nLab G-structure

Idea

A G-structure on an $n$-manifold $M$, for a given structure group $G$, is a $G$-subbundle of the frame bundle (of the tangent bundle) of $M$.

Equivalently, this means that a $G$-structure is a choice of reduction of the canonical structure group GL(n) of the principal bundle to which the tangent bundle is associated along the given inclusion $G \hookrightarrow GL(n)$.

More generally, one can consider the case $G$ is not a subgroup but equipped with any group homomorphism $G \to GL(n)$. If this is instead an epimorphism one speaks of a lift of structure groups.

Both cases, in turn, can naturally be understood as special cases of twisted differential c-structures, which is a notion that applies more generally to principal infinity-bundles.

In this language, a $G$-structure is a lift in classifying stacks (differentiable delooping stacks $\mathbf{B}G$ of Lie groups $G$) of the classifying map of the tangent bundle

$$\array{ && \mathbf{B}G \\ & {}^{ \mathllap{G structure} }\nearrow & \big\downarrow^{ str } \\ X &\underset{ \vdash T X }{\longrightarrow}& \mathbf{B} GL(n) }$$

Beware the distinction to tangential structure on $X$, where such a lift is considered (only) at the level of underlying classifying spaces.

Definition

General

Given a smooth manifold $X$ of dimension $n$ and given a Lie subgroup $G \hookrightarrow GL(n)$ of the general linear group, then a $G$-structure on $X$ is a reduction of the structure group of the frame bundle of $X$ to $G$.

There are many more explicit (and more abstract) equivalent ways to say this, which we discuss below. There are also many evident variants and generalizations.

Notably one may consider reductions of the frames in the $k$th order jet bundle. (e. g. Alekseevskii) This yields order $k$ $G$-structure and the ordinary $G$-structures above are then first order.

Moreover, the definition makes sense for generalized manifolds modeled on other base spaces than just Cartesian spaces. In particular there are evident generalizations to supermanifolds and to complex manifolds.

In terms of subbundles of the frame bundle

Definition

Given a smooth manifold $X$ of dimension $n$ with frame bundle $Fr(X)$, and given a Lie group monomorphism

$$G \longrightarrow GL(\mathbb{R}^n)$$

into the general linear group, then a $G$-structure on $X$ is an $G$-principal bundle $P \to X$ equipped with an inclusion of fiber bundles

$$\array{ P &&\hookrightarrow&& Fr(X) \\ & \searrow && \swarrow \\ && X }$$

which is $G$-equivariant.

(Sternberg 64, section VII, def. 2.1).

Remark

From this perspective, a $G$-structure consists of the collection of all $G$-frames on a manifold. For instance for an orthogonal structure it consists of all those frames which are pointwise an orthonormal basis of the tangent bundle (with respect to the Riemannian metric which is defined by the orthonormal structure).

Accordingly:

Definition

Given $G \hookrightarrow GL(n)$ and given any one frame field $\sigma \colon X \to Fr(X)$ over a manifold $X$, then acting with $G$ on $\sigma$ at each point produces a $G$-subbundle. This is called the $G$-structure generated by the frame field $\sigma$.

In terms of Cartan connections

A $G$-structure equipped with compatible connection data is equivalently a Cartan connection for the inclusion $(G \hookrightarrow \mathbb{R}^n \rtimes G)$.

See at Cartan connection – Examples – G-structures

In higher differential geometry

$G$-structure on a $K$-principal bundle

We give an equivalent definition of $G$-structures in terms of higher differential geometry (“from the nPOV”). This serves to clarify the slightly subtle but important difference between existence and choice of $G$-structure, and seamlessly embeds the notion into the more general context of twisted differential c-structures.

Definition

Let $G \to K$ be a homomorphism of Lie groups. Write

$$\mathbf{c} : \mathbf{B}G \to \mathbf{B}K$$

for the morphism of delooping Lie groupoids ( the smooth moduli stacks of smooth $K$- and $G$-principal bundles, respectively).

For $X$ a smooth manifold (or generally an orbifold or Lie groupoid, etc.) Let $P \to X$ be a $K$-principal bundle and let

$$k \colon X \longrightarrow \mathbf{B}K$$

be any choice of morphism modulating it.

Write $\mathbf{H}(X, \mathbf{B}G)$ etc. for the hom-groupoid of smooth groupoids / smooth stacks . This is equivalently the groupoid of $G$-principal bundles over $X$ and smooth gauge transformations between them.

Then the groupoid of $G$-structure on $P$ (with respect to the given morphism $G \to K$) is the homotopy pullback

$$\mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{k\} \,.$$

$$\array{ \mathbf{c}Struc_{[P]}(X) &\longrightarrow& \ast \\ \downarrow & \swArrow_\simeq& \downarrow^{\mathrlap{k}} \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{H}(X, \mathbf{c})}{\longrightarrow}& \mathbf{H}(X, \mathbf{B}K) }$$

(the groupoid of twisted c-structures).

Remark

If here $k$ is trivial in that it factors through the point, $k \colon X \to \ast \to \mathbf{B}K$ then this homotopy fiber product is $\mathbf{H}(X,K/G)$, where $K/G$ is the coset space (Klein geometry) which itself sits in the homotopy fiber sequence

$$K/G \to \mathbf{B}G \to \mathbf{B}K \,.$$

Example

Specifically, when $X$ is a smooth manifold of dimension $n$, the frame bundle $Fr(X)$ is modulated? by a morphism $\tau_X \colon X \to \mathbf{B} GL(n)$ into the moduli stack for the general linear group $K := GL(n)$. Then for any group homomorphism $G \to GL(n)$, a $G$-structure on $X$ is a $G$-structure on $Fr(X)$, as above.

$G$-Structure on an etale $\infty$-groupoid

We discuss the concept in the generality of higher differential geometry, formalized in differential cohesion.

See at differential cohesion – G-Structure

Properties

Integrability of $G$-structure

Definition

A $G$-structure on a manifold $X$ is called locally flat (Sternberg 64, section VII, def. 24) or integrable (e.g. Alekseevskii) if it is locally equivalent to the standard flat $G$-structure, def. .

This means that there is an open cover $\{U_i \to X\}$ by open subsets of the Cartesian space $\mathbb{R}^n$ such that the restriction of the $G$-structure to each of these is equivalent to the standard flat $G$-structure.

See at integrability of G-structures for more on this

The obstruction to integrability of $G$-structure is the torsion of a G-structure. See there for more.

Relation to special holonomy

The existence of $G$-structures on tangent bundles of Riemannian manifolds is closely related to these having special holonomy.

Theorem

Let $(X,g)$ be a connected Riemannian manifold of dimension $n$ with holonomy group $Hol(g) \subset O(n)$.

For $G \subset O(n)$ some other subgroup, $(X,g)$ admits a torsion-free G-structure precisely if $Hol(g)$ is conjugate to a subgroup of $G$.

Moreover, the space of such $G$-structures is the coset $G/L$, where $L$ is the group of elements suchthat conjugating $Hol(g)$ with them lands in $G$.

This appears as (Joyce prop. 3.1.8)

Examples

Canonical G-structures

Definition

For $G \hookrightarrow GL(n)$ a subgroup, the standard flat $G$-structure on the Cartesian space $\mathbb{R}^n$ is the $G$-structure which is generated, via def. , from the canonical frame field on $\mathbb{R}^n$ (the one which is the identity at each point, under the defining identifications).

Example

For $H \subset G$ any Lie subgroup-inclusion, the canonical quotient space coprojection $G \to G/H$ to the coset space is an $H$-principal bundle that exhibits $H$-structure on $G/H$.

(e.g. Čap-Slovak 09, p. 53)

Reduction of tangent bundle structure

• For the subgroup of $GL(n, \mathbb{R})$ of matrices of positive determinant, a $GL(n, \mathbb{R})^+$-structure defines an orientation.

• For the orthogonal group, an $O(n)$-structure defines a Riemannian metric. (See the discussion at vielbein and at

• For the special linear group, an $SL(n,R)$-structure defines a volume form.

• For the trivial group, an $\{e\}$-structure consists of an absolute parallelism? of the manifold.

• For $n = 2 m$ even, a $GL(m, \mathbb{C})$-structure defines an almost complex structure on the manifold. It must satisfy an integrability condition to be a complex structure.

Lift of tangent bundle structure

An example for a lift of structure groups is

• for the spin group $spin(n)$, a $G$-structure is a spin structure.

This continues with lifts to the

• string group giving string structure;

• fivebrane group giving fivebrane structure.

Reduction of more general bundle structure

• For general $G \to K$, the corresponding notion of Cartan geometry involves $G$-structure on $K$-principal bundles (not necessarily underlying a tangent bundle).

• A $U(n,n) \hookrightarrow O(2n,2n)$-structure is a generalized complex structure;

• For $H_n \to E_{n(n)}$ the inclusion of the maximal compact subgroup into the split real form of an exceptional Lie group, the corresponding structure is an exceptional generalized geometry.

Complex geometric examples

• The choice of $SO(n, \mathbb{C})$ as subgroup of $GL(n, \mathbb{C})$, determines a complex Riemannian structure;

• $CO(n, \mathbb{C}) \hookrightarrow GL(n, \mathbb{C})$, a complex conformal structure;

• $Sp(2n, \mathbb{C})\hookrightarrow GL(2n, \mathbb{C})$, an almost symplectic structure;

• $GL(2, \mathbb{C}) GL(n, \mathbb{C}) \hookrightarrow GL(2n, \mathbb{C}), n \geq 3$, determines an almost quaternionic structure;

• more generally a $GL(m, \mathbb{C}) GL(n, \mathbb{C})$-structure on a $m n$-dimensional manifold is locally identical to a Grassmannian spinor structure.

Special holonomy examples

$\;$normed division algebra$\;$	$\;\mathbb{A}\;$	$\;$Riemannian $\mathbb{A}$-manifolds$\;$	$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$	$\;\mathbb{R}\;$	$\;$Riemannian manifold$\;$	$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$	$\;\mathbb{C}\;$	$\;$Kähler manifold$\;$	$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$	$\;\mathbb{H}\;$	$\;$quaternion-Kähler manifold$\;$	$\;$hyperkähler manifold$\;$
$\;$octonions$\;$	$\;\mathbb{O}\;$	$\;$Spin(7)-manifold$\;$	$\;$G₂-manifold$\;$

(Leung 02)

$G$-Structures on 8-manifolds

For discussion of G-structures on closed 8-manifolds see there.

Higher geometric examples

See the list at twisted differential c-structure.

Related concepts

• structure group

• reduction of structure groups

• special holonomy

• integrability of G-structures

• homothety

• tangential structure

References

Traditional

The concept originates around the work of Eli Cartan (Cartan geometry) and

• Shiing-Shen Chern, The geometry of G-structures, Bull. Amer. Math. Soc. 72(2): 167–219. 1966 (pdf, Euclid)

Textbook accounts:

• Shlomo Sternberg, chapter VII of: Lectures on differential geometry, Prentice-Hall (1964), AMS (1983) [ISBNJ:978-0-8218-1385-0, ams:chel-316, ark:/13960/t1pg9dv6k]

• Shoshichi Kobayashi, Transformation Groups in Differential Geometry 1972, reprinted as: Classics in Mathematics Vol. 70, Springer 1995 (doi:10.1007/978-3-642-61981-6)

• Pierre Molino, Theorie des G-Structures: Le Probleme d’Equivalence, Lecture Notes in Mathematics, Springer (1977) (ISBN:978-3-540-37360-5)

• Andreas Čap, Jan Slovák, Chapter 1 of: Parabolic Geometries I – Background and General Theory, AMS 2009 (ISBN:978-1-4704-1381-1)

See also:

• Pierre Molino, Sur quelques propriétés des G-structures, J. Differential Geom. Volume 7, Number 3-4 (1972), 489-518 (euclid:jdg/1214431168)

Lecture notes:

• Marius Crainic, Chapters 3 and 4 of: Differential geometry course, 2015 (pdf, pdf)

• Federica Pasquotto, Linear $G$-structures by examples (pdf, pdf)

Surveys include

• Dmitry Vladimirovich Alekseevsky, $G$-structure on a manifold in M. Hazewinkel (ed.) Encyclopedia of Mathematics, Volume 4 (eom:G-structure)

See also

• Wikipedia, G-structure

Discussion with an eye towards special holonomy is in

• Dominic Joyce, section 2.6 of Compact manifolds with special holonomy , Oxford Mathematical Monogrophs (200)

Discussion with an eye towards torsion constraints in supergravity is in

• John Lott, The Geometry of Supergravity Torsion Constraints, Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)

Discussion of G-structures more generally on orbifolds:

• A. V. Bagaev, N. I. Zhukova, The Automorphism Groups of Finite Type $G$-Structures on Orbifolds, Siberian Mathematical Journal 44, 213–224 (2003) (doi:10.1023/A:1022920417785)

• Robert Wolak, Orbifolds, geometric structures and foliations. Applications to harmonic maps, Rendiconti del seminario matematico - Universita politecnico di Torino vol. 73/1 , 3-4 (2016), 173-187 (arXiv:1605.04190)

• Sebastian Daza, $G$-Structures on Orbifolds (2021) [pdf, pdf]

On $G$-structure for $G = Br_\infty$ the infinite braid group:

• Frederick R. Cohen, Braid orientations and bundles with flat connections, Inventiones mathematicae 46 (1978) 99–110 [doi:10.1007/BF01393249]

• Jonathan Beardsley, On Braids and Cobordism Theories, Glasgow (2022) [notes: pdf, pdf]

In supergeometry

Discussion of $G$-structures in supergeometry includes

• Dmitri Alekseevsky, Vicente Cortés, Chandrashekar Devchand, Uwe Semmelmann, Killing spinors are Killing vector fields in Riemannian Supergeometry (arXiv:dg-ga/9704002)

  (on Killing spinors as super-Killing vectors)

In supergravity

Discussion of G-structures in supergravity and superstring theory:

In relation to torsion constraints in supergravity:

• John Lott, The Geometry of Supergravity Torsion Constraints, Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)

As a way of speaking about Calabi-Yau structure and generalized Calabi-Yau structure:

• Daniël Prins, On flux vacua, $SU(n)$-structures and generalised complex geometry, Université Claude Bernard – Lyon I, 2015. (arXiv:1602.05415, tel:[phone])

In relation to BPS states/partial reduction of number of supersymmetries under KK-compactification:

• Paul Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59: 169-242, 2011 (arXiv:1006.1536)

  (with application to generalized complex geometry)

• Jerome Gauntlett, Dario Martelli, Stathis Pakis, Daniel Waldram, G-Structures and Wrapped NS5-branes, Commun.Math.Phys. 247 (2004) 421-445 (arxiv:hep-th/0205050)

  (application to flux compactifications)

• Jérôme Gaillard, On $G$-structures in gauge/string duality, 2011 (cronfa:42569 spire:1340775, pdf)

  (with application to holographic QCD)

• Ulf Danielsson, Giuseppe Dibitetto, Adolfo Guarino, KK-monopoles and $G$-structures in M-theory/type IIA reductions, JHEP 1502 (2015) 096 (arXiv:1411.0575)

  (with application to D6-branes/KK-monopoles in M-theory)

• Ruben Minasian, Daniël Prins, Hagen Triendl, Supersymmetric branes and instantons on curved spaces, JHEP 10 (2017) 159 (arXiv:1707.07002)

and specifically so for M-theory on 8-manifolds:

• Chris Isham, Christopher Pope, Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings, Class. Quant. Grav. 5 (1988) 257 (spire:251240, doi:10.1088/0264-9381/5/2/006)

  (focus on Spin(7)-structure)

• Chris Isham, Christopher Pope, Nicholas Warner, Nowhere-vanishing spinors and triality rotations in 8-manifolds, Classical and Quantum Gravity, Volume 5, Number 10, 1988 (cds:185144, doi:10.1088/0264-9381/5/10/009)

  (focus on Spin(7)-structure)

• Cezar Condeescu, Andrei Micu, Eran Palti, M-theory Compactifications to Three Dimensions with M2-brane Potentials, JHEP 04 (2014) 026 (arxiv:1311.5901)

• Daniël Prins, Dimitrios Tsimpis, IIA supergravity and M-theory on manifolds with $SU(4)$ structure, Phys. Rev. D 89.064030 (arXiv:1312.1692)

• Elena Babalic, Calin Lazaroiu, Singular foliations for M-theory compactification, JHEP 03 (2015) 116 (arXiv:1411.3497)

• Elena Babalic, Calin Lazaroiu, Foliated eight-manifolds for M-theory compactification, JHEP 01 (2015) 140 (arXiv:1411.3148)

• Carlos S. Shahbazi, M-theory on non-Kähler manifolds, JHEP 09 (2015) 178 [arXiv:1503.00733]

• Elena Babalic, Calin Lazaroiu, The landscape of $G$-structures in eight-manifold compactifications of M-theory, JHEP 11 (2015) 007 (arXiv:1505.02270)

• Elena Babalic, Calin Lazaroiu, Internal circle uplifts, transversality and stratified $G$-structures, JHEP 11 (2015) 174 (arXiv:1505.05238)

• Robin Terrisse, Dimitrios Tsimpis, $SU(3)$ structures on $S^2$ bundles over four-manifolds, JHEP09 (2017) 133 (arXiv:1707.04636)

• Vikas Yadav, Aalok Misra, On M-Theory Dual of Large-$N$ Thermal QCD-Like Theories up to $\mathcal{O}(R^4)$ and $G$-Structure Classification of Underlying Non-Supersymmetric Geometries (arXiv:2004.07259)

  (relation to holographic QCD)

• Mateo Galdeano, The Geometry and Superconformal Algebras of String Compactifications with a $G$-structure [arXiv:2212.13781]

  (relation to superconformal algebras)

See also

• Johannes Held, Section 3 of: Non-SupersymmetricFlux Compactifications of Heterotic String- and M-theory (pdf)

• George Papadopoulos, Geometry and symmetries of null $G$-structures (arXiv:1811.03500)

• Daniel Waldram, Fluxes, holography and uses of Exceptional Generalized Geometry, talk at Strings 2022 [indico:4940880, slides, video ]

  on generalized G-structures and exceptional generalized geometry of flux compactifications and in AdS/CFT

In complex geometry

• Sergey Merkulov, On group theoretic aspects of the non-linear twistor transform, (pdf)

and his chapter A in

• Yuri Manin, Gauge Field Theory and Complex Geometry, Springer.

• Norman Wildberger, On the complexication of the classical geometries and exceptional numbers, (pdf)

• Jun-Muk Hwang, Rational curves and prolongations of G-structures,arXiv:1703.03160

In higher geometry

Some discussion in higher differential geometry is in section 4.4.2 of

• differential cohomology in a cohesive topos

Formalization in modal homotopy type theory is in

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory, 2017

• Felix Wellen, Cartan Geometry in Modal Homotopy Type Theory (arXiv:1806.05966)