**Higher Cartan Geometry: Motivation.** The first session of the higher Cartan geometry seminar tomorrow will informally survey motivation and examples for the theory.

http://ncatlab.org/nlab/show/higher+Cartan+geometry#MotivationWe consider some motivation for higher Cartan geometry from phenomena and open problems visible already in traditional geometry.

1) Pre-quantization of symplectic geometry

2) Higher pre-quantization and Parameterized WZW terms

3) Interlude: Super Cartan geometry

4) Higher pre-quantization and Globalized WZW terms

5) Higher Cartan connections and stacky Cartan geometry

6) Combining it all: Definite higher WZW terms on stacky Cartan geometries

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 below proceeds. In the Examples we discuss how this abstract theory indeed serves to inform the motivating phenomena lister here.

**1) Pre-quantization of symplectic geometry**While a symplectic manifold structure (X,ω) is an example of an (integrable) G-structure, hence of a Cartan geometry, in many applications symplectic forms ω are to be refined to circle-bundles with connection ∇ (with curvature F∇=ω), a refinement known as prequantization. Notice that while two differential forms on X are either equal or not, two principal connections on X may be different and still equivalent: 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,∇) involves higher geometry, namley homotopy n-types for n>0.

In this way the pair (X,∇) 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 has an atlas by charts isomorphic to (R2n,dpi∧dqi), so every prequantum line bundle ∇ on X refining ω is equivalent over this atlas to the U(1)-principal connection given by the globally defined connection for θ≔pi∧dqi.

Moreover, just like the affine symplectic group is the stabilizer group of the local model dpi∧dqi under the canonical Euclidean group-action on R2n, so the homotopy stabilizer group of θ covering this is the extended affine symplectic group which is the semidirect product of the Heisenberg group and the metaplectic group. In this sense metaplectic quantization is a pre-quantized higher analog of symplectic structure.

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 considerd below.

**2) Higher pre-quantization and Parameterized WZW terms**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). These may be understood as the transgression to loop space of a higher-degree analog of traditional pre-quantization down on G: the canonical left invariant differential 3-form ω3=⟨−,[−,−]⟩ lifts to a circle 2-bundle with connection ∇G. This 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.

Since the 2-connections on G form a 2-groupoid hence a homotopy 2-type, the pair (G,∇G) may be regarded as being an object in yet a bit higher differential geometry.

Now given a G-principal bundle P→X, then a natural question is whether there is a definite parameterization of ∇G to a 2-form connection on P which restricts fiberwise to ∇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 ∇G under the right G-action on itself, and this turns out to be the string 2-group String(G).

While this class of examples is not yet Cartan geometry proper (higher or not) since the bundle here is not a tangent bundle, it contains in it the key aspect of definitie 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.

More generally, one considers this situation for WZW terms on coset spaces G/H (the gauged WZW model), and their definite parameterization over G/H-fiber bundles.

**3) Interlude: Super-Cartan geometry**Before further motiviating 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 geoemtry in the generality of smooth spaces, and next over sites other than that of smooth manifolds.

One key example for this is supergeometry. Where a major application of traditonal 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. 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.

**4) Higher pre-quantization and Globalized WZW terms**Given a vector space V equipped with a (constant, i.e. translationally left invariant) differential (p+2)-form

ϕ∈Ωp+2(V)

a natural question to ask is for a V-manifold X (i.e. an n-dimensional manifold if V≃Rn) to carry a differential form

ω∈Ω2(X)

which is a definite form, definite on ϕ, in that its restriction to each tangent space is equal, up to a GL(V)-transformation, to ϕ.

Standard theory of G-structures easily shows that such definite forms correspond to StabGL(V)(ϕ)-structures on X, for StabGL(V)(ϕ) the stabilizer group of ϕ under the canonical GL(V)-action (by pullback of differential forms).

For instance if V=R7 and ϕ∈Ω3(V) is the associative 3-form, then StabGL(V)(ϕ)=G2 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 ϕ is refined to a prequantum (p+1)-bundle ∇. A definite globalization of this over a V-manifold X should be a circle (p+1)-connection on X which suitably, up to the relavant higher gauge transformations, restrics locally to ∇.

This problem indeed appears in the formulation of super p-brane sigma models on target super-spacetimes. Here V is a super Minkowski spacetime, ϕ 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-qauntized) to a higher WZW term, a p-form connection. The supergravity equations of motion imply a definite globalization ω of ϕ 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, and hence they imply higher G-structure along the above lines.

It is here that developing a theory of higher Cartan geometry has real potential, since, while the globalizations of the forms ϕ have been extensively studied in the literature, the globalization of their pre-quantized refinement to higher WZW-terms 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 geoemtric space. This leads us to higher Cartan geometry proper. This we turn to now.

**5) Higher Cartan connections and Stacky Cartan geometries**A traditonal Cartan connection, being a principal connection satisfying some extra conditions, is locally (on some chart U→X) in particular a Lie algebra valued differential form A∈Ω1(U,g). Following Cartan, this is equivalently a homomorphism of dg-algebras of the form

Ω∙(U)⟵W(g):A

from the Weil algebra of the Lie algebra g to the de Rham complex of U, equivalently a homomorphism of just graded algebras

Ω∙(U)⟵CE(g):A

from the Chevalley-Eilenberg algebra of g. (Requiring this second morphism to also respect the dg-algebra structure, hence the differential, is equivalent to requiring the curvature form FA to vanish, hence to the connection being a flat connection).

In particular for the description of supergravity superspacetimes one considers this for g=Iso(Rd−1,1|N) the super Poincaré Lie algebra of some super Minkowski spacetime Rd−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(Iso(R10,1|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 {ea}11a=1 and {ωab} of degree (1,even);

and elements {ψα}32α=1 of degree (1,odd)

and as a differential graded algebra its differential dCE is determined by the equations

dCEωab=ωab∧ωbc

dCEea=ωab∧eb+i2ψ¯Γaψ.

An algebra homomorphism as above sends these generators to differential forms of the corresponding degree, the vielbein

Ea≔A(ea)∈Ω(1,even)(U),

whe spin connection

Ωab≔A(ωab)∈Ω(1,even)(U)

and the gravitino

Ψα≔A(ψα)∈Ω(1,odd)(U).

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

a single generator c3 of degree (3,even)

and extend the differential to that by the formula

dCEc3=12ψ¯Γab∧ψ∧ea∧eb.

This still squares to zero due to the remarkable property of 11d super Minkowski spacetime by which 12ψ¯Γab∧ψ∧ea∧eb∈CE4(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, me may write the above generalized CE-algebra with the extra degree-3 generator c3 as the CE-algebra CE(m2brane)

of the supergravity Lie 3-algebra m2brane.

Now a morphism

Ω∙(U)←−−CE(m2brane):A

encodes graviton and gravitino fields as above, but in addition it encodes a 3-form

C3≔A(c3)∈Ω(3,even)(U)

whose curvature

G4=dC3+12Ψ¯Γab∧Ψ∧Ea∧Eb

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

Rˆ10,1|N=1≔g/h=m2brane/o(R10,1|N=1)

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.

**6) Combining it all: Definite higher WZW terms on stacky Cartan geometries**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 7-form

ω7≔12ψ¯∧Γa1⋯a5ψ∧ea1∧⋯∧ea5+132ψ¯Γa1a2ψ∧ea1∧ea2∧c3∈CE7(Rˆ10,1|N=1)

on the above extended super-Minkowski spacetime, where c3 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,LX) 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.

There is much interesting structure known to be hidden in this pair. One finds (dcct, sections 1.2.11.3 and 1.2.15.3.3) that the higher analog of symplectomorphisms, namely higher quantomorphisms

ϕ:X−→−≃X

η:ϕ∗LX−→−≃L

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. “wall crossing”), but one sees ehre that it is but the local and infinitesimal shadow of a much richer structure: higher isometries in higher super-Cartan geometry.