nLab
Cartan geometry, Supergravity and Branes

under construction – These are presently just notes to go along with a talk here

This entry is going to become an exposition to the topics of

  1. Cartan geometry;

  2. higher dimensional supergravity;

  3. higher dimensional WZW models;

  4. the completed brane scan;

  5. anomaly cancellation in higher WZW models via higher string lifts.

and an indication of how these all naturally flow out of a common source when considered in higher differential supergeometry.

Contents

Cartan geometry

Smooth manifolds, such as spacetimes, are modeled on the real line and its products, the Cartesian spaces d\mathbb{R}^d (see also at geometry of physics – coordinate systems).

It is clear that here it is crucial that n\mathbb{R}^n is regarded with its smooth structure.

But d\mathbb{R}^d also has (smooth) group structure. As such it is the translation group. In Newtonian mechanics this group structure plays a key role, as it allows for instance to say what in means that a particle propagates along a straight trajectory.

With the dawn of general relativity and its “general covariance” it was finally understood that this is a bit too naive: free particles propagate on straight lines in n\mathbb{R}^n only infinitesimally. After every infinitesimal translation, the notion of “straight” needs to be adjusted, due to the “force” of gravity.

This has a straightforward visualization in 2-dimensions: visualize a bumpy surface XX embedded in 3\mathbb{R}^3 (for instance a 2-sphere). Pick a point xXx \in X and visualize the tangent plane to that point, which is an 2\mathbb{R}^2. Now given a curve in XX that emanates from xx, then one may visualize the plane to be “rolling without sliding” on XX such that the point where it touches XX follows the curve.

Under this map from curves in the plane to curves in the XX, straight lines in 2\mathbb{R}^2 now map to geodesics in XX. These are the actual paths that free particles in XX follow.

The most popular way to formalize this is the modern concept of the Levi-Civita connection of the Riemannian metric on XX (which in the above example is induced from the canonical one on the embedding space 3\mathbb{R}^3).

But Élie Cartan‘s original way of speaking about affine connections much closer resembles this picture of a “local model space with group structure” doing “rolling without slipping” over spacetime. This is now mostly called a Cartan connection.

Here one regards the Euclidean group Iso(n)Iso(n) of all isometries of d\mathbb{R}^d. Inside this is the rotation group SO(d)Iso(d)SO(d) \hookrightarrow Iso(d). The quotient group is Cartesian space

dIso(d)/SO(d). \mathbb{R}^d \simeq Iso(d)/SO(d) \,.

The structure of “rolling without sliding” is then formalized in the concept of a Cartan connection by saying

  1. rolling – there is an Iso(d)Iso(d)-principal connection on XX (the group Iso(d)Iso(d), via its action on d\mathbb{R}^d, rolls and slides the Cartesian space around);

  2. without sliding – such that there is a reduction of the structure group to SO(d)SO(d) (this makes the original Iso(n)Iso(n)-bundle be have associated to it the actual d\mathbb{R}^d-fiber bundle);

and such that under this reduction the connection at each point infinitesimally identifies the tangent space of XX with the abstract copy of dIso(d)/SO(d)\mathbb{R}^d \simeq Iso(d)/SO(d).

This has an evident generalization where we consider any inclusion HGH \hookrightarrow G of Lie groups. If one considers instead of Cartesian space d\mathbb{R}^d the Minkowski space d1,1\mathbb{R}^{d-1,1}, then Iso(d1,1)Iso(d-1,1) is the Poincaré group. and SO(d1,1)SO(d-1,1) is the Lorentz group. Now the geodesics via parallel transport along a (SO(d1,1)Iso(d1,1))(SO(d-1,1)\hookrightarrow Iso(d-1,1))-Cartan connection reflect the force of gravity in the theory of general relativity.

Many other combinations (HG)(H \hookrightarrow G) may be considered:

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/HKlein geometryCartan geometryCartan connection
examplesEuclidean group Iso(d)Iso(d)rotation group O(d)O(d)Cartesian space d\mathbb{R}^dEuclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group Iso(d1,1)Iso(d-1,1)Lorentz group O(d1,1)O(d-1,1)Minkowski spacetime d1,1\mathbb{R}^{d-1,1}Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group O(d1,2)O(d-1,2)O(d1,1)O(d-1,1)anti de Sitter spacetime AdS dAdS^dAdS gravity
de Sitter group O(d,1)O(d,1)O(d1,1)O(d-1,1)de Sitter spacetime dS ddS^ddeSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group O(d,t+1)O(d,t+1)conformal parabolic subgroupMöbius space S d,tS^{d,t}conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group GGsubgroup (monomorphism) HGH \hookrightarrow Gquotient (“coset space”) G/HG/Hsuper Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N}Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group GG2-monomorphism HGH \to Ghomotopy quotient G//HG//HKlein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) HGH \to Ghomotopy quotient G//HG//H of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

Higher dimensional supergravity

In the contemporary physics literature this concept of Cartan connection may seem to play an orphaned role, at least if one compares the number of occurrences of the explicit term Cartan connection in the physics literature over that of “Levi-Civita connection”, and certainly when compared to the number of contemporary articles that speak of geodesics without any concept of connection made explicit.

However, this is partly an illusion. Examination of the literature shows that at least as soon as authors consider supergravity, then everything is really secretly formulated in terms of super Cartan geometry, as only in this formulation is the incorporation of fermions and of supersymmetry really natural.

In fact, higher dimensional supergravity such as type II supergravity, heterotic supergravity and 11-dimensional supergravity crucially includes fields which are higher gauge fields – the B2-field, the B6-field, the C3-field, the C6-field, and also the RR-field if one digs deeper.

The only proposal in the physics literature for how to deal with such higher gauge higher supergravity theories geometrically is the D'Auria-Fre formulation of supergravity. And this is secretly higher Cartan geometry.

Here super L-∞ algebra is the joint generalization of Lie algebra to super Lie algebra and to L-∞ algebra.

Liealgebras superLiealgebras L algebras superL algebras. \array{ && Lie\;algebras \\ &\swarrow && \searrow \\ super \;Lie \; algebras && && L_\infty-algebras \\ & \searrow && \swarrow \\ && super\; L_\infty-algebras } \,.

Let 𝕘\mathbb{g} be a finite dimensional vector space and write \wedge^\bullet for the Grassmann algebra of its dual vector space. A differential in this algebra is a map d CE: 𝔤 𝔤d_{CE} \colon \wedge^\bullet \mathfrak{g} \longrightarrow \wedge^\bullet \mathfrak{g} which is a graded derivation of degree 1 and squares to 0, (d CE) 2=0(d_{CE})^2 = 0. One finds that choices of such differentials are equivalent to Lie algebra structures on 𝔤\mathfrak{g}: a degree 1 derivation d CEd_{CE} on 𝔤 *\wedge^\bullet \mathfrak{g}^\ast is equivalently a skew bilinear bracket, and then the condition (d CE) 2=0(d_{CE})^2 = 0 is equivalent to the Jacobi identity, hence makes this bracket a Lie bracket. The resulting dg-algebra (CE(𝔤),d CE)(CE(\mathfrak{g}), d_{CE}) is called the Chevalley-Eilenberg algebra of this Lie algebra.

If we her allow 𝔤\mathfrak{g} to be a super vector space so that 𝔤 *\wedge^\bullet \mathfrak{g}^\ast is now (,/2)(\mathbb{Z}, \mathbb{Z}/2)-bigraded, and require d CEd_{CE} to be of degree (1,even)(1,even) (see at ) then in the same way we find that this is equivalent to the structure of a super Lie algebra.

Now more generally, let 𝔤\mathfrak{g} be a \mathbb{N}-graded super vector space (degreewise finite dimensional, hence of finite type). Then choices of degree (1,even)(1,even) differentials d CEd_{CE} on 𝔤 *\wedge^\bullet \mathfrak{g}^\ast are equivalent to choices of super L-∞ algebra structures on 𝔤\mathfrak{g}.

Finally, still a bit more generally, let 𝔞 0=C (X)\mathfrak{a}_0 = C^\infty(X) be the algebra of functions on some (super) manifold XX, and let 𝕒\mathbb{a} be an \mathbb{N}-graded projective module over 𝕒 0\mathbb{a}_0 which in degree 0 is 𝔞 0\mathfrak{a}_0. Write now Sym 𝔞 0 (𝔞 *)Sym^\bullet_{\mathfrak{a}_0}(\mathfrak{a}^\ast) for the graded-symmetric algebra of the 𝔞 0\mathfrak{a}_0-dual of 𝔞\mathfrak{a}. Now a choice of differential d CEd_{CE} in this (mnathbbR\mnathbb{R}-linear, not necesssarily 𝔞 0\mathfrak{a}_0-linear) gives the structure of a (super) L-∞ algebroid. We write again

CE(𝔞)(Sym 𝔞 0 𝔞 *,d CE) CE(\mathfrak{a}) \coloneqq (Sym^\bullet_{\mathfrak{a}_0} \mathfrak{a}^\ast, d_{CE})

for this Chevalley-Eilenberg algebra.

The full subcategory

sL AlgdsdgAlg op s L_\infty Algd \hookrightarrow sdgAlg^{op}

of that of super-dg-algebras whose underlying graded algebra is free on a graded super vector space in this way we call that of super L L_\infty-algebroids.

examples

super Poincaré Lie algebra

super Minkowski spacetime:

d1,1|N=𝔰ℑ𝔰𝔬(d1,1|N)/𝔰𝔬\mathbb{R}^{d-1,1\vert N} = \mathfrak{sIso}(d-1,1\vert N)/\mathfrak{so}

in fact for higher dimensional supergravity we need extended super Minkowski spacetimes… To motivate these we now consider WZW models.

Higher Wess-Zumino-Witten-type sigma-models

Cohomology

A miracle happens when one passes from Lorentzian geometry to Lorentzian supergeometry.

While the cohomology of Cartesian space and Minkowski spacetime is fairly trivial…

…the cohomology of super-Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert N} turns out to have “exceptional” cohomology classes of degree (p+2)(p+2) for some special combinations of

  1. the dimension dd of spacetime;

  2. the real spin representation NN;

  3. the degree (p+2)(p+2) (where, see in a few lines below, pp turns out to be the dimension of a super p-brane propagating through spacetime).

More precisely, it is the SO(d1,1)SO(d-1,1)-invariant group cohomology of Iso( d1,1|N)Iso(\mathbb{R}^{d-1,1\vert N}) which has these exceptional cocycles, and hence the super Lie algebra cohomology of super Minkowski spacetime. We come to this below.

But the thing is: ever since Paul Dirac it is known that given a cocycle in differential cohomology of degree (p+2)(p+2) on spacetime XX – such as a line bundle with connection representing the electromagnetic field – then this serves as the background field for a sigma-model describing the propagation of a p-brane in XX which is charged under this higher gauge field and feels its forces – such as, for p=0p = 0, an electron subject to the Lorentz force.

Moreover, the consideration of Dirac monopoles and other instantons and black branes shows that given any such higher gauge field, it automatically induces p-branes which are charged under it: the p-brane sigma model turns out to give the perturbation theory for which the black branes are the non-perturbative effects.

This is how most of the p-branes in superstring theory and M-theory were originally found by looking at black brane solutions in higher dimesnional supergravity.

More in detail, the higher dimensional analog of the Lorentz force, felt by these p-branes is given by interaction action functional which is the higher parallel transport (higher volume holonomy) of the background gauge field over the worldvolume of the p-brane. This is also known as the WZW term in higher dimensional WZW theory.

Locally this is a simple phenomenon, and this local picture is what most of the physics textbook will ever consider:

there is a differential curvature (p+2)-form ω\omega on the local model space G/HG/H, and if that is like Minkowski space then it is contractible space hence there is a “higher vector potentialAΩ p+1(G/H)A \in \Omega^{p+1}(G/H) such that ω=dA\omega = \mathbf{d}A, and the action functional in question is just that given by integration of differential forms.

However, already in local model spaces such as super Minkowski spacetime, each ω\omega may not have such a potential that is HH-invariant. Worse, once the p-brane leaves a given local model space G/HXG/H \hookrightarrow X of spacetime, then one needs a higher gauge transformation to connect the interaction terms on the two patches. Still worse, these gauge transformation need to glue (need to “descent” along the cover iG/HX\coprod_i G/H \to X) to a globally defined (“non-perturbative”, free of classical anomalies) higher gauge field on all of XX.

For the traditional 2d WZW model this was eventually fairly widely appreciated, for higher dimensional WZW models this is still much of an open secret.

Before proceeding to the explanation of the global higher WZW terms, we consider some boundary field theory now.

Higher WZW terms

Observation: L L_\infty-cocycle is

𝔤B \mathfrak{g} \longrightarrow \mathbf{B}^{}
Proposition

These have Lie integration to global WZW term

L WZW:G˜B p+1(/Γ) conn \mathbf{L}_{WZW} \;\colon\; \tilde G \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}

(FSS 13, section 5)

Brane intersection laws and Brane condensates in extended super-spacetime

We discuss now a way (FSS 13) to find the boundary conditions and brane condensates via homotopy theory of super L-∞ algebras and via the cobordism hypothesis for local prequantum boundary field theory.

So given a (p+2)(p+2)-cocycle on super Minkowski spacetime, which is just a homomorphism of super L-∞ algebras of the form

d1,1|Nμ p+2B p+1 \mathbb{R}^{d-1,1 \vert N} \stackrel{\mu_{p+2}}{\longrightarrow} \mathbf{B}^{p+1} \mathbb{R}

then we discussed how this is the local (“rational”) expression for a higher WZW term for a sigma-model of a p-brane. A field configuration of that sigma-model is hence a map as on the left of

Σ p+1ϕ d1,1|Nμ p+2B p+1 \array{ \Sigma_{p+1} \stackrel{\phi}{\longrightarrow} \mathbb{R}^{d-1,1 \vert N} \stackrel{\mu_{p+2}}{\longrightarrow} \mathbf{B}^{p+1} \mathbb{R} }

and the composite is (the curvature of) the local Lagrangian of the gauge interaction. In (FSS 13) it is show how to use Lie integration to produce from this the full higher WZW term/prequantum n-bundle/interaction local Lagrangian

L WZW: d1,1|NB p+1(/Γ) conn \mathbf{L}_{WZW} \;\colon\; \mathbb{R}^{d-1,1 \vert N} \longrightarrow \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}

But for the moment it is helpful for simplicity to stay at the rational/infinitesimal/L L_\infty-algebraic level. All what we find here lifts as expected under Lie integration.

So, now according to the cobordism hypothesis for local prequantum boundary field theory, a boundary condition for this Lagrangian is a diagram of the form

Q * κ 0 Σ p+1 ϕ d1,1|N μ p+2 B p+1, \array{ && Q &\longrightarrow& \ast \\ && \downarrow &^{\mathllap{\simeq}}\swArrow_{\mathrlap{\kappa}}& \downarrow^{\mathrlap{0}} \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,,

hence is some space QQ inside spacetime such that the pullback of the higher WZW term/prequantum n-bundle/interaction local Lagrangian for the bulk fields to this space is equipped with a choice of gauge trivialization κ\kappa. Moreover, given this then the fields of the sigma-model on a worldvolume Σ p+1\Sigma_{p+1} with boundary (Σ) pΣ p+1(\partial \Sigma)_p \hookrightarrow \Sigma_{p+1} is a diagram as on the left of

(Σ) p+1 ϕ bdr Q * κ 0 Σ p+1 ϕ d1,1|N μ p+2 B p+1, \array{ (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}& Q &\longrightarrow& \ast \\ \downarrow && \downarrow &^{\mathllap{\simeq}}\swArrow_{\mathrlap{\kappa}}& \downarrow^{\mathrlap{0}} \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,,

We want to understand what kind of object this QQ is that the boundary of the pp-brane may stick to. To that end, observe that by the universal property of homotopy pullbacks, we may decompose the diagram on the right into two diagrams, where the intermediate stage is the extended super Minkowski spacetime ^ d1,1|N{\widehat{\mathbb{R}}}^{d-1,1 \vert N} which, as a super L-∞ algebra, is the homotopy fiber of μ p+2\mu_{p+2}

(Σ) p+1 ϕ bdr Q Φ ^ d1,1|N * 0 Σ p+1 ϕ d1,1|N id d1,1|N μ p+2 B p+1. \array{ (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}& Q &\stackrel{\Phi}{\longrightarrow} & {\widehat{\mathbb{R}}}^{d-1,1 \vert N} &\longrightarrow& \ast \\ \downarrow && \downarrow &{}^{\mathllap{\simeq}}\swArrow& \downarrow &^{\mathllap{\simeq}}\swArrow& \downarrow^{\mathrlap{0}} \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{id}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,.

This is the higher L-∞ extension classified by the cocycle in generalization to the familiar fact that 2-coycles classify plain Lie algebra extensions.

Proposition

The Chevalley-Eilenberg algebras of these ^ d1,1|N{\widehat{\mathbb{R}}}^{d-1,1 \vert N} are precisely the dg-algebras used at least since (D’Auria-Fré 82) in the D'Auria-Fré formulation of supergravity.

This is shown in (FSS 13) using a characterization of homotopy pullbacks in a model structure for L-infinity algebras derived in (FRS 13b).

But now from this we see that on QQ there is itself a sigma-model field Φ\Phi that exhibits QQ itself as a brane, propagating in this extended super Minkowski spacetime. Or rather: that would exhibit this if there were an action functional for this sigma model. By repeating the reasoning, this is given in turn by (if it exists nontrivially) a higher WZW term given by a higher L-∞ cocycle μ p˜+2\mu_{\tilde p + 2} of some further degree p˜+2\tilde p + 2.

(Σ) p+1 ϕ bdr Σ p˜+1 Φ ^ d1,1|N μ p˜+2 B p˜+1 Σ p+1 ϕ d1,1|N id d1,1|N μ p+2 B p+1. \array{ (\partial \Sigma)_{p+1} &\stackrel{\phi_{bdr}}{\longrightarrow}& \Sigma_{\tilde p + 1} &\stackrel{\Phi}{\longrightarrow} & {\widehat{\mathbb{R}}}^{d-1,1 \vert N} &\stackrel{\mu_{\tilde p + 2}}{\longrightarrow}& \mathbf{B}^{\tilde p + 1} \\ \downarrow && \downarrow \\ \Sigma_{p+1} &\stackrel{\phi}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{id}{\longrightarrow}& \mathbb{R}^{d-1,1 \vert N} &\stackrel{\mu_{p+2}}{\longrightarrow}& \mathbf{B}^{p+1} \mathbb{R} } \,.

The completed brane scan

This gives a curious direct identification between L-∞ algebra cohomology and brane intersection laws: every L L_\infty-extension classified by an L L_\infty-cocycle together with a further cocycle on the extension gives a higher WZW sigma-model for some super p-brane wich may end on a super-p˜\tilde p-brane. It turns out that this reasoning reproduces the completed brane scan of superstring theory/M-theory, including notably the D-branes of type II string theory together with the information that the fundamental string may end on them, as well as the sigma-model for the M5-brane with its tensor multiplet fields (FSS 13) and the information that M2-brane may end on it. These maps out much of the key statements about M-theory (and does so in a precise/rigorous way).

𝔇(2p)𝔟𝔯𝔞𝔫𝔢 𝔇(2p+1)𝔟𝔯𝔞𝔫𝔢 𝔰𝔱𝔯𝔦𝔫𝔤 IIA 𝔰𝔱𝔯𝔦𝔫𝔤 IIB 𝔰𝔡𝔰𝔱𝔯𝔦𝔫𝔤 10|N=(1,1) 10|N=(2,0) 𝔰𝔱𝔯𝔦𝔫𝔤 het 6|N=(2,0) 10|N=2 10|N=(1,0) 0|N 11|N=1 𝔪2𝔟𝔯𝔞𝔫𝔢 𝔪5𝔟𝔯𝔞𝔫𝔢 \array{ && &\mathfrak{D}(2p)\mathfrak{brane} &&& \mathfrak{D}(2p+1)\mathfrak{brane} \\ &&& \downarrow && & \downarrow \\ && & \mathfrak{string}_{IIA} && & \mathfrak{string}_{IIB} \\ && & \searrow && \swarrow \\ \mathfrak{sdstring} & && \mathbb{R}^{10 \vert N=(1,1)} & & \mathbb{R}^{10 \vert N=(2,0)} &&& \mathfrak{string}_{het} \\ & \searrow & && \downarrow && & \swarrow \\ && \mathbb{R}^{6 \vert N=(2,0)} && \mathbb{R}^{10 \vert N=2} && \mathbb{R}^{10 \vert N=(1,0)} \\ && & \searrow & \downarrow & \swarrow \\ && && \mathbb{R}^{0\vert N} &\leftarrow& \mathbb{R}^{11\vert N=1} &\leftarrow& \mathfrak{m}2\mathfrak{brane} &\leftarrow& \mathfrak{m}5\mathfrak{brane} }

The brane scan.

The Green-Schwarz type super pp-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):

=d\stackrel{d}{=}p=p =123456789
11M2M5
10D0F1, D1D2D3D4NS5, D5D6D7D8D9
9*\ast
8*\ast
7M2 top{}_{top}
6F1 little{}_{little}, S1 sd{}_{sd}S3
5*\ast
4**
3*

(The first columns follow the exceptional spinors table.)

The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):

=d\stackrel{d}{=}p=p =123456789
11Ψ 2E 2\Psi^2 E^2 on sIso(10,1)Ψ 2E 5+Ψ 2E 2C 3\Psi^2 E^5 + \Psi^2 E^2 C_3 on m2brane
10Ψ 2E 1\Psi^2 E^1 on sIso(9,1)B 2 2+B 2Ψ 2+Ψ 2E 2B_2^2 + B_2 \Psi^2 + \Psi^2 E^2 on StringIIA\cdots on StringIIBB 2 3+B 2 2Ψ 2+B 2Ψ 2E 2+Ψ 2E 4B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4 on StringIIAΨ 2E 5\Psi^2 E^5 on sIso(9,1)B 2 4++Ψ 2E 6B_2^4 + \cdots + \Psi^2 E^6 on StringIIA\cdots on StringIIBB 2 5++Ψ 2E 8B_2^5 + \cdots + \Psi^2 E^8 in StringIIA\cdots on StringIIB
9Ψ 2E 4\Psi^2 E^4 on sIso(8,1)
8Ψ 2E 3\Psi^2 E^3 on sIso(7,1)
7Ψ 2E 2\Psi^2 E^2 on sIso(6,1)
6Ψ 2E 1\Psi^2 E^1 on sIso(5,1)Ψ 2E 3\Psi^2 E^3 on sIso(5,1)
5Ψ 2E 2\Psi^2 E^2 on sIso(4,1)
4Ψ 2E 1\Psi^2 E^1 on sIso(3,1)Ψ 2E 2\Psi^2 E^2 on sIso(3,1)
3Ψ 2E 1\Psi^2 E^1 on sIso(2,1)

Proposition

We may construct the prequantum (p+1)-bundle

L WZW:^ d1,1|NL WZWB p+1(/Γ) conn \mathbf{L}_{WZW} \;\colon\; {\widehat \mathbb{R}}^{d-1,1\vert N} \stackrel{\mathbf{L}_{WZW}}{\longrightarrow} \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}

for all super p-brane sigma-models via a kind of Lie integration. (FSS13)

Anomaly-free higher sigma-models via higher string-lifts of Cartan connections

All would be done and said if spacetime were fixed to be a give extended super Minkowski spacetime.

But of course the key now is that in supergravity instead spacetime XX only locally looks this way, and globally is a Cartan connection for the super Poincaré group or its higher analogs acting on an extended super Minkowski spacetime.

What we need to solve the gluing problem for the higher WZW term and hence cancel its classical anomaly is that on XX itself there is a WZW term

L WZW global:XB p+1(/Γ) conn \mathbf{L}_{WZW}^{global} \colon X \longrightarrow \mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}

such that for a given cover by local model space as given by the Cartan connection background gauge field of gravity and B-field, C-field, etc; this restricts to the canonical one described above, i.e. we need a commuting diagram of the form

\array{ {\widehat{\mathbb{R}}}^{d-1,1\vert N} \\ \downarrow & \searrow^{\mathrlap{\stackrel{\mathbf{L}_{WZW}}} \\ X & \stackrel{\mathbf{L}_{WZW}^{global}}{\longrightarrow} & \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} } \,.

I claim that the solution to this globalization problem works as follows, though I have not written this down in full detail yet. Notice that while I write super-Minkowski spacetimes here just for the heck of it, this may be considered much more generally in higher Cartan geometry.

Consider the restriction of the WZW term to any formal disk (this is synthetically the L \array{ {\widehat{\mathbb{R}}}^{d-1,1\vert N} \\ \downarrow & \searrow^{\mathrlap{\stackrel{\mathbf{L}_{WZW}}}L_\infty-algebra, really), in the sense discussed at Lie differentiation.

L WZW formal:𝔻^ d1,1|N^ d1,1|NL WZWB p+1(/Γ) conn \mathbf{L}_{WZW}^{formal} \;\colon\; {\widehat{\mathbb{D}}}^{d-1,1\vert N} \hookrightarrow {\widehat \mathbb{R}}^{d-1,1\vert N} \stackrel{\mathbf{L}_{WZW}}{\longrightarrow} \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn}

Consider then the quantomorphism n-group of this formal WZW term (FRS 13a), defined by sitting in the homotopy fiber sequence

QuantMorph(L WZW formal)Aut(𝔻 d1,1|N)GL(d|N)L WZW forma()(B p+1(/Γ))Conn(𝔻 d1,1|N) QuantMorph(\mathbf{L}_{WZW}^{formal}) \longrightarrow \mathbf{Aut}(\mathbb{D}^{d-1,1\vert N}) \simeq GL(d\vert N) \stackrel{\mathbf{L}_{WZW}^{forma} \circ (-)}{\longrightarrow} (\mathbf{B}^{p+1} (\mathbb{R}/\Gamma))\mathbf{Conn}(\mathbb{D}^{d-1,1\vert N})

where the rightmost term is the differential concretification of the mapping stack [B p+1(/Γ) conn[\mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}, hence the higher moduli stack of connections on the given formal disk inside extended super Minkowski spacetime.

Intuitively this smooth super ∞-group looks as follows (FRS 13a):

QuantMorph(L WZW formal)={𝔻 d1,1|N 𝔻 d1,1|N L WZW formal L WZW formal B p+1(/Γ) conn} QuantMorph(\mathbf{L}_{WZW}^{formal}) \;=\; \left\{ \array{ \mathbb{D}^{d-1,1\vert N} && \stackrel{\simeq}{\longrightarrow} && \mathbb{D}^{d-1,1\vert N} \\ & {}_{\mathllap{\mathbf{L}_{WZW}^{formal}}}\searrow &\simeq& \swarrow_{\mathrlap{\mathbf{L}_{WZW}^{formal}}} \\ && \mathbf{B}^{p+1} (\mathbb{R}/\Gamma)_{conn} } \right\}

An element in this group is precisely the datum needed to change tangent spaces in a Cartan connection while carrying also the WZWZ term along! See also at parameterized WZW model.

By the discussion in (FRS 13a), this is the general higher and super-analog of string 2-group, fivebrane 6-group, ninebrane 10-group. Hence it makes sense to give it a name like so:

pBrane(d|N)QuantMorph(L WZW formal) p Brane(d\vert N) \coloneqq QuantMorph(\mathbf{L}_{WZW}^{formal})

Now the claim is that the obstruction to globally anomlay free super pp-brane WZW models with respect to a given supergravity Cartan backround is a “super pp-brane”-structure, hence a lift

B(pBrane(d|N)) τ^ X X τ X BGL(d,N) \array{ && \mathbf{B}(p Brane(d\vert N)) \\ & {}^{\mathllap{\widehat {\tau}_X}}\nearrow & \downarrow \\ X &\stackrel{\tau_X}{\longrightarrow} & \mathbf{B}GL(d,N) }

of the map that classifies the frame bundle as discussed at differential cohesion in the section Differential cohesion – Frame bundles.

By the above picture of pBrane(d|N)p Brane(d\vert N) it should be plausbile that this is precisely the data needed to carry the WZW turn around with Cartan’s “rolling without slipping”.

References

Last revised on January 18, 2015 at 21:02:37. See the history of this page for a list of all contributions to it.