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Higher Cartan geometry is supposed to be the generalization of Cartan geometry to higher geometry; hence the theory of geometric homotopy types (manifolds, orbifolds, Lie groupoids, geometric stacks, smooth groupoids, smooth infinity-groupoids, …) which are locally modeled on homotopy quotients $G/H$ of geometric infinity-groups – the globalized version of higher Klein geometry (see also the survey table below).
More in detail, this means that given a morphism $H \to G$ of suitably geometric ∞-groups then a higher Cartan geometry modeled on the homotopy quotient $G/H$ is a higher geometric space $X$ such as an orbifold, geometric ∞-stack, derived scheme etc. which, in some suitable sense, has its tangent spaces identified with an infinitesimal neighbourhood in $G/H$.
Just like traditional Cartan geometry (in particular in the guise of G-structures) captures a plethora of relevant kinds of geometries ((pseudo-)Riemannian geometry (Cartan 23), conformal geometry, … complex geometry, symplectic geometry, …, parabolic geometry) so higher Cartan geometry is supposed to similarly govern types of higher differential geometry.
A class of examples where aspects of higher Cartan geometry may be seen to secretly underlie traditional discussion is the theory of super p-brane sigma-models on supergravity target-super-spacetimes. This we consider in the examples below. See also at super-Cartan geometry.
It is therefore maybe curious to note that while Cartan geometry as originating in (Cartan 23) drew its motivation from the mathematical formulation of the theory of Einstein gravity, higher Cartan geometry is well motivated by higher dimensional supergravity such as 10d type II supergravity and heterotic supergravity as well as 11-dimensional supergravity.
Here we informally survey motivation for higher Cartan geometry from phenomena and open problems visible in traditional geometry.
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Alternatively, higher Cartan geometry may be motivated intrinsically simply as the result of synthetically formulating Cartan geometry in homotopy type theory. This is the way in which the definition below proceeds. In the Examples we discuss how this abstract theory indeed serves to inform the motivating phenomena listed here.
While a symplectic manifold structure $(X,\omega)$ is an example of an (integrable) G-structure, hence of a Cartan geometry, in many applications symplectic forms $\omega$ are to be refined to complex line bundles with connection, equivalently circle-bundles with connection $\mathbf{L}$ (with curvature $F_{\mathbf{L}} = \omega$), a refinement known as geometric prequantization.
While two differential forms on $X$ are either equal or not, two principal connections on $X$ may be different and still equivalent. The connection $\nabla$ may have non-trivial automorphisms, while a differential form $\omega$ does not. (Readers may be more familiar with this kind of phenomenon from the discussion of the moduli stack of elliptic curves.)
Hence while there is just a set and hence a homotopy 0-type of symplectic forms on $X$, there is a groupoid and hence a homotopy 1-type of principal connections on $X$. It is in this sense that the pair $(X,\nabla)$ involves higher geometry, namely homotopy n-types for $n \gt 0$.
This implies notably that where $\omega$ has a stabilizer group under the diffeomorphism action on $X$ – the symplectomorphism group –, $\mathbf{L}$ instead has a “homotopy stabilizer group”,
consisting of pairs of a diffeomorphism $\phi$ and an isomorphism $\phi^\ast \mathbf{L} \stackrel{\simeq}{\to} \mathbf{L}$. This is called the quantomorphism group.
Hence the prequantum geometry $(X,\mathbf{L})$ is still clearly a geometry of sorts, but not a Cartan geometry. On the other hand, it is still similar enough to be usefully regarded form this perspective:
Just like, by the Darboux theorem, every symplectic manifold $(X,\omega)$ has an atlas by charts isomorphic to a symplectic vector space
so every prequantum line bundle $\mathbf{L}$ on $X$ refining $\omega$ is equivalent over this atlas to the $U(1)$-principal connection given by the globally defined connection for
Moreover, just like the symplectic group $Sp(V,\omega_V)$ is the stabilizer group of $\omega_V$ under the canonical general linear group-action on $V$, so the homotopy stabilizer group of $\mathbf{L}_V$ (the part of the quantomorphism group $QuantMorph(\mathbf{L}_V)$ covering this) is the Mp^c-group, $Mp^c(V,\omega_V) = Mp(V,\omega_V)\underset{\mathbb{Z}/2\mathbb{Z}}{\times}U(1)$, the $U(1)$-version of the metaplectic group $Mp(V,\omega_V)$ ,
In this sense metaplectic quantization is a higher analog of symplectic geometry.
While one may well reason, evidently, about pre-quantization of symplectic manifolds without a general theory of higher Cartan geometry in hand, this class of examples serves as a first blueprint for what higher Cartan geometry should be like, and points the way to its higher-degree generalizations considered below.
In particular, recurring themes are
circle n-bundles with connection $\mathbf{L}$ higher prequantizing definite forms $\omega$;
their homotopy stabilizer groups/higher quantomorphism groups and the infinity-group extension they form and the higher G-structures associated with them.
A particularly interesting example of a pre-quantization as above is the Kac-Moody central extension of loop groups of compact semisimple Lie group $G$ (see here).
Loop groups are naturally symplectic geometries, whose symplectic form is the transgression of the canonical left invariant differential 3-form $\omega_3 = \langle-,[-,-]\rangle$ on $G$:
Similarly their central extension is the transgression to loop space of a higher-degree analog of traditional pre-quantization down on $G$: the canonical left invariant differential 3-form $\omega_3 = \langle-,[-,-]\rangle$ lifts to a circle 2-bundle with connection $\mathbf{L}_3$, whose curvature 3-form is $F_{\mathbf{L}_3} = \omega_3$:
This $\mathbf{L}_3$ is also called the WZW gerbe or WZW term, as its volume holonomy serves as the gauge interaction action functional for the Wess-Zumino-Witten sigma model with target space $G$.
Now the 2-connections on $G$ form a 2-groupoid hence a homotopy 2-type, the pair $(G,\nabla^G)$ may be regarded as being an object in yet a bit higher differential geometry.
Now given a $G$-principal bundle
then a natural question is whether there is a definite parameterization $\mathbf{L}_3^P$ of $\mathbf{L}_3$ to a 2-form connection on $P$ which restricts fiberwise to $\nabla^G$ in a suitable sense up to gauge transformation. Such parameterized WZW terms play a key role in heterotic string theory and equivariant elliptic cohomology.
One finds that such definite parameterizations are equivalent to lifts of structure group of the bundle from $G$ to the homotopy stabilizer group of $\mathbf{L}_3$ under the right $G$-action on itself, and this turns out to be the string 2-group $String(G)$, which is itself the homotopy quotient of the group of based paths of $G$ by the Kac-Moody loop group of $G$. By the above we may also think of this as a Heisenberg 2-group:
Hence a definite parameterization of $\mathbf{L}_3$ over $P$ is a string structure on $P$. The obstruction to that is
for $G = SU(N)$: the second Chern class $\c_2(P)$
for $G = Spin$: the first fractional Pontryagin class $\tfrac{1}{2}p_1(P)$.
These are the obstructions famous from Green-Schwarz anomaly cancellation in heterotic supergravity.
While this class of examples is not yet Cartan geometry proper (higher or not) since the bundle $P$ here is not a tangent bundle, it contains in it the key aspect of definite parameterizations of higher pre-quantized forms related to higher G-structures. Such definite parameterizations turn out to be part of genuine examples of higher Cartan geometry, to which we turn below and key ingredients of higher Cartan geometry apply to both cases.
But more generally, one considers this situation for WZW terms on coset spaces $G/H$, relevant in gauged WZW model.
Provide obstruction classes for definite parameterizations of higher WZW terms.
This we consider below
Often one wants to consider definite parameterizations as above along the tangent bundle of a $V$-manifold $X$, such that the parameterization comes from a global $\mathbf{L}$ on $X$, a definite globalization of a WZW term.
Given a vector space $V$ equipped with a (constant, i.e. translationally left invariant) differential (p+2)-form
a natural question to ask is for a $V$-manifold $X$ (i.e. an $n$-dimensional manifold if $V \simeq \mathbb{R}^n$) to carry a differential form
which is a definite form, definite on $\omega_V$, in that its restriction to each tangent space is equal, up to a $GL(V)$-transformation, to $\omega_V$.
Standard theory of G-structures easily shows that such definite forms correspond to $Stab_{GL(V)}(\omega_V)$-structures on $X$, for $Stab_{GL(V)}(\omega_V)$ the stabilizer group of $\omega_V$ under the canonical $GL(V)$-action (by pullback of differential forms).
For instance if $V = \mathbb{R}^7$ and $\omega_V \in \Omega^3(V)$ is the associative 3-form, then $Stab_{GL(V)}(\omega_V) = G_2$ is the exceptional Lie group G2 and this yields G2-structures.
But in view of the above discussion one is led to re-state this question for the case that $(V,\omega_V)$ is refined to a prequantum (p+1)-bundle $(V,\mathbf{L}_{p+2})$.
Just as a 1-connection is precisely the data needed to define line holonomy, so an $(p+1)$-connection is precisely the data needed to define $(p+1)$-volume holonomy
A definite globalization of such $\mathbf{L}_{p+2}$ over a $V$-manifold $X$ should be a circle (p+1)-connection $\mathbf{L}_{p+1}^X$ on $X$ which suitably, up to the relevant higher gauge transformations, restricts locally to $\mathbf{L}_V$.
For instance for first-order integrable such globalizations one would require that (in particular) for each infinitesimal disk $\mathbb{D}$ in a $V$-cover $U$ we have an equivalence
This problem indeed appears in the formulation of super p-brane sigma models on target super-spacetimes. Here $V$ is a super Minkowski spacetime, $\omega_V$ is an exceptional super Lie algebra cocycle of degree $(p+2)$ and the formulation of the Green-Schwarz sigma model requires that it is refined (higher pre-quantized) to a higher WZW term, a $p$-form connection. The supergravity equations of motion imply a definite globalization $\omega$ of $\omega_V$ of a super-spacetime, but to globally define the GS-WZW model one hence needs to lift this globalization to a $(p+1)$-connection, too (thereby “canceling the classical anomaly” of the model).
These definite globalizations are in particular definite parameterizations, as above, of the restriction of the higher WZW term to the infinitesimal disk-bundle of spacetime,
Notice that for the infinitesimal disk every diffeomorphism is a linear transformation, hence
and therefore by the above a definite globalization determines a G-structure for $G = Stab^h_{GL(V)}(\mathbf{L}_{p+2})$. Conversely, the obstruction to such a structure is an obstruction to a definite globalization.
This construction extends to forgetful functor (an (infinity,1)-functor)
(
For instance in the case of applications to supergravity that we turn to below, these structures are extensions of strutures given by solutions to the super-Einstein equations.
It is here that developing a theory of higher Cartan geometry has much potential, since, while the globalizations of the forms $\omega_V$ have been extensively studied in the literature, the globalization of their pre-quantized refinement to higher WZW-terms $\mathbf{L}_{p+2}$ has traditionally received almost no attention yet. A brief mentioning of the necessity of considering appears for instance in (Witten 86, p. 17), but traditional tools do get one very far in this question.
More precisely, this is the situation for all those branes in the old brane scan which have no tensor-multiplets on the worldvolume, equivalently those on which no other branes may end (such as the string or the M2-brane, but not the D-branes and not the M5-brane). For more general branes, it turns out that the target space itself is a higher geometric space. This leads us to higher Cartan geometry proper. This we turn to below.
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Accordingly, now the symmetries of $\mathbf{L}_{p+2}^X$ form an extension of the isometries of the induced $Stab^h_{GL(V)}(\mathbf{L}_{p+2}|_{\mathbb{D}})$-structure.
One finds that after Lie differentiation these extensions are of the kind konwn in the phyiscs literature as BPS charge extensions.
Classify these refined and generalized BPS states.
This we turn to belowsometryGroups).
So far these examples point to higher Cartan geometry modeled on homomorphisms
where the homotopy stabilizer group $Stab^h_{GL(V)}(\mathbf{L}_{p+2})$ is a infinity-group, but where $V$ is still an ordinary manifold. We now turn (below) to examples that also turn the local model space $V$ into an higher geometric homotopy type. But first we need a little interlude.
Before further motivating ever higher Cartan geometry, it serves to pause and realize that while passing from manifolds to stacks, we are in particular first of all generalizing to sheaves. So even before going higher in homotopy degree, one may ask how much of Cartan geometry may be formulated in sheaf toposes, first over the site of smooth manifolds itself, which leads to Cartan geometry in the generality of smooth spaces, and next over sites other than that of smooth manifolds – super-Cartan geometry.
One key example for this is supergeometry. Where a major application of traditional Cartan geometry is its restriction to orthogonal structures encoding (pseudo-)Riemannian geometry of particular relevance in the theory of gravity, the analogous orthogonal structures in supergeometry serve to set up the theory of supergravity.
More in detail, after picking a dimension $d\in \mathbb{N}$ and writing $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ for the Poincaré Lie algebra, then a choice of “number of supersymmetries” is a choice of real spin representation $N$. Then the direct sum
regarded as a super vector space with $N$ in odd degree becomes a super Lie algebra by letting the $[even,odd]$ bracket to be given by the defining action and by letting the $[odd,odd]$ bracket be given by a canonically induced bilinear and $\mathfrak{o}$-equivariant pairing – the super Poincaré Lie algebra. This still canonical contains the Lorentz Lie algebra $\mathfrak{o}(\mathbb{R}^{d-1,1})$ and the quotient
is called super Minkowski spacetime (equipped with its super translation Lie algebra structure).
From this, a super-Cartan geometry is defined in direct analogy to the Cartan formulation of Riemannian geometry
Cartan geometry | $\mathfrak{g}$ | $\mathfrak{h}$ | $\mathfrak{g}/\mathfrak{h}$ |
---|---|---|---|
pseudo-Riemannian geometry/Einstein gravity | $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1}$ |
supergravity | $\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1\vert N}$ |
Indeed, all the traditional literature on supergravity (e.g. (Castellani-D’Auria-Fré 91)) is phrased, more or less explicitly, in terms of Cartan connections for the inclusion of the Lorentz group into the super Poincaré group, this being the formalization of what physicists mean when saying that they pass to “local supersymmetry”.
It so happens that from within such super-Cartan geometry there appear some of the most interesting examples of what should be higher Cartan geometry, hence higher super-Cartan geometry. This we turn to below.
A traditional Cartan connection, being a principal connection satisfying some extra conditions, is locally (on some chart $U \to X$) in particular a Lie algebra valued differential form $A \in \Omega^1(U,\mathfrak{g})$. Following Cartan, this is equivalently a homomorphism of dg-algebras of the form
from the Weil algebra of the Lie algebra $\mathfrak{g}$ to the de Rham complex of $U$, equivalently a homomorphism of just graded algebras
from the Chevalley-Eilenberg algebra of $\mathfrak{g}$. (Requiring this second morphism to also respect the dg-algebra structure, hence the differential, is equivalent to requiring the curvature form $F_A$ to vanish, hence to the connection being a flat connection).
In particular for the description of supergravity superspacetimes one considers this for $\mathfrak{g} = \mathfrak{Iso}(\mathbb{R}^{d-1,1|N})$ the super Poincaré Lie algebra of some super Minkowski spacetime $\mathbb{R}^{d-1|N}$. This serves to encode a Levi-Civita connection as for ordinary gravity modeled by ordinary orthogonal structure Cartan geometry, together with the gravitino field.
In detail, the Chevalley-Eilenberg algebra $CE(\mathfrak{Iso}(\mathbb{R}^{10,1|N=1}))$ for 11-dimensional Minkowski spacetime turned super via the unique irreducible 32-dimensional spin representation (see here) is freely generated as a graded commutative superalgebra on
elements $\{e^a\}_{a = 1}^{11}$ and $\{\omega^{ a b}\}$ of degree $(1,even)$;
and elements $\{\psi^\alpha\}_{\alpha = 1}^{32}$ of degree $(1,odd)$
and as a differential graded algebra its differential $d_{CE}$ is determined by the equations
An algebra homomorphism as above sends these generators to differential forms of the corresponding degree, the vielbein
whe spin connection
and the gravitino
But a key aspect of higher dimensional supergravity theories is that their field content necessarily includes, in addition to the graviton and the gravitino, higher differential n-form fields, notably the 2-fom B-field of 10-dimensional type II supergravity and heterotic supergravity as well as the 3-form C-field of 11-dimensional supergravity.
This means that these higher dimensional supergravity theories are not in fact entirely described by super-Cartan geometry. This is to be contrasted with the fact that the very motivation for Cartan geometry, in the original article (Cartan 23), was the mathematical formulation of the theory of gravity (general relativity).
Now a key insight due to (D’Auria-Fré-Regge 80, D’Auria-Fré 82) was that the “tensor multiplet” fields of higher dimensional supergravity theories as above are naturally brought into the previous perspective if only one allows more general Chevalley-Eilenberg algebras.
Namely, we may add to the above CE-algabra
and extend the differential to that by the formula
This still squares to zero due to the remarkable property of 11d super Minkowski spacetime by which $\frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \in CE^4(\mathfrak{Iso}(10,1|N=1))$ is a representative of an exception super-Lie algebra cohomology class. (The collection of all these exceptional classes constitutes what is known as the brane scan).
In the textbook (Castellani-D’Auria-Fré 91) a beautiful algorithm for constructing and handling higher supergravity theories based on such generalized CE-algebras is presented, but it seems fair to say that the authors struggle a bit with the right mathematical perspective to describe what is really happening here.
But from a modern perspective this becomes crystal clear: these generalized CE algebras are CE-algebras not of Lie algebras but of strong homotopy Lie algebra, hence of L-infinity algebras, in fact of Lie (p+1)-algebras for $(p+1)$ the degree of the relevant differential form field.
Specifically, we may write the above generalized CE-algebra with the extra degree-3 generator $c_3$ as the CE-algebra $CE(\mathfrak{m}2\mathfrak{brane})$
of the supergravity Lie 3-algebra $\mathfrak{m}2\mathfrak{brane}$.
Now a morphism
encodes graviton and gravitino fields as above, but in addition it encodes a 3-form
whose curvature
satisfies a modified Bianchi identity, crucial for the theory of 11-dimensional supergravity (D’Auria-Fré 82).
So this collection of differential form data is no longer a Lie algebra valued differential form, it is an L-infinity algebra valued differential form, with values in the supergravity Lie 3-algebra.
The quotient
is known as extended super Minkowski spacetime.
The Lie integration of this is a smooth 3-group $G$ which receives a map from the Lorentz group.
This means that a global description of the geometry which (Castellani-D’Auria-Fré 91) discuss locally on charts has to be a higher kind of Cartan geometry which is locally modeled not just on cosets, but on the homotopy quotients of (smooth, supergeometric, …) infinity-groups.
Once such a higher Cartan super-spacetime $X$ as above has been obtained, then we are back to the above question of constructing definite globalizations of WZW terms over it.
Indeed, the super p-brane sigma-models of the D-branes and the M5-brane have WZW terms defined not on plain super Minkowski spacetimes, but on the above extended super Minkowski spacetimes. For instance the WZW term of the M5-brane sigma model is a higher prequantization of the following 7-form (D’Auria-Fré 82)
on the above extended super Minkowski spacetime, where $c_3$ is the extra degree-3 generator discussed above.
Under Lie integration this becomes (FSS 13) a degree-7 WZW term defined on a supergeometric 3-group $G/H$ and defining the M5-brane sigma model on a curved supergravity target space means to construct definite globalizations of this over higher Cartan geometries $X$ modeled on this homotopy quotient $G/H$.
The result $(X,\mathbf{L}^X_7)$ is a pair which is still analogous to the symplectic geometries that we started with, but is now in higher geometric homotopy theory in every possible sense.
Computing for this case the higher extensions of isometries as above, one finds (dcct, sections 1.2.11.3 and 1.2.15.3.3) the quantomorphism n-group for the supergeometric 7-group with is the Lie integration of the M-theory Lie algebra of $X$, witnessing the degree of $X$ being a “BPS state” of 11d supergravity. These BPS states are known to be an immensely rich mathematical topic (e.g. via their “wall crossing phenomena”), but one sees here that it is but the local and infinitesimal shadow of a much richer structure: higher isometries in higher super-Cartan geometry.
In terms of the physics this refinement corresponds to classical anomaly-cancellation of super p-brane sigma models, a problem that is by and large open.
Provide classical anomaly cancellation for super p-brane sigma-models such as the M5-brane.
(…)
see at differential cohesion the section structures.
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Notice that ordinary gravity can be understood as the theory of $(O(d,1) \hookrightarrow Iso(d,1))$-Cartan geometry, where $Iso(d,1)$ is the Poincare group and $O(d,1)$ the orthogonal group of Minkowski space. This is called the first order formulation of gravity.
One can read the D'Auria-Fre formulation of supergravity as saying that higher dimensional supergravity is analogously given by higher Cartan supergeometry. See there and see the examples at higher Klein geometry for more on this.
Traditional Cartan geometry goes back to
There is secretly a good bit of higher super-Cartan geometry in the supergravity textbook
based on results and observations due to
Riccardo D'Auria, Pietro Fré Tullio Regge, Graded Lie algebra, cohomology and supergravity, Riv. Nuov. Cim. 3, fasc. 12 (1980) (spire)
Riccardo D'Auria, Pietro Fré Geometric Supergravity in D=11 and its hidden supergroup, Nuclear Physics B201 (1982) 101-140
Mentioning of the need for definite globalizations of WZW terms is (ever so briefly) in
That there ought to be a systematic study of higher Klein geometry and higher Cartan geometry has been amplified by David Corfield since 2006.
Construction of the higher WZW terms on homotopy quotients $G/H$ of higher super-gorups is due to
with more details in (dcct).
A formalization of higher Cartan geometry via differential cohesion is in
Urs Schreiber, Higher Cartan Geometry, Harmonic analysis seminar, Charles University Prague, 2015
Formalization of this in homotopy type theory is in