uni-hamburg.de/sgstructures/hamburg-talk-2.pdf))

* {#Hitchin} [[Nigel Hitchin]], _Special holonomy and beyond_, Clay Mathematics Proceedings ([pdf](https://people.maths.ox.ac.uk/hitchin/hitchinlist/clay.pdf))

* {#Bryant05} [[Robert Bryant]], _Some remarks on $G_2$-structures_, [Proceedings of the 12th Gökova Geometry-Topology Conference 2005](http://gokovagt.org/proceedings/2005), pp. 75-109 ([arXiv:math/0305124](https://arxiv.org/abs/math/0305124), [webpage](http://gokovagt.org/proceedings/2005/bryant.html), [pdf](http://gokovagt.org/proceedings/2005/ggt05-bryant.pdf))

The relation to [[multisymplectic geometry]]/[[2-plectic geometry]] is mentioned explicitly in

* {#Ibort} Alberto Ibort, _Multisymplectic geometry: generic and exceptional_, _[Proceedings of the IX Fall workshop on geometry and physics](http://rsme.es/public/publi3.htm)_ ([[IbortMultisymplectic.pdf:file]])

(but beware of some mistakes in that article...)

For more see the references at _[[exceptional geometry]]_.

[[!include G2-conifolds -- references]]

### $G_2$-Orbifolds
 {#ReferencesG2Orbifolds}

Discussion of [[G2-orbifolds]] includes

* {#Acharya98} [[Bobby Acharya]], _M theory, Joyce Orbifolds and Super Yang-Mills_, Adv.Theor.Math.Phys. 3 (1999) 227-248 ([arXiv:hep-th/9812205](http://arxiv.org/abs/hep-th/9812205))

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys.Rept.392:121-189,2004 ([arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191))

* {#Barrett06} Adam B. Barrett, _M-Theory on Manifolds with $G_2$ Holonomy_, 2006 ([arXiv:hep-th/0612096](http://arxiv.org/abs/hep-th/0612096))

* {#Reidegeld15} [[Frank Reidegeld]], _$G_2$-orbifolds from K3 surfaces with ADE-singularities_ ([arXiv:1512.05114](http://arxiv.org/abs/1512.05114), [spire:1409963](https://inspirehep.net/literature/1409963), [doi:10.17877/DE290R-18940](http://dx.doi.org/10.17877/DE290R-18940))

* [[Frank Reidegeld]], _K3 surfaces with a pair of commuting non-symplectic involutions_ ([arXiv:1809.07501](https://arxiv.org/abs/1809.07501))

* [[Bobby Acharya]], [[Andreas Braun]], Eirik Eik Svanes, Roberto Valandro,  _Counting Associatives in Compact $G_2$ Orbifolds_ ([arXiv:1812.04008](https://arxiv.org/abs/1812.04008))

* Daniel Platt, _Existence of torsion-free $G_2$-structures on resolutions of $G_2$-orbifolds using weighted Hölder norms_ ([arXiv:2011.00482](https://arxiv.org/abs/2011.00482))

### Moduli

Discussion of the [[moduli space]] of $G_2$-structures:

* {#GrigorianYau08} [[Sergey Grigorian]], [[Shing-Tung Yau]], _Local geometry of the $G_2$ moduli space_, Commun.Math.Phys.287:459-488,2009 ([arXiv:0802.0723](http://arxiv.org/abs/0802.0723))

* [[Spiro Karigiannis]], [[Naichung Conan Leung]], _Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps_, Proceedings of the London Mathematical Society (3) 99, 297-325 (2009) [[arXiv:0709.2987](http://arxiv.org/abs/0709.2987), [talk slides pdf](http://www.math.uwaterloo.ca/~karigian/talks/g2modulispace.pdf)]

Relating to moduli of [[flat connections]] on [[tori]]:

* [[Bobby S. Acharya]], Daniel A. Baldwin, *Ricci Flat Metrics, Flat Connections and $G_2$-Manifolds* [[arXiv:2312.12311](https://arxiv.org/abs/2312.12311)]

### Variants and generalizations

Discussion of the more general concept of Riemannian manifolds equipped with [[covariant derivative|covariantly constant]] 3-forms is in

* [[Hông Vân Lê]] , _Geometric structures associated with a simple Cartan 3-form_, J. of Geometry and Physics 70 (2013) 205--223 [arXiv:1103.1201](https://arxiv.org/abs/1103.1201)

### Relation to Killing spinors

Discussion of $G_2$-structures in view of the existence of [[Killing spinors]] includes

* {#FriedrichIvanov01} [[Thomas Friedrich]], Stefan Ivanov, _Parallel spinors and connections with skew-symmetric torsion in string theory_, AsianJ.Math.6:303-336,2002 ([arXiv:math/0102142](http://arxiv.org/abs/math/0102142))

### Application in supergravity

The following references discuss the role of $G_2$-manifolds in [[M-theory on G2-manifolds]]:

* {#Duff} [[Mike Duff]], _M-theory on manifolds of G2 holonomy: the first twenty years_ ([arXiv:hep-th/0201062](http://arxiv.org/abs/hep-th/0201062))

A survey of the corresponding [[string phenomenology]] for [[M-theory on G2-manifolds]] (see there for more) is in

* [[Bobby Acharya]], _$G_2$-manifolds at the CERN Large Hadron collider and in the Galaxy_, talk at _$G_2$-days_ (2012) ([pdf](http://www.mth.kcl.ac.uk/~tbmadsen/acharya.pdf))

See also

* {#BGGG01} [[Andreas Brandhuber]], [[Jaume Gomis]], [[Steven Gubser]], [[Sergei Gukov]], _Gauge Theory at Large N and New $G_2$ Holonomy Metrics_, Nucl.Phys. B611 (2001) 179-204 ([arXiv:hep-th/0106034](http://arxiv.org/abs/hep-th/0106034))

Weak $G_2$-holonomy is discussed in

* {#BilalDerendingerSfetsos} [[Adel Bilal]], J.-P. Derendinger, K. Sfetsos, _(Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry_, Nucl.Phys. B628 (2002) 112-132 ([arXiv:hep-th/0111274](http://arxiv.org/abs/hep-th/0111274))

* {#BilalMetzger03} [[Adel Bilal]], Steffen Metzger, _Compact weak $G_2$-manifolds with conical singularities_ ([arXiv:hep-th/0302021](http://arxiv.org/abs/hep-th/0302021))

* {#HouseMicu04} Thomas House, [[Andrei Micu]], _M-theory Compactifications on Manifolds with $G_2$ Structure_ ([arXiv:hep-th/0412006](http://arxiv.org/abs/hep-th/0412006))

For more on this see at _[[M-theory on G2-manifolds]]_

### Cohomology

An analysis of the [[de Rham complex]] of $G_2$ manifolds, and an analogue of [[Dolbeault cohomology]] is in

* Marisa Fernández, Luis Ugarte. *Dolbeault Cohomology for $G2$-Manifolds*. Geometriae Dedicata 70, 57–86 (1998). ([doi](https://doi.org/10.1023/A:1004940807017))

[[!redirects G2 manifolds]]
[[!redirects G2-manifold]]
[[!redirects G2-manifolds]]

[[!redirects G2 structure]]
[[!redirects G2-structure]]
[[!redirects G2 structures]]
[[!redirects G2-structures]]

[[!redirects G2-holonomy]]
[[!redirects G2-holonomies]]

[[!redirects weak G2-holonomy]]
[[!redirects weak G2 holonomy]]

[[!redirects twisted connected sum G2-manifold]]
[[!redirects twisted connected sum G2-manifolds]]

[[!redirects compact twisted connected sum G2-manifold]]
[[!redirects compact twisted connected sum G2-manifolds]]

[[!redirects compact twisted connected sum G2-manifold]]
[[!redirects compact twisted connected sum G2-manifolds]]


G2-conifold

[[!redirects G2-conifolds]]

#Contents#
* table of contents
{:toc}

## Idea

A [[G2-manifold]] with [[conical singularities]] is also called a _$G_2$-conifold_.

## References

[[!include G2-conifolds -- references]]


G2-conifolds -- references

### $G_2$-conifolds

On [[G2-conifolds]] ([[G2-manifolds]] with [[conical singularities]]):

Survey:

* [[Spiro Karigiannis]], _$G_2$-conifolds: A survey_, 2014 ([[KarigiannisG2Conifolds.pdf:file]])

* Mark Haskins, _Exotic Einstein metrics on $S^6$ and $S^3 \times S^3$ nearly Kähler 6-manifolds and $G_2$-holonomy cones_, 2016 ([[HaskinsG2Conifolds.pdf:file]])

Three [[simply connected topological space|simply connected]] $G_2$-cones are known: the

1. [[metric cone on complex projective 3-space]]

1. metric cone on $S^3 \times S^3$

1. metric cone on $SU(3) / (U(1) \times U(1))$

* {#BryantSalamon89} [[Robert Bryant]], [[Simon Salamon]], _On the construction of some complete metrics with exceptional holonomy_, Duke Math. J. Volume 58, Number 3 (1989), 829-850 ([euclid:euclid.dmj/1077307681](https://projecteuclid.org/euclid.dmj/1077307681))

* [[Gary Gibbons]], [[Don Page]], [[Christopher Pope]], _Einstein metrics on $S^3$, $\mathbb{R}^3$ and $\mathbb{R}^4$ bundles_, Comm. Math. Phys. Volume 127, Number 3 (1990), 529-553 ([euclid:cmp/1104180218](https://projecteuclid.org/euclid.cmp/1104180218))

More on the [[metric cone over complex projective 3-space]] as a [[G2-manifold]]:

* [[Bobby Acharya]], [[Robert Bryant]], [[Simon Salamon]], _A circle quotient of a $G_2$ cone_, Differential Geometry and its Applications Volume 73, December 2020, 101681  ([arXiv:1910.09518](https://arxiv.org/abs/1910.09518), [doi:10.1016/j.difgeo.2020.101681](https://doi.org/10.1016/j.difgeo.2020.101681))


G2-MSSM

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

#Contents#
* table of contents
{:toc}

## Idea

The _$G_2$-MSSM_ is an [[MSSM]] [[model (in particle physics)]] motivated from [[Kaluza-Klein mechanism|KK-compactification]] of [[M-theory]] on a 7-dimensional singular [[G2-manifold]], hence a model in [[string phenomenology]] based on [[M-theory on G2-manifolds]].

The G2-MSSM is derived in the context of [[effective QFT|effective]] [[N=1 D=4 supergravity]]. The model is not based on a specific example of a [[G2 manifold]] but rather, it relies on the most general known properties of [[G2 compactifications of M-theory]]. For example, the Kähler potential is taken to be of the most general form consistent with [[G2]] [[holonomy]].

**Moduli stabilization.** Unlike [[Calabi-Yau manifold]] compactifications of [[string theory]] (see _[[supersymmetry and Calabi-Yau manifolds]]_) where one must [[moduli stabilization|stabilize]] different classes of [[moduli field|moduli]], which often require a different [[moduli stabilization|stabilization]] mechanism for each class, in [[M-theory on G2-manifolds|G2 compactifications of M-theory]] all the geometric moduli are on an equal footing. In other words, there is only one class of moduli - the [[periods]] of the [[associative 3-form]] over the basis 3-[[cycles]].

{#GaugeHierarchy} **Gauge Hierarchy.** Furthermore, each geometric modulus is paired up with an [[axion|axionic]] partner to form [[complex scalar fields]], which possess a Pecchei-Quinn-type shift symmetry. This symmetry originates from the [[gauge symmetry]] of the [[supergravity C-field]] in [[11-dimensional supergravity]]. The shift symmetry is exact to all orders in [[perturbative quantum field theory|perturbation theory]] and can only be [[broken symmetry|broken]] by [[non-perturbative effects]]. Therefore, in the absence of [[fluxes]] the [[superpotential]] in the corresponding [[N=1 d=4 supergravity]] is purely [[non-perturbative effect|non-perturbative]]. This unique feature of the superpotential in [[G2 compactifications of M-theory]] provides a natural mechanism to generate the [[hierarchy problem|hierarchy]] between the [[Planck scale]] and the [[scale]] of [[supersymmetry breaking]], and ultimately the [[electroweak field|electroweak]] breaking scale in the context of the G2-MSSM, via the exponential nature of [[non-perturbative effects]] such as [[membrane instantons]]. This solution to the [[hierarchy problem]] was pointed out in ([Acharya-Kane-Kumar 12, section  V.A.2 (p. 10-11)](#AcharyaKaneKumar12)).

**Axiverse.** Another crucial feature that distinguishes the G2-MSSM from other known scenarios is the exponentially large split between masses of the geometric moduli and their [[axion|axionic]] partners once the moduli are stabilized. This happens because the two dominant terms in the superpotential depend only on a single linear combination of all the moduli. This linear combination represents the volume of an associative three-cycle that supports the [[hidden sector]] [[Yang-Mills theory|non-Abelian gauge theory]], which is taken to be [[Poincare duality|Poincaré dual]] (up to a positive real number) to the [[co-associative 4-form]] of the [[G2-manifold]]. Therefore, the model not only provides a natural [[QCD]] [[axion]] candidate, but also explicitly realizes the [String Axiverse scenario](axion#StringyAxionPhenomenology) proposed by [Arvanitaki et.al. 09](axion#ADDKM09). Incidentally, this very feature of the superpotential ultimately explains the absence of large CP-violating phases in the [[soft supersymmetry breaking terms]].

**Slightly split supersymmetry.** Another distinctive feature of the G2-MSSM is a [[split supersymmetry|slightly split spectrum of superpartners]]. The [[gaugino]] masses are suppressed relative to the [[gravitino]] mass by a factor of $\mathcal{O}(60)$ due to the dynamics of the [[moduli stabilization]]. On the other hand, the scalar masses and the trilinear couplings are as heavy as the [[gravitino]] mass. The [[higgsinos]] may or may not be suppressed relative to the gravitino mass depending on the details of the solution to the [[doublet-triplet splitting problem]].

Apart from the very specific form of the [[soft supersymmetry breaking terms]], the difference between the G2-MSSM and a generic [[bottom-up model building|bottom-up]] [[split supersymmetry|Split Supersymmetry]] scenario is that in [[split supersymmetry|Split SUSY]] both [[gaugino]] masses and trilinear couplings are [[protection from quantum corrections|protected]] by an [[R-symmetry]] and therefore remain light, while the scalar masses can be as heavy as the [[GUT]] [[scale]].

The slightly split G2-MSSM spectrum naturally accommodates the [[Higgs particle]] at the observed [[energy]] of around 125 [[GeV]] together with reasonable [[dark matter]] candidates in the form of [[axions]] and/or W-ino [[WIMPs]], which are generated non-thermally via late-time moduli decays.

## Properties

### Hierarchy (non-)problem
 {#HierarchyProblem}

It has been argued that ([Acharya-Kane-Kumar 12, section  V.A.2 (p. 10-11)](#AcharyaKaneKumar12)) [[model (physics)|models]] in [[M-theory on G2-manifolds]] provide a generic solution to the [[hierarchy problem]] due to control over [[non-perturbative effects]] from [[membrane instantons]], which due to their exponential nature, provide the required exponential hierarchy. See also [above](#GaugeHierarchy).

## Related concepts

* [[G2]], [[G2-manifold]]

* [[split supersymmetry]]

[[!include KK-compactifications of M-theory -- table]]

## References

A popular exposition is in

* {#Kane17} [[Gordon Kane]], _String theory and the real world_, Morgan & Claypool, 2017 (doi:0.1088/978-1-6817-4489-6)

A comprehensive account is in

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _The $G_2$-MSSM - An $M$ Theory motivated model of Particle Physics_ ([arXiv:0801.0478](http://arxiv.org/abs/0801.0478))

Original articles include

* {#AcharyaWitten01} [[Bobby Acharya]], [[Edward Witten]], _Chiral Fermions from Manifolds of $G_2$ Holonomy_ ([arXiv:hep-th/0109152](http://arxiv.org/abs/hep-th/0109152))

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Jing Shao, _Explaining the Electroweak Scale and Stabilizing Moduli in M Theory_ ([arXiv:hep-th/0701034](http://xxx.lanl.gov/abs/hep-th/0701034))

* [[Bobby Acharya]], Konstantin Bobkov, [[Gordon Kane]], [[Piyush Kumar]], Diana Vaman, _An M theory Solution to the Hierarchy Problem_ ([arXiv:hep-th/0606262](http://arxiv.org/abs/hepth/0606262))

* [[Bobby Acharya]], Konstantin Bobkov, _Kähler Independence of the $G_2$-MSSM_, JHEP ([arXiv:0810.3285](http://arxiv.org/abs/0810.3285))

A review of the specific [[phenomenology|phenomenological]] properties is in

* {#AcharyaKaneKumar12} [[Bobby Acharya]], [[Gordon Kane]], Piyush Kumar, _Compactified String Theories -- Generic Predictions for Particle Physics_,  Int. J. Mod. Phys. A, Volume 27 (2012) 1230012 ([arXiv:1204.2795](http://arxiv.org/abs/1204.2795))

For more references see at _[[M-theory on G2-manifolds]]_.

Discussion of the [[Higgs mechanism]] in this model includes

* {#Kane11} [[Gordon Kane]], _String theory and generic predictions for our world – superpartner masses, LHC signatures, dark matter, EWSB, cosmological history of universe, etc_, talk at String phenomenology 2011, August 2011 ([pdf](http://conferencing.uwex.edu/conferences/stringpheno2011/documents/kane.pdf))

* [[Gordon Kane]], Piyush Kumar, Ran Lu, Bob Zheng, _Higgs Mass Prediction for Realistic String/M Theory Vacua_, Phys. Rev. D 85, 075026 ([arXiv:1112.1059](http://arxiv.org/abs/1112.1059))

(a useful informed comment is [here](http://www.math.columbia.edu/~woit/wordpress/?p=4262&cpage=1#comment-101525))

More discussion of experimental signatures is in

* {#EllisKaneZheng014} Sebastian A.R. Ellis, [[Gordon Kane]], Bob Zheng, _Superpartners at LHC and Future Colliders: Predictions from Constrained Compactified M-Theory_ ([arXiv:1408.1961](http://arxiv.org/abs/1408.1961))

* {#BuschmannGonzalesKane18} Malte Buschmann, Eric Gonzalez, [[Gordon Kane]], _Revisiting Gluinos at LHC_ ([arXiv:1803.04394](https://arxiv.org/abs/1803.04394))

Discussion of [[Yukawa couplings]]:

* Eric Gonzalez, [[Gordon Kane]], Khoa Dang Nguyen, [[Malcolm Perry]], _Quark and lepton mass matrices from localization in M-theory on $G_2$ orbifold_ ([arXiv:2002.11820](https://arxiv.org/abs/2002.11820))


G2-orbifold

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{:toc}

## Idea

A $G_2$-orbifold is a [[Riemannian orbifold]] with [[G2-holonomy]], also called a _Joyce orbifold_.

This concept is the generalization from [[manifolds]] to [[orbifolds]] of the concept of [[G2-manifolds]].

for the moment see at _[G2-manifold -- With ADE orbifold structure](G2+manifold#WithADEOrbifoldStructure)_

## Related concepts

* [[M-theory on G2-manifolds]]

## References

* {#Acharya98} [[Bobby Acharya]], _M theory, Joyce Orbifolds and Super Yang-Mills_, Adv.Theor.Math.Phys. 3 (1999) 227-248 ([arXiv:hep-th/9812205](http://arxiv.org/abs/hep-th/9812205))

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]] _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177))

* {#AcharyaGukov04} [[Bobby Acharya]], [[Sergei Gukov]], _M theory and Singularities of Exceptional Holonomy Manifolds_, Phys.Rept.392:121-189,2004 ([arXiv:hep-th/0409191](http://arxiv.org/abs/hep-th/0409191))

* {#Barrett06} Adam B. Barrett, _M-Theory on Manifolds with $G_2$ Holonomy_, 2006 ([arXiv:hep-th/0612096](http://arxiv.org/abs/hep-th/0612096))

* {#Reidegeld15} [[Frank Reidegeld]] _$G_2$-orbifolds from K3 surfaces with ADE-singularities_ ([arXiv:1512.05114](http://arxiv.org/abs/1512.05114))

* [[Frank Reidegeld]], _K3 surfaces with a pair of commuting non-symplectic involutions_ ([arXiv:1809.07501](https://arxiv.org/abs/1809.07501))

* [[Bobby Acharya]], [[Andreas Braun]], Eirik Eik Svanes, Roberto Valandro,  _Counting Associatives in Compact $G_2$ Orbifolds_ ([arXiv:1812.04008](https://arxiv.org/abs/1812.04008))

[[!redirects G2-orbifolds]]

[[!redirects G2 orbifold]]
[[!redirects G2 orbifolds]]

[[!redirects Joyce orbifold]]
[[!redirects Joyce orbifolds]]


G2/SU(3) is the 6-sphere

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#Contents#
* table of contents
{:toc}

## Statement

The [[coset space]] of the [[exceptional Lie group]] [[G2]] by its [[special unitary group|special unitary]] [[subgroup]] [[SU(3)]] is [[diffeomorphism|diffeomorphic]] to the [[6-sphere]]:

$$
  G_2/SU(3)
  \;\simeq\;
  S^6
  \,.
$$

## Related concepts

[[!include coset space structure on n-spheres -- table]]

## References

According to [Gray-Green 70, p. 2](#GrayGreen70) the statement is originally due to

* {#MontogomerySamelson43} Deane Montgomery, Hans Samelson, _Transformation Groups of Spheres_, Annals of Mathematics Second Series, Vol. 44, No. 3 (Jul., 1943), pp. 454-470 ([jstor:1968975](https://www.jstor.org/stable/1968975))

According to [ABF 17](#ABF17) the statement is originally due to to

* T. Fukami, S. Ishihara, _Almost Hermitian structure on $S^6$_, Tohoku Math J. 7 (1955), 151–156 ([doi:10.2748/tmj/1178245052](https://doi.org/10.2748/tmj/1178245052))

Review:

* {#GrayGreen70} [[Alfred Gray]], Paul S. Green, p. 2 of _Sphere transitive structures and the triality automorphism_, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 ([euclid:1102976640](https://projecteuclid.org/euclid.pjm/1102976640))

* {#Bryant98} [[Robert Bryant]], Section 1.3 of _Manifolds with $G_2$-Holonomy_, lecture notes, 1998 ([web](http://www.math.utah.edu/~mckay/g2.pdf))

* [[Robert Bryant]], Section 2.3 of _Some remarks on $G_2$-structures_, Proceedings of 12th Gokova Geometry-Topology Conference, 2005 ([pdf](https://pdfs.semanticscholar.org/67b2/9d0b56c386401805c593126323a777ae2a0b.pdf))

* [[Simon Salamon]], p. 6 of _A tour of exceptional geometry_, Milanj. math.72 (2002)1-0 ([web](http://calvino.polito.it/~salamon/G/COR/tour.pdf))

* A. J. MacFarlane, _The sphere $S^6$ viewed as a $G_2/SU(3)$-coset space_,  International Journal of Modern Physics A Vol. 17, No. 19, pp. 2595-2613 (2002)  ([doi:10.1142/S0217751X02010650](https://doi.org/10.1142/S0217751X02010650))

* A. Gyenge, Section 4.2 of _On the topology of the exceptional Lie group $G_2$_, 2011 ([pdf](https://www.math.ubc.ca/~agyenge/thesis_mathmsc.pdf))

* {#ABF17} [[Ilka Agricola]], Aleksandra Borówka, [[Thomas Friedrich]], _$S^6$ and the geometry of nearly Kähler 6-manifolds_ ([arXiv:1707.08591](https://arxiv.org/abs/1707.08591))


Gabe Dijkstra

* [personal page](https://gdijkstra.github.io/)

## Selected writings

On [[quotient types]] and [[higher inductive types]]:

* [[Thorsten Altenkirch]], [[Paolo Capriotti]], [[Gabe Dijkstra]], [[Nicolai Kraus]], [[Fredrik Forsberg]], *Quotient inductive-inductive types*, Foundations of Software Science and Computation Structures (FoSSaCS 2018), Lecture Notes in Computer Science **10803** (2018) [[doi:10.1007/978-3-319-89366-2_16](https://doi.org/10.1007/978-3-319-89366-2_16), [arXiv:1612.02346](https://arxiv.org/abs/1612.02346)]

## Related concepts

* [[quotient inductive type]]

category: people


Gabor Etesi

* [webpage](http://www.math.bme.hu/~etesi/)

## related $n$Lab entries

* [[cosmic censorship hypothesis]]

* [[computability in physics]]

* [[Malament–Hogarth spacetime]]

category: people


Gabor Kunstatter

* [webpage](http://physics.uwinnipeg.ca/gkunstatter/)

category: people


Gabriel and Zisman > history

< [[Gabriel and Zisman]]


Gabriel Braun

[[!redirects Gabriel Brau]]

* [webpage](http://gabrielbraun.free.fr)

## related $n$Lab entries

* [[Euclidean geometry]], [[synthetic geometry]]

* [[Coq]]

category: people


Gabriel C. Drummond-Cole

[[!redirects Gabriel Drummond-Cole]]
[[!redirects Gabriel C Drummond-Cole]]
[[!redirects Gabriel C. Drummond-Cole]]

[website](http://cgp.ibs.re.kr/~gabriel/)

category: people


Gabriel Catren

* [webpage](http://www.sphere.univ-paris-diderot.fr/spip.php?article675&lang=fr)

* [publications](http://sppm.hypotheses.org/publications)

> My research interests span a range of topics from [[quantization]] of [[gauge theories]] such as [[general relativity]] and [[Yang-Mills theory]] to [[philosophy|philosophical]] [[foundations]] of both gauge theories and [[quantum mechanics]]. With respect to the philosophy of gauge theories, I analyzed the physical content of the heuristic "gauge argument" in Yang-Mills theory and the ontological assumptions entailed by the passage from a global to a [[local symmetry]], I distinguished the two independent first principles of Yang-Mills theory—namely internal relativity and internal background independence—, and I proposed a conceptual interpretation of the [[BRST formalism]]. With respect to the philosophy of quantum mechanics, I'm working on a formalism which had never been addressed—up to my knowledge—by philosophers of quantum mechanics, namely [[geometric quantization]]. I showed that the corresponding technical shift in the presentation of quantum mechanics is the formal counterpart of a radically different conceptual comprehension of quantum physics. Indeed, this new avenue for research in the philosophy of quantum mechanics has already led to unexpected results. In particular, I showed that quantum mechanics endorses a realistic quantum ontology of [[physical systems]]. This conclusion contests the widespread idea according to which the passage from classical mechanics to quantum mechanics comes hand in hand with a weakening of the notion of physical objectivity. According to the proposed quantum ontology, quantum mechanics provides—unlike classical mechanics—a complete description of all the objective properties of physical systems.

## Some writings

* {#Catren13} _Towards a Group-Theoretical Interpretation of Mechanics_, 2013  ([PhilSci Archive](http://philsci-archive.pitt.edu/10116/))

## Further links

* _[[New Spaces for Mathematics and Physics]]_

* Gabriel Catren, _[Nature Absolue -- Speculative Philosophy of the Physico-Mathematical](http://sppm.hypotheses.org)_

* _Philosophy of Mechanics: Mathematical Foundations_ Workshop Feb 12-14, 2014 ([website](http://phil.physico-math.gie.im) [poster](http://phil.physico-math.gie.im/static/PhilosophyofMechanics25size.jpg))

* _["Translation: the philosopher's task", interview with Gabriel Catren](http://ifverso.com/en/node/628253)_

* [Gabriel Catren on Mechanics as Formal Ontology](http://enemyindustry.net/blog/?p=3815)

* [Seminar of Philosophy and mathematical physics](http://www.sphere.univ-paris-diderot.fr/spip.php?rubrique124&lang=en)

* [[philosophy of physics]]

category: people


Gabriel composition of filters

Given a [[filter]] $F$ of ideals in a ring $R$, a left $R$-[[module]] $M$ is __$F$-torsion__ ([[Pierre Gabriel|Gabriel]]'s terminology: $F$-negligible) if for any $m\in M$ there exists a $L$ in $F$ which annihilates it: $L m=0$.

Given two filters $F,G$ in the [[lattice]] $I_l R$ of left ideals on $R$, one defines their __Gabriel composition__ (or __Gabriel product__) $F\bullet G$ as the set of all left $R$-[[ideal]]s $L$ such that there is $K$ in $G$ such that $K/L$ is $F$-torsion. Gabriel composition of filters corresponds to the [[Gabriel multiplication]] of the corresponding torsion classes considered as strictly full subcategories. A Gabriel composition of [[uniform filter]]s is uniform.


Gabriel filter

## Definition

A __Gabriel filter__ $G$ is a [[uniform filter]] of left ideals in a [[ring]] $R$ which is [[idempotent]] under [[Gabriel composition of filters]].

## Terminology

In our definition this notion is equivalent to [[topologizing filter]]; though for some authors the latter notion slightly differs. Stenstroem
says __Gabriel topology__ instead of Gabriel filter, because all Gabriel filters form a basis of neighborhoods of $0$ for a topology on $R$; on the other hand the Gabriel filter itself is an additive analogue of Grothendieck topology, see also [[enriched sheaf]].

## Properties

If $L$ and $L'$ are left ideals in a Gabriel filter $F$, then the set $L L'$ (of all products $l l'$ where $l\in L, l'\in L'$) is an element on $F$. Any uniform filter $F$ is contained in a minimal Gabriel filter $G$ (said to be *generated by $F$*), namely the intersection of all Gabriel filters containing $F$. Given a Gabriel filter $G$, the class of all $G$-[[torsion]] modules (see [[uniform filter]]) is a hereditary torsion class.

## Literature

* [[Pierre Gabriel]], _[[Des catégories abéliennes]]_
* [[Bo Stenstroem]], _Rings of quotients_, Springer 1975.
* [[Zoran Škoda]], _Noncommutative localization in noncommutative geometry_, London Math. Society Lecture Note Series __330__, ed. A. Ranicki; pp. 220–313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276).
* [[Wendy Lowen]], _A generalization of the Gabriel–Popescu theorem_, J. Pure Appl. Alg. __190__:1–3 (2004) 197-211 [doi](https://doi.org/10.1016/j.jpaa.2003.11.016)

A version for [[quantale]]s:

* [[Kimmo Rosenthal|Kimmo I. Rosenthal]], _A general approach to Gabriel filters on quantales_, Comm. Algebra __20__, n.11 (1992) 3393--3409 [doi](https://doi.org/10.1080/00927879208824522)

[[!redirects Gabriel topology]]
[[!redirects Gabriel filters]]


Gabriel localization

#Contents#
* table of contents
{:toc}

## Idea

[[Pierre Gabriel]] introduced a number of constructions in localization theory, mostly in [[abelian category|abelian]] context in his thesis published as

* [[Pierre Gabriel]], [[Des Categories Abeliennes]]

and in general context in his [[Gabriel-Zisman|book]] with Zisman. By __Gabriel localization__ one usually means a specific class of localizations of rings and the corresponding localization of categories of modules over rings.

Given a (possibly noncommutative and nonunital) ring $R$ and a [[Gabriel filter]] $\mathcal{F}$ of left ideals in $R$, a Gabriel localization endofunctor

$$
G_{\mathcal{F}} : {}_R Mod\to {}_R Mod
$$

is defined in one of the number of equivalent ways.

For example, for any [[uniform filter]] $\mathcal{F}$ of left ideals in $R$ one defines a subfunctor of the identity functor $\sigma_{\mathcal{F}}$ on the category of left $R$-modules

$$
M\mapsto \sigma_{\mathcal{F}}(M) =
\sigma_{\mathcal{L}} = \{m\in M \,|\, m\in M, \exists m\in M,\, Jm = 0\}\subset M
$$

In a later work of Goldman $\sigma_{\mathcal{L}}$ was called a radical functor. If $\mathcal{F}$ is not only uniform but in fact a [[Gabriel filter]] then the radical $\sigma_{\mathcal{F}}$ is idempotent, i.e. $\sigma_{\mathcal{F}}^2 = \sigma_{\mathcal{F}}$. If $R$ is unital, $\sigma_{\mathcal{F}}$ is equivalent to the functor given on objects by

$$
\sigma'_{\mathcal{L}}(M) = colim_{J\in\mathcal{F}} Hom_R(R/J,M)
$$

For each uniform fiter $\mathcal{F}$ one also defines the endofunctor $H_{\mathcal{F}}$
on ${}_R Mod$ by

$$
H_{\mathcal{F}}(M) = colim_{J\in\mathcal{F}} Hom_R(J,M)
$$

(the colimit is over downward directed family of ;eft ideals in $\mathcal{F}$ and is a colimit of a functor with values in the category of abelian groups; the uniformness condition however gurantees that there is a canonical structure of an $R$-module on the colimit group $H_{\mathcal{F}}(M)$).

Finally, for the Gabriel filter $\mathcal{F}$ one defines the Gabriel (endo)functor $G_{\mathcal{F}}$ on objects by

$$
G_{\mathcal{F}}(M) := H_{\mathcal{F}}(M/\sigma_{\mathcal{F}}(M))
= colim_{J\in\mathcal{L}}Hom_R(J,M/\sigma_{\mathcal{F}}(M))
$$

The essential image of the functor $G_{\mathcal{F}}$ is the localized category. The left $R$-module $G_{\mathcal{F}}(R)$ has a canonical structure of a ring over $R$; there is a natural forgetful functor from the localized category to the category of left $G_{\mathcal{F}}(R)$-modules. Under strong assumptions on the filter this functor is in fact an equivalence of categories, e.g. when the localization is [[Ore localization|Ore]].

## Related entries

* _[[Calculus of fractions and homotopy theory]]_

## References

* [[Zoran Škoda]], _Localizations for construction of quantum coset spaces_, in "Noncommutative geometry and Quantum groups", W.Pusz,  P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003, [math.QA/0301090](http://arxiv.org/abs/math.QA/0301090).
* [[Zoran Škoda]], _Noncommutative localization in noncommutative geometry_, London Math. Society Lecture Note Series __330__, ed. [[A. Ranicki]]; pp. 220–313, [math.QA/0403276](http://arxiv.org/abs/math.QA/0403276)


Gabriel Minian

Elias Gabriel Minian is an Argentine mathematician based at Universidad de Buenos Aires. He works in algebraic topology, in particular in the homotopy theory of [[finite topological spaces]]

* [Webpage](http://mate.dm.uba.ar/~gminian/research.html)

## Publications:

* see [webpage of the algebraic topology research group](http://mate.dm.uba.ar/~gminian/algtopgroup/publications.html)

##Related pages

* [[Jonathan Barmak]]

* [[finite topological spaces]]

category: people


Gabriel multiplication

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## Definition

For any two [[full subcategory|full subcategories]] $S$ and $T$ of an [[abelian category]] $A$, define their __Gabriel product__ $S\bullet T$ as the full subcategory of $A$ generated by all objects $M$ such that $M$ fits in a short [[exact sequence]] of the
form

$$
  0\to N\to M\to P\to 0
$$

where $N$ is an object in $S$ and $P$ is an object in $T$.

In the case of the abelian category of modules over a ring, the Gabriel multiplication is sometimes expressed as [[Gabriel composition of filters]] of ideals, rather than in terms of abelian subcategories.

An analogous notion in the triangulated setup is the [[Verdier product]].

## Properties

When restricted to the class of [[topologizing subcategory|topologizing subcategories]], Gabriel multiplication is [[associativity|associative]]; if $A$ is [[small category|small]] then the topologizing subcategories make a [[semiring]] with respect to the commutative operation $\cap$ and Gabriel multiplication; in particular $\bullet$ is left and right distributive with respect to intersection of topologizing subcategories.

category: algebra


Gabriel spectrum

Given a Noetherian [[Grothendieck category]] $A$, [[Pierre Gabriel|Pierre Peter Gabriel]] considered the set of isomorphism classes of indecomposable [[injective object]]s, together with an appropriate topology on its set. This topological space is called the spectrum of indecomposable injectives or Gabriel spectrum (or Gabriel's spectrum).

* [[Pierre Gabriel]], [[Des Catégories Abéliennes]], Bulletin de la Société Mathématique de France 90 (1962) 323-448, [numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0)

For the category of quasicoherent sheaves over quasicompact Noetherian scheme, Gabriel's spectrum with appropriate construction of structure sheaf reconstructs the scheme.

A refinement is the spectrum of indecomposable pure injectives introduced by Ziegler and studied by Herzog, Krause, Prest and others, often in relation to model theory, called [[Ziegler spectrum]]. Herzog has generalized its construction to locally coherent Grothendieck categories.

Gabriel's spectrum of the category of Abelian presheaves on the category of finitely presentable $R$-modules is related to Ziegler's spectrum of the category of fp $R$-modules, see

* Ivo Herzog, _Contravariant functors on the category of finitely presented modules_, Israel J. Math. __167__, 347–410 (2008) [doi](https://doi.org/10.1007/s11856-008-1052-8)


Gabriel's theorem

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#Contents#
* table of contents
{:toc}

## Statement

[[Gabriel's theorem]] ([Gabriel 72](#Gabriel72)) says that connected [[quivers]] with a [[finite number]] of [[indecomposable object|indecomposable]] [[quiver representations]] over an [[algebraically closed field]] are precisely the _[[Dynkin quivers]]_: those whose [[underlying]] [[undirected graph]] is a [[Dynkin diagram]] in the [[ADE classification|ADE series]].

Moreover, the [[indecomposable object|indecomposable]] [[quiver representations]] in this case are in [[bijection]] with the [[positive number|positive]] [[root (in representation theory)|roots]] in the [[root system]] of the [[Dynkin diagram]].

## Examples

In the following we write $\mathbb{K}$ for the given [[ground field]] (typically $\mathbb{K} = \mathbb{C}$ the [[complex numbers]], but much of the following works for general fields, eg. $\mathbb{K} = \mathbb{R}$ the [[real numbers]] or $\mathbb{K} = \mathbb{F}_p$ a [[finite field]]).

Given a [[quiver]] $Q$ and a [[quiver representation]] $\rho$, we denote for any [[edge]] $v \xrightarrow{\;e\;} v'$ in $Q_1$ the corresponding value of $\rho$ by

\begin{tikzcd}[column sep = small, row sep=small]
  v
  \ar[r, "{e}"]
  &
  v'
  \\
  V_e
  \ar[r, "{ \rho(e) }"]
  &
  V_{e'}
\end{tikzcd}

### A-type quivers
  {#ATypeQuivers}

For $n \in \mathbb{N}_+$ consider the $\mathbb{A}_{n}$-quiver, hence with this underlying undirected graph:

\begin{tikzcd}[column sep = small, row sep=small]
  1
  \ar[r, -]
  &
  2
  \ar[r, -]
  &
  3
  \ar[r,-]
  &
  \cdots
  \ar[r,- ]
  &
  n
\end{tikzcd}

\begin{example}
\label{IndecomposableRepsOfAn}
The [[indecomposable object|indecomposable]] [[quiver representations]] of $\mathbb{A}_n$ are labeled by [[pairs]] $a,b \in \mathbb{N}^2$ with $1 \leq a \leq b \leq n$ and are given as follows (see also [Carlsson & de Silva 2010](#CarlssonDeSilva10), reviewed in [Oudot 15, p. 17](#Oudot15)):

\begin{tikzcd}[column sep = small, row sep=small]
  1
  \ar[r,-]
  &
  \cdots
  \ar[r,-]
  &
  (a-1)
  \ar[r,-]
  &
  a
  \ar[r,-]
  &
  \cdots
  \ar[r,-]
  &
  b
  \ar[r,-]
  &
  (b+1)
  \ar[r,-]
  &
  \cdots
  \ar[r,-]
  &
  n
  \\
  \mathbb{K}^0
  \ar[r, -, "{0}"]
  &
  \cdots
  \ar[r, -, "{0}"]
  &
  0
  \ar[r,-, "{0}"]
  &
  \mathbb{K}^1
  \ar[r,-, "{\mathrm{id}}"]
  &
  \cdots
  \ar[r,-, "{\mathrm{id}}"]
  &
  \mathbb{K}^1
  \ar[r,-,"{0}"]
  &
  \mathbb{K}^0
  \ar[r,-,"{0}"]
  &
  \cdots
  \ar[r,-,"{0}"]
  &
  \mathbb{K}^0
\end{tikzcd}

for any given [[orientation]] of the [[edges]].

So for instance if the quiver is the *linear* $\mathbb{A}_n$-quiver

\begin{tikzcd}[column sep = small, row sep=small]
  1
  \ar[r]
  &
  2
  \ar[r]
  &
  3
  \ar[r]
  &
  \cdots
  \ar[r]
  &
  n
\end{tikzcd}

then its indecomposable representations are of this form:

\begin{tikzcd}[column sep = small, row sep=small]
  1
  \ar[r]
  &
  \cdots
  \ar[r]
  &
  (a-1)
  \ar[r]
  &
  a
  \ar[r]
  &
  \cdots
  \ar[r]
  &
  b
  \ar[r]
  &
  (b+1)
  \ar[r]
  &
  \cdots
  \ar[r]
  &
  n
  \\
  \mathbb{K}^0
  \ar[r, "{0}"]
  &
  \cdots
  \ar[r, "{0}"]
  &
  0
  \ar[r, "{0}"]
  &
  \mathbb{K}^1
  \ar[r, "{\mathrm{id}}"]
  &
  \cdots
  \ar[r, "{\mathrm{id}}"]
  &
  \mathbb{K}^1
  \ar[r, "{0}"]
  &
  0
  \ar[r, "{0}"]
  &
  \cdots
  \ar[r, "{0}"]
  &
  0
\end{tikzcd}
\end{example}

\begin{remark}
\label{RelationToPersistencyModules}
Example \ref{IndecomposableRepsOfAn} is of key relevance in the discussion of [[persistent homology]] (notably in [[topological data analysis]]):
Here an $\mathbb{A}_n$-[[quiver representation]] is called a *[[zigzag persistence|zig-zag]] [[persistence module]]* and the statement of Ex. \ref{IndecomposableRepsOfAn} is interpreted as saying that every persistence module is spanned by elements which

* *appear* at some *resolution* $a$,

* *persist* across a range $b - a$,

* *disappear* at some resolution $b$.

The [[multiset]] of these [[pairs]] $(a,b)$ is then called the *[[barcode]]* or *[[persistence diagram]]* of the [[persistence module]].

This relation between quiver representation theory and persistent homolology was originally highlighted by [Carlsson & de Silva 2010](#CarlssonDeSilva10).
\end{remark}

## Related concepts

* [[ADE classification]]

* [[Dynkin quiver]], [[McKay quiver]]

## References

The result is due to

* {#Gabriel72} [[Peter Gabriel]], _Unzerlegbare Darstellungen. I_, Manuscripta Mathematica 6: 71–103, (1972) $[$[doi:10.1007/BF01298413](https://link.springer.com/article/10.1007/BF01298413)$]$

Lecture notes:

* _Quiver representations and Gabriel's theorem_ ([pdf](https://www2.math.ethz.ch/education/bachelor/seminars/hs2009/clualg/Cluster-A5.pdf))

See also:

* Wikipedia, _[Gabriel's theorem](http://en.wikipedia.org/wiki/Gabriel%27s_theorem)_

Re-proof of Gabriel's theorem in the case of A-type quivers and relation to [[persistent homology]]:

* {#CarlssonDeSilva10} [[Gunnar Carlsson]], [[Vin de Silva]], *Zigzag Persistence*, Found Comput Math **10** (2010) 367–405 $[$[arXiv:0812.0197](https://arxiv.org/abs/0812.0197), [doi:10.1007/s10208-010-9066-0](https://doi.org/10.1007/s10208-010-9066-0)$]$

reviewed in:

* {#Oudot15} [[Steve Y. Oudot]], *Persistence Theory: From Quiver Representations to Data Analysis*, Mathematical Surveys and Monographs **209** AMS (2015)
$[$[ISBN:978-1-4704-3443-4](https://bookstore.ams.org/surv-209/), [pdf](https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf)$]$

[[!redirects Gabriel theorem]]


Gabriel-Oberst duality

The category [[Ab]] of discrete abelian groups is (as a case of [[Pontrjagin duality]] for locally compact Hausdorff groups) [[dual|dual category]] to the category of compact Hausdorff abelian groups.

Gabriel-Roos-Oberst duality refers to several generalizations of this duality for more general [[Grothendieck categories]] in terms of linearly compact topological rings and modules.

In Gabriel's thesis

* [[Pierre Gabriel]], [[Des catégories abéliennes]], Bulletin de la Société Mathématique de France 90 (1962), 323-448 (numdam)

the dual of any locally finite Grothendieck category has been found. Roos has
generalized this to locally noetherian Grothendieck categories in

* [[Jan Erik Roos|J. E. Roos]], _Locally noetherian categories and generalized strictly linearly compact rings. Applications._, in: Cat. theory homology theory and their appl., "Battelle Institute Conference 1968", vol. 2, Springer 1969, pp. 197-227

The case of (dual of) a general Grothendieck category is found in

* U. Oberst, _Duality theory for Grothendieck categories_, Bull. Amer. Math. Soc. __75__, (1969) 1401--1408 [pdf](https://projecteuclid.org/download/pdf_1/euclid.bams/1183530938); _Duality theory for Grothendieck categories and linearly compact rings_, J. Alg. __15__ (1970) 473--542 [journal](http://www.sciencedirect.com/science/article/pii/0021869370900517) doi

A textbook exposition is in the chapter 6, _Duality_ of

* N. Popescu, _Abelian categories with applications to rings and modules_, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)

[[!redirects Gabriel-Roos-Oberst duality]]


Gabriel-Popescu theorem

# Giraud's subcategories and Gabriel-Popescu theorem

The __Gabriel–Popescu theorem__, also quoted as Popesco–Gabriel theorem because of that spelling in the original published paper (cf. the spelling in that order, which is sometimes considered wrong when in the order of the authors!) asserts that every [[Grothendieck category]] can be represented as a reflective subcategory of a category ${}_R Mod$ of modules over a unital ring, where the localization functor is exact (one calls such subcategories of module categories Giraud subcategories). The right adjoint (reflection) is in this setup sometimes called the section functor.

This is the [[Ab-enriched]] analogue of the fact that [[sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes]]. See a more general statement at [[enriched sheaf]].

# Literature

* [[Nicolae Popescu|Nicolae Popesco]], [[Pierre Gabriel]], _Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes_, Les Comptes rendus de l'Académie des sciences Paris __258__: 4188–4190 [MR 0166241](http://www.ams.org/mathscinet-getitem?mr=0166241)
* wikipedia [Gabriel-Popesco theorem](http://en.wikipedia.org/wiki/Gabriel%E2%80%93Popesco_theorem)
* [[Nicolae Popescu]], Abelian categories with applications to rings and modules, London Math. Soc. Monographs 3, Academic Press 1973. xii+467 pp. [MR0340375](http://www.ams.org/mathscinet-getitem?mr=0340375)
* [[Mitsuhiro Takeuchi]], _A simple proof of Gabriel and Popesco’s theorem_, J. Alg. 18,  112-113 (1971) [pdf](https://core.ac.uk/download/pdf/82301148.pdf)

There are various generalizations, e.g.

* Wendy Lowen, _A generalization of the Gabriel-Popescu theorem_, Journal of Pure and Applied Algebra 190 (1) (2004): 197--211, [doi](https://doi.org/10.1016/j.jpaa.2003.11.016) [MR2043328](https://www.ams.org/mathscinet-getitem?mr=2043328)

N. Kuhn had related results in the study of [[Steenrod algebra]], cf. also

* Nicholas J. Kuhn, _Generic representations of the finite general linear groups and the Steenrod algebra. I_, Amer. J. Math. 116 (2): 327-360 (1994) [doi](https://doi.org/10.2307/2374932)
* [[Jacob Lurie]], _[A theorem of Gabriel-Kuhn-Popesco](http://www-math.mit.edu/~lurie/917notes/Lecture8.pdf)_, MIT math 917 notes, lecture 8

An analogue in triangulated setup: every triangulated category which is well generated in the sense of [[Amnon Neeman]] and algebraic (over ground ring $k$) in the sense of [[Bernhard Keller]] is $k$-linearly triangle equivalent to a localization of the derived category of a small pretriangulated dg-category by a localizing subcategory generated by a set of objects:

* M. Porta, _The Popescu-Gabriel theorem for triangulated categories_, Adv. Math. __225__ (2010) 1669--1715 [doi](https://doi.org/10.1016/j.aim.2010.04.002)

[[!redirects Gabriel–Popesco theorem]]
[[!redirects Popesco-Gabriel theorem]]
[[!redirects Popesco–Gabriel theorem]]
[[!redirects Giraud subcategory]]
[[!redirects Giraud subcategories]]
[[!redirects Gabriel-Popescu embedding theorem]]


Gabriel-Rosenberg theorem

#Contents#
* table of contents
{:toc}

## Idea

The __Gabriel-Rosenberg reconstruction theorem__ says: an [[algebraic scheme]] can be reconstructed up to an isomorphism from the abelian category of quasicoherent sheaves over that scheme. If a scheme is quasicompact quasiseparated, then $X$ is the 'geometric center' of the category of quasicoherent sheaves $Qcoh_X$.  The geometric center is a construction obtained as a composition of two functors: first Rosenberg's [[spectrum of an abelian category]] as a topological space with a stack of local categories and then a certain construction of a center, which will yield a topological space with a sheaf of local commutative unital rings.

## History and versions

This theorem has been, under some restrictions proved, in [[Pierre Gabriel]]'s thesis and subsequent article [[Des catégories abéliennes]] using the notion of indecomposable spectrum (or [[Gabriel's spectrum of indecomposables]]). The case of general [[quasicompact schemes]] is proved in Max Planck 1996 preprint ([pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948)) of [[Alexander Rosenberg]] published in 1998 (a short sketch is also in the 1998 paper _Noncommutative schemes_ of Rosenberg) using his construction of a [[spectrum of an abelian category]] generalizing straightforwardly his 1982 construction of a left spectrum of a ring). The quasicompactness condition has been noted in this construction after a counterexample exhibited by [[Ofer Gabber]]. Few years later (see 2003 MPI preprint), Rosenberg got rid of quasicompactness condition inventing yet another notion of a spectrum.  Another proof, using a slightly modified version of Rosenberg's spectrum due to [[Ofer Gabber]], has been written down in [Brandenburg 2013](#Brandenburg13).

A [[moduli]]-theoretic proof, and an extension to [[algebraic spaces]], has been proved in [Calabrese-Groechenig 13](#CalabreseGroechenig13).

## Literature

* [[Pierre Gabriel]], __[[Des catégories abéliennes]]__, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 ([numdam](http://www.numdam.org/item?id=BSMF_1962__90__323_0))

* A. L. Rosenberg, _The spectrum of abelian categories and reconstructions of schemes_, in Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. __197__, Marcel Dekker, New York, 257--274, 1998; MR99d:18011; and Max Planck Bonn preprint _Reconstruction of Schemes_, MPIM1996-108, [pdf](http://www.mpim-bonn.mpg.de/preprints/send?bid=3948) (1996).

* A. L. Rosenberg, _Spectra of noncommutative spaces_, MPIM2003-110 [ps](http://www.mpim-bonn.mpg.de/preprints/send?bid=1946) [dvi](http://www.mpim-bonn.mpg.de/preprints/send?bid=1945) (2003)

* A. L. Rosenberg, _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125 ([doi](http://dx.doi.org/10.1023/A:1000479824211))

* A. L. Rosenberg, _The left spectrum, the Levitzki radical, and noncommutative schemes_, Proc. Nat. Acad. Sci. U.S.A. __87__  (1990),  no. 21, 8583--8586.

* A. L.  Rosenberg, _Noncommutative algebraic geometry and representations of quantized algebras_, MIA __330__, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

* [[Martin Brandenburg]], _Rosenberg's reconstruction theorem (after Gabber)_, 2013, ([arXiv:1310.5978](http://arxiv.org/abs/1310.5978)).
 {#Brandenburg13}

* John Calabrese, Michael Groechenig.  _Moduli problems in abelian categories and the reconstruction theorem._  2013.  ([arXiv:1310.6600](http://arxiv.org/abs/1310.6600)).
 {#CalabreseGroechenig13}

* related MO question: [what is the center of QCohX](http://mathoverflow.net/questions/41789/what-is-the-center-of-qcohx)

category: algebraic geometry, noncommutative geometry
[[!redirects Gabriel-Rosenberg reconstruction theorem]]
[[!redirects Gabriel-Rosenberg's theorem]]
[[!redirects Gabriel-Rosenberg reconstruction]]
[[!redirects Gabriel-Rosenberg's reconstruction]]


Gabriele Honecker

* [former homepage in Bonn](http://www.th.physik.uni-bonn.de/nilles/people/honecker.html)

* [former COST grant page](https://www.prisma.uni-mainz.de/gabriele-honecker-is-one-of-two-german-representatives-in-the-european-string-theory-network/)

## Selected writings

On [[intersecting D-brane models]] on [[toroidal orbifolds]]/[[orientifolds]] with [[RR-field tadpole cancellation]] with [[branes at angles]]:

* {#Honecker02} [[Gabriele Honecker]], _Non-supersymmetric orientifolds and chiral fermions from intersecting D6- and D8-branes_, thesis 2002 ([[HoneckerIntersectingDBraneModels02.pdf:file]])

* {#ForsteHoneckerSchreyer01} Stefan Forste, [[Gabriele Honecker]], Ralph Schreyer, _Orientifolds with branes at angles_, JHEP 0106:004, 2001 ([arXiv:hep-th/0105208](https://arxiv.org/abs/hep-th/0105208))

* {#Honecker01} [[Gabriele Honecker]], _Non-supersymmetric Orientifolds with D-branes at Angles_, Fortsch.Phys. 50 (2002) 896-902 ([arXiv:hep-th/0112174](https://arxiv.org/abs/hep-th/0112174))

* {#Honecker02} [[Gabriele Honecker]], _Intersecting brane world models from D8-branes on $(T^2 \times T^4/\mathbb{Z}_3)/\Omega\mathcal{R}_1$ type IIA orientifolds_, JHEP 0201 (2002) 025 ([arXiv:hep-th/0201037](https://arxiv.org/abs/hep-th/0201037))

Specifically on [[MSSM]] [[string phenomenology]] of such models:

* Jill Ecker, [[Gabriele Honecker]], Wieland Staessens, _D6-Brane Model Building on $\mathbb{Z}_2 \times \mathbb{Z}_6$: MSSM-like and Left-Right Symmetric Models_, Nuclear Physics B Volume 901, December 2015, Pages 139-215,  ([arXiv:1509.00048](https://arxiv.org/abs/1509.00048))

* [[Gabriele Honecker]], _From Type II string theory towards BSM/dark sector physics_, International Journal of Modern Physics A Vol. 31 (2016) 1630050 ([arXiv:1610.00007](https://arxiv.org/abs/1610.00007))

Specifically on [[landscape of perturbative string theory vacua|the landscape]] of such models:

* [[Ralph Blumenhagen]], [[Florian Gmeiner]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _The Statistics of Supersymmetric D-brane Models_, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](https://arxiv.org/abs/hep-th/0411173))

* {#GBHLW05} [[Florian Gmeiner]], [[Ralph Blumenhagen]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _One in a Billion: MSSM-like D-Brane Statistics_, JHEP 0601:004, 2006 ([arXiv:hep-th/0510170](https://arxiv.org/abs/hep-th/0510170))

## Related $n$Lab entries

* [[axion]]

category: people


Gabriele Lobbia

* [personal page](https://globbia.github.io/)

* [institute page](https://www.muni.cz/en/people/247277-gabriele-lobbia)

## Selected writings

On [[distributive laws]] for [[relative monads]]:

* [[Gabriele Lobbia]], *Distributive Laws for Relative Monads*, Applied Categorical Structures **31** 19 (2023)  [[doi:10.1007/s10485-023-09716-1](https://doi.org/10.1007/s10485-023-09716-1), [arXiv:2007.12982](https://arxiv.org/abs/2007.12982)]

[[skew monoidal|Skew monoidal]] closed structures on the category of [[Gray categories]] are considered in:

* [[John Bourke]], [[Gabriele Lobbia]], _A skew approach to enrichment for Gray-categories_, (2022). [[arXiv:2212.12358](https://arxiv.org/abs/2212.12358)]

category: people


Gabriele Veneziano

* [Wikipedia entry](http://en.wikipedia.org/wiki/Gabriele_Veneziano)

### Selected writings

On the [[theta angle]] in [[QCD]]:

* {#CrewtherDiVecchiaVenezianoWitten79} [[Rodney Crewther]], [[Paolo Di Vecchia]], [[Gabriele Veneziano]], [[Edward Witten]], _Chiral estimate of the electric dipole moment of the neutron in quantum chromodynamics_, Phys. Lett. B 88 (1979) 123-127 ([CERN](http://cds.cern.ch/record/133382)). See also

On the early history of [[string theory]] as a theory of [[hadrons]] ([[Regge trajectories]], [[Veneziano amplitude]]):

* [[Gabriele Veneziano]], *Rise and fall of the hadronic string*, Section 2 in: [[Paolo Di Vecchia]] et al. (eds.), *The Birth of String Theory* (2012) [[doi:10.1017/CBO9780511977725.004](https://doi.org/10.1017/CBO9780511977725.004), lecture video:[15073](https://lecture2go.uni-hamburg.de/en/l2go/-/get/v/15073)]

## Related entries

* [[Veneziano amplitude]]

category: people


Gabriele Vezzosi

* [website](http://www.dma.unifi.it/~vezzosi)

(The picture [shows](http://owpdb.mfo.de/detail?photo_id=7420) Vezzosi on the left and [[Bertrand Toen]] on the right during a research in pairs stay at [Oberwolfach](http://www.mfo.de/) in 2002).

Together with [[Bertrand Toën]], and based partially on earlier ideas of [[Carlos Simpson]], Gabriele Vezosi has laid foundations of what is now called [[derived geometry]]. Another collaborator in parts of that work has been [[Michel Vaquié]].

## Selected writings

On [[derived algebraic geometry]]:

* {#ToenVezzosi04} [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry II: geometric stacks and applications_, Memoirs of the AMS **193** (2008) [[arXiv:math/0404373](http://arxiv.org/abs/math/0404373), [ams:memo-193-902](https://bookstore.ams.org/memo-193-902)]

category: people

[[!redirects G. Vezzosi]]


Gabriella Böhm

__Gabriella Böhm__ is a Hungarian [[mathematical physicist]]. She studies symmetries appearing in [[quantum field theories]], inclusions of [[subfactor]]s, [[bialgebroid]]s, [[Hopf algebroid]]s, [[entwining structure]]s, [[corings]], Hopf cyclic homology and some related categorical structures like [[monads]], weak monads and so on.

* [webpage](http://www.rmki.kfki.hu/~bgabr)

##Selected writings

* Gabriella Böhm, [[Joost Vercruysse]], _Morita theory for comodules over corings_,  3207-3247 (2009)  Commun. Alg. __37__:9 (2009) ([doi:10.1080/00927870902747993](https://doi.org/10.1080/00927870902747993))

* [[Gabriella Böhm]], _Internal bialgebroids, entwining structures and corings_, [math.QA/0311244](http://arxiv.org/abs/math/0311244), in: Algebraic structures and their representations, 207–226,  Contemp. Math. __376__, Amer. Math. Soc. 2005.

* G. Böhm, _Hopf algebroids_, in: Handbook of algebra, Vol. __6__, ed. by M. Hazewinkel, Elsevier 2009, 173--236 [arXiv:math.RA/0805.3806](https://arxiv.org/abs/0805.3806)
* G. Böhm, [[K. Szlachányi]], _Hopf algebroids with bijective antipodes: axioms, integrals and duals_, Comm. Algebra __32__ (11) (2004) 4433 - 4464 [math.QA/0305136](https://arxiv.org/abs/math.QA/0305136)

[[!redirects Gabi Bohm]]
[[!redirects Gabi Böhm]]
[[!redirects G. Böhm]]

[[!redirects Gabriella Bohm]]


Gabriel–Ulmer duality

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

* table of contents
{: toc}

## The idea

**Gabriel--Ulmer duality** says that there is an [[equivalence of categories|equivalence]] of [[2-categories]] (or in other words, a [[biequivalence]])

$$
\begin{matrix}
Lex^{op} & \to &  LFP \\
  C      & \mapsto & Lex(C, Set)
\end{matrix}
$$

where [[Lex]] is the 2-category of:

* small [[finitely complete categories]],

* [[finite limit]]$\:$ [[preserved limit|preserving]] [[functors]], and

* [[natural transformations]],

and LFP is the 2-category of

* [[locally finitely presentable categories]],

* finitary [[right adjoint|right]] [[adjoint functors]] and

* [[natural transformations]].

The idea is that an object $C \in Lex$ can be thought of as an [[essentially algebraic theory]], which has a category of [[model|models]] $Lex(C,Set)$.  Gabriel--Ulmer duality says that this category of models is locally finitely presentable, all LFP categories arise in this way, and we can recover the theory $C$ from its category of models. There are similar dualities for other classes of theory such as [[regular theories]].

A version of Gabriel--Ulmer duality for [[enriched category theory]] was proved by [[Max Kelly]] (see [LackTendas](#LackTendas)). Let $\mathcal{V}$ be a symmetric monoidal closed complete and cocomplete category which is locally finitely presentable as a closed category. Then let $\mathcal{V}$-$Lex$ be the 2-category of finitely complete $\mathcal{V}$-categories ($\mathcal{V}$-categories with finite weighted limits), finite limit preserving $\mathcal{V}$-functors, and $\mathcal{V}$-natural transformations, and $\mathcal{V}$-$LFP$ the 2-category of locally finitely presentable $\mathcal{V}$-categories, right adjoint $\mathcal{V}$-functors that preserve filtered colimits, and $\mathcal{V}$-natural transformations. Then there is a biequivalence
$$
\begin{matrix}
\mathcal{V}-Lex^{op} & \to &  \mathcal{V}-LFP \\
  C      & \mapsto & Lex(C, \mathcal{V}).
\end{matrix}
$$

For instance, in the [[truth value]]-enriched case, the duality is between [[meet semilattices]] and [[algebraic lattices]].

Gabriel-Ulmer duality is a duality exhibited by the 2-[[Chu construction]], $Chu(Cat,Set)$.

## References

The original source is:

* [[Peter Gabriel]], [[Friedrich Ulmer]], _Lokal Praesentierbare Kategorien_, Springer Lecture Notes in Mathematics **221**, Berlin, 1971.

A careful discussion and proof of the biequivalence is in

* [[Jiri Adamek]], Hans-Eberhard Porst, _Algebraic Theories of Quasivarieties_ ,  J. Algebra **208** (1998) pp.379-398.

Some other general treatments of Gabriel-Ulmer duality (and generalizations to other [[doctrine|doctrines]]):

* C. Centazzo, [[E. M. Vitale]], _A duality relative to a limit doctrine_, Theory and Appl. of Categories __10__, No. 20, 2002, 486–497, [pdf](http://www.tac.mta.ca/tac/volumes/10/20/10-20.pdf)

* [[Stephen Lack]], [[John Power]], _Gabriel--Ulmer duality and Lawvere Theories enriched over a general base_, [pdf](http://maths.mq.edu.au/~slack/papers/jfp.pdf)

* [[M. Makkai]], [[A. Pitts]], _Some results on locally finitely presentable categories_, Trans. Amer. Math. Soc. __299__ (1987), 473-496, [MR88a:03162](http://www.ams.org/mathscinet-getitem?mr=869216), [doi](http://dx.doi.org/10.2307/2000508), [pdf](http://www.ams.org/journals/tran/1987-299-02/S0002-9947-1987-0869216-2/S0002-9947-1987-0869216-2.pdf)

For a 2-dimensional analogue see the slides from a 2010 talk by Makkai: [pdf](http://www.math.mcgill.ca/makkai/scans/2010Washington0001.pdf)

A formal-categorical account using [[KZ-doctrines]] can be found in

* {#DL18} [[Ivan Di Liberti]], [[Fosco Loregian]], _Accessibility and Presentability in 2-Categories_, Journal of Pure and Applied Algebra 227.1 (2023)([arXiv:1804.08710](https://arxiv.org/abs/1804.08710))

For a discussion of Gabriel--Ulmer duality and related dualities in the context of [[enriched category theory]] see

* {#LackTendas} [[Stephen Lack]], [[Giacomo Tendas]], _Enriched regular theories_, J. Pure Appl. Alg. __226__, n. 6 (2020) 106268, ([arXiv:1907.02301](https://arxiv.org/abs/1907.02301), [doi:10.1016/j.jpaa.2019.106268](https://doi.org/10.1016/j.jpaa.2019.106268))

This discusses (see Theorem 2.1) Kelly's original result for $V$-[[enriched categories]], where $V$ is a closed symmetric monoidal category whose underlying category $V_0$ is locally small, complete and cocomplete, in section 9 (cf. theorem 9.8) of

* [[Max Kelly]], _Structures defined by finite limits in the enriched context_, Cahiers de Topologie et Géométrie Différentielle catégoriques, **23** no. 1 (1982), pp. 3-42, [MR648793](http://www.ams.org/mathscinet-getitem?mr=648793),[numdam](http://www.numdam.org/item?id=CTGDC_1982__23_1_3_0)

For an extension of Gabriel--Ulmer duality to a duality between [[Cauchy-complete]] [[clans]] and [[locally finitely presentable categories]] equipped with a well-behaved kind of [[weak factorization system]] see

* [[Jonas Frey]], _Duality for Clans: a Refinement of Gabriel-Ulmer Duality_ ([arXiv:2308.11967](https://arxiv.org/abs/2308.11967))

For a connection to [[Tannaka duality]] theory see

* nCafé discussion [here](http://golem.ph.utexas.edu/category/2011/07/doctrinal_and_tannakian_recons.html)

* [[Brian Day]], _Enriched Tannaka duality_, JPAA __108__ (1996) pp.17-22, [MR97d:18008](http://www.ams.org/mathscinet-getitem?mr=1382240) doi

For a discussion of an $\infty$-version of Gabriel-Ulmer duality between finitely complete and idempotent complete $(\infty, 1)$-categories and locally finitely presentable $(\infty, 1)$-categories see this [MO discussion](https://mathoverflow.net/q/293031/447).

[[!redirects Gabriel-Ulmer duality]]
[[!redirects Gabriel–Ulmer duality]]
[[!redirects Gabriel--Ulmer duality]]


Gaeta topos

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

#Contents#
* table of contents
{:toc}

## Idea

The **Gaeta topos construction** assigns a [[Grothendieck topos]] to a small [[extensive category]].

The construction arises in attempts (Lawvere [1976](#Law76), [1991](#Law91)) to build a [[gros topos]] of spaces with an inherent space-quantity 'duality'. The terminology is apparently a tribute to the Spanish algebraic geometer [[Federico Gaeta]] who in his lecture notes of [[Alexander Grothendieck|A. Grothendieck's]] Buffalo 1973 colloquium (Gaeta [1974](#Gaeta74)) has pointed out the utility of the extensivity property for functorial algebraic geometry.[^gaeta]

[^gaeta]: _"Gaeta's notes of Grothendieck's lecture series at Buffalo point out that $A$_ (i.e. the opposite category of finitely presented algebras over a field $k$ -nLab) _is more closely suited than most categories to serve as a site for a geometric category, because it is what is now called 'extensive' "_ , (Lawvere [2003](#Law03)).

## Definition
Let $\mathcal{C}$ be a small [[finitary category|finitary]] [[extensive category]]. The _Gaeta topology_[^extensive] on $\mathcal{C}$ is the [[Grothendieck topology]] $J_G$ generated by all finite families $(X_i\to X)$ such that $\sum_i X_i\to X$ is an isomorphism. The _Gaeta topos_ $G(\mathcal{C})$ is the topos of sheaves for the [[site]] $(\mathcal{C}, J_G)$.

[^extensive]: $J_G$ is also called the _finite disjoint covering topology_. The nLab elsewhere uses the term _extensive topology_ (cf. [[extensive category]]). $(\mathcal{C},J_G)$ is in fact an example of a [[superextensive site]].

## Examples

* The [[bornological topos]] is the Gaeta topos on the category of countable sets.

## Properties

The Gaeta topology $J_G$ is naturally induced by the extensivity of $\mathcal{C}$:

+-- {: .num_prop #gaeta_topology}
###### Proposition
The Gaeta topology $J_G$ as defined above is indeed a Grothendieck topology on a small extensive category $\mathcal{C}$.
=--

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

...still to do!

=--

The next proposition shows that $G(\mathcal{C})$ is not only a topos but a multisorted variety as well as it coincides with the category of finite product preserving presheaves on $\mathcal{C}$:

+-- {: .num_prop #gaeta_topos}
###### Proposition
Let $\mathcal{C}^{op}$ be a small Cauchy complete category with finite products. The category $Prod(\mathcal{C}^{op},Set)$ of finite product preserving functors $\mathcal{C}^{op}\to Set$ is a topos iff $\mathcal{C}$ is extensive. Moreover, in this case $Prod(\mathcal{C}^{op},Set)$ coincides with $G(\mathcal{C})$.
=--

Cf. Carboni-Pedicchio-Rosicky ([2001](#CPR01), p.72) and Lawvere ([2008](#Law08), p.498).

From an algebraic perspective, $Prod(\mathcal{C}^{op},Set)$ is the category $\mathcal{C}^{op}-Mod$ of algebras of the multisorted [[algebraic theory]] $\mathcal{C}^{op}$ .

Note, since they are naturally equivalent to $Hom_\mathcal{C}(-,X)$ which preserve all limits (cf. [[hom-functor]]), representable presheaves preserve finite products, in particular, and hence are sheaves for $J_G$; in other words, the Gaeta topology is [[subcanonical topology|subcanonical]].

## Buffalo style algebraic geometry

...

## Related entries

* [[space and quantity]]

* [[functorial geometry]]

* [[bornological topos]]

* [[extensive category]]

* [[superextensive site]]

* [[gros topos]]

* [[Nullstellensatz]]

* [[Zariski topos]]

## Link

* {#Law03}[[F. William Lawvere]], _Grothendieck's 1973 Buffalo Colloquium_, posting to the mailing list _categories@mta.ca_, march 2003. ([link](http://permalink.gmane.org/gmane.science.mathematics.categories/2228))

## References

* {#CPR01}[[Aurelio Carboni]], [[Maria Cristina Pedicchio]], [[Jiri Rosicky]], _Syntactic characterization of various classes of locally presentable categories_ , JPAA **161** (2001) pp.65-90.([zbMath entry](https://zbmath.org/0982.18006))

* {#Gaeta74}[[Federico Gaeta]], _Introduction to Functorial Algebraic Geometry - After a Summer Course by A. Grothendieck: I Affine Algebraic Geometry_ , Suny Buffalo 1974. ([pdf scan](http://matematicas.unex.es/~navarro/res/ifag.pdf))

* {#Law76}[[F. William Lawvere]], _Variable quantities and variable structures in topoi_ , pp.101-131 in Heller, Tierney (eds.), _Algebra, Topology and Category Theory_, Academic Press New York 1976. (pp.109-111)

* {#Law86}[[F. William Lawvere]], _Taking Categories Seriously_, Revista Colombiana de Matemáticas **XX** (1986) pp.147-178. Reprinted as TAC Reprint no.8 (2005) pp.1-24. ([pdf](ftp://ftp.tac.mta.ca/tac/html/tac/reprints/articles/8/tr8.pdf))

* {#Law91}[[F. William Lawvere]], _Some Thoughts on the Future of Category Theory_ , pp.1-13 in LNM **1488** Springer Heidelberg (1991). (pp.3-6)

* {#Law00}[[F. William Lawvere]], _Comments on the development of topos theory_ , pp.715-734 in Jean-Pierre Pier (ed.), _Development of Mathmatics 1950-2000_ , Birkhäuser Basel 2000. Reprinted with author comment as TAC Reprint no.24 (2014). ([abstract](http://www.tac.mta.ca/tac/reprints/articles/24/tr24abs.html); p.727)

* {#Law08}[[F. William Lawvere]], _Core Varieties, Extensivity, and Rig Geometry_ , TAC **20** no.14 (2008) pp. 497–503. ([pdf](http://www.tac.mta.ca/tac/volumes/20/14/20-14.pdf))

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

* {#Menni14a}[[Matías Menni]], _Sufficient Cohesion over Atomic Toposes_ , Cah. Top. Géom. Diff. Cat. **LV** no.2 (2014) pp.113-149. ([preprint](https://sites.google.com/site/matiasmenni/SufCohesion12.pdf?attredirects=0))

* {#Menni22}[[Matías Menni]], *Decidable objects and molecular toposes* (2022) [[arXiv:arXiv:2212.03647](http://arxiv.org/abs/2212.03647)]

[[!redirects gaeta topos]]
[[!redirects Gaeta topology]]
[[!redirects disjoint covering topology]]