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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
Beware the distinction to tangential structure on $X$, where such a lift is considered (only) at the level of underlying classifying spaces.
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.
Given a smooth manifold $X$ of dimension $n$ with frame bundle $Fr(X)$, and given a Lie group monomorphism
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
which is $G$-equivariant.
(Sternberg 64, section VII, def. 2.1).
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:
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 $\sigma$.
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
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.
Let $G \to K$ be a homomorphism of Lie groups. Write
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
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
(the groupoid of twisted c-structures).
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
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.
We discuss the concept in the generality of higher differential geometry, formalized in differential cohesion.
See at differential cohesion – G-Structure
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.
The existence of $G$-structures on tangent bundles of Riemannian manifolds is closely related to these having special holonomy.
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)
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).
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)
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.
An example for a lift of structure groups is
This continues with lifts to the
string group giving string structure;
fivebrane group giving fivebrane 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.
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.
$\;$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)
For discussion of G-structures on closed 8-manifolds see there.
See the list at twisted differential c-structure.
The concept originates around the work of Eli Cartan (Cartan geometry) and
Textbook accounts:
Shlomo Sternberg, chapter VII of: Lectures on differential geometry, Prentice-Hall 1964,
2nd edition AMS 1983 (ISBN:978-0-8218-1385-0)
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:
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
See also
Discussion with an eye towards special holonomy is in
Discussion with an eye towards torsion constraints in supergravity is in
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)
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]
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)
Discussion of G-structures in supergravity and superstring theory:
In relation to torsion constraints in supergravity:
As a way of speaking about Calabi-Yau structure and generalized Calabi-Yau structure:
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
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
Some discussion in higher differential geometry is in section 4.4.2 of
Formalization in modal homotopy type theory is in
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