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
higher dimensional supergravity;
the completed brane scan;
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.
Smooth manifolds, such as spacetimes, are modeled on the real line and its products, the Cartesian spaces $\mathbb{R}^d$ (see also at geometry of physics – coordinate systems).
It is clear that here it is crucial that $\mathbb{R}^n$ is regarded with its smooth structure.
But $\mathbb{R}^d$ 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 $\mathbb{R}^n$ 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 $X$ embedded in $\mathbb{R}^3$ (for instance a 2-sphere). Pick a point $x \in X$ and visualize the tangent plane to that point, which is an $\mathbb{R}^2$. Now given a curve in $X$ that emanates from $x$, then one may visualize the plane to be “rolling without sliding” on $X$ such that the point where it touches $X$ follows the curve.
Under this map from curves in the plane to curves in the $X$, straight lines in $\mathbb{R}^2$ now map to geodesics in $X$. These are the actual paths that free particles in $X$ follow.
The most popular way to formalize this is the modern concept of the Levi-Civita connection of the Riemannian metric on $X$ (which in the above example is induced from the canonical one on the embedding space $\mathbb{R}^3$).
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 $Iso(n)$ of all isometries of $\mathbb{R}^d$. Inside this is the rotation group $SO(d) \hookrightarrow Iso(d)$. The quotient group is Cartesian space
The structure of “rolling without sliding” is then formalized in the concept of a Cartan connection by saying
rolling – there is an $Iso(d)$-principal connection on $X$ (the group $Iso(d)$, via its action on $\mathbb{R}^d$, rolls and slides the Cartesian space around);
without sliding – such that there is a reduction of the structure group to $SO(d)$ (this makes the original $Iso(n)$-bundle be have associated to it the actual $\mathbb{R}^d$-fiber bundle);
and such that under this reduction the connection at each point infinitesimally identifies the tangent space of $X$ with the abstract copy of $\mathbb{R}^d \simeq Iso(d)/SO(d)$.
This has an evident generalization where we consider any inclusion $H \hookrightarrow G$ of Lie groups. If one considers instead of Cartesian space $\mathbb{R}^d$ the Minkowski space $\mathbb{R}^{d-1,1}$, then $Iso(d-1,1)$ is the Poincaré group. and $SO(d-1,1)$ is the Lorentz group. Now the geodesics via parallel transport along a $(SO(d-1,1)\hookrightarrow Iso(d-1,1))$-Cartan connection reflect the force of gravity in the theory of general relativity.
Many other combinations $(H \hookrightarrow G)$ may be considered:
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.
Let $\mathbb{g}$ 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 $d_{CE} \colon \wedge^\bullet \mathfrak{g} \longrightarrow \wedge^\bullet \mathfrak{g}$ which is a graded derivation of degree 1 and squares to 0, $(d_{CE})^2 = 0$. One finds that choices of such differentials are equivalent to Lie algebra structures on $\mathfrak{g}$: a degree 1 derivation $d_{CE}$ on $\wedge^\bullet \mathfrak{g}^\ast$ is equivalently a skew bilinear bracket, and then the condition $(d_{CE})^2 = 0$ is equivalent to the Jacobi identity, hence makes this bracket a Lie bracket. The resulting dg-algebra $(CE(\mathfrak{g}), d_{CE})$ is called the Chevalley-Eilenberg algebra of this Lie algebra.
If we her allow $\mathfrak{g}$ to be a super vector space so that $\wedge^\bullet \mathfrak{g}^\ast$ is now $(\mathbb{Z}, \mathbb{Z}/2)$-bigraded, and require $d_{CE}$ to be of degree $(1,even)$ (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 $\mathfrak{g}$ be a $\mathbb{N}$-graded super vector space (degreewise finite dimensional, hence of finite type). Then choices of degree $(1,even)$ differentials $d_{CE}$ on $\wedge^\bullet \mathfrak{g}^\ast$ are equivalent to choices of super L-∞ algebra structures on $\mathfrak{g}$.
Finally, still a bit more generally, let $\mathfrak{a}_0 = C^\infty(X)$ be the algebra of functions on some (super) manifold $X$, and let $\mathbb{a}$ be an $\mathbb{N}$-graded projective module over $\mathbb{a}_0$ which in degree 0 is $\mathfrak{a}_0$. Write now $Sym^\bullet_{\mathfrak{a}_0}(\mathfrak{a}^\ast)$ for the graded-symmetric algebra of the $\mathfrak{a}_0$-dual of $\mathfrak{a}$. Now a choice of differential $d_{CE}$ in this ($\mnathbb{R}$-linear, not necesssarily $\mathfrak{a}_0$-linear) gives the structure of a (super) L-∞ algebroid. We write again
for this Chevalley-Eilenberg algebra.
The full subcategory
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 $L_\infty$-algebroids.
examples
$\mathbb{R}^{d-1,1\vert N} = \mathfrak{sIso}(d-1,1\vert N)/\mathfrak{so}$
in fact for higher dimensional supergravity we need extended super Minkowski spacetimes… To motivate these we now consider WZW models.
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 $\mathbb{R}^{d-1,1\vert N}$ turns out to have “exceptional” cohomology classes of degree $(p+2)$ for some special combinations of
the real spin representation $N$;
the degree $(p+2)$ (where, see in a few lines below, $p$ turns out to be the dimension of a super p-brane propagating through spacetime).
More precisely, it is the $SO(d-1,1)$-invariant group cohomology of $Iso(\mathbb{R}^{d-1,1\vert N})$ 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 $(p+2)$ on spacetime $X$ – 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 $X$ which is charged under this higher gauge field and feels its forces – such as, for $p = 0$, 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 $\omega$ on the local model space $G/H$, and if that is like Minkowski space then it is contractible space hence there is a “higher vector potential” $A \in \Omega^{p+1}(G/H)$ such that $\omega = \mathbf{d}A$, 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 $\omega$ may not have such a potential that is $H$-invariant. Worse, once the p-brane leaves a given local model space $G/H \hookrightarrow X$ 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 $\coprod_i G/H \to X$) to a globally defined (“non-perturbative”, free of classical anomalies) higher gauge field on all of $X$.
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.
Observation: $L_\infty$-cocycle is
These have Lie integration to global WZW term
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 $(p+2)$-cocycle on super Minkowski spacetime, which is just a homomorphism of super L-∞ algebras of the form
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
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
But for the moment it is helpful for simplicity to stay at the rational/infinitesimal/$L_\infty$-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
hence is some space $Q$ 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 $\kappa$. Moreover, given this then the fields of the sigma-model on a worldvolume $\Sigma_{p+1}$ with boundary $(\partial \Sigma)_p \hookrightarrow \Sigma_{p+1}$ is a diagram as on the left of
We want to understand what kind of object this $Q$ is that the boundary of the $p$-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 ${\widehat{\mathbb{R}}}^{d-1,1 \vert N}$ which, as a super L-∞ algebra, is the homotopy fiber of $\mu_{p+2}$
This is the higher L-∞ extension classified by the cocycle in generalization to the familiar fact that 2-coycles classify plain Lie algebra extensions.
The Chevalley-Eilenberg algebras of these ${\widehat{\mathbb{R}}}^{d-1,1 \vert N}$ 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 $Q$ there is itself a sigma-model field $\Phi$ that exhibits $Q$ 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 $\mu_{\tilde p + 2}$ of some further degree $\tilde p + 2$.
This gives a curious direct identification between L-∞ algebra cohomology and brane intersection laws: every $L_\infty$-extension classified by an $L_\infty$-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-$\tilde p$-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).
The brane scan.
The Green-Schwarz type super $p$-brane sigma-models (see at table of branes for further links and see at The brane bouquet for the full classification):
$\stackrel{d}{=}$ | $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
11 | M2 | M5 | ||||||||
10 | D0 | F1, D1 | D2 | D3 | D4 | NS5, D5 | D6 | D7 | D8 | D9 |
9 | $\ast$ | |||||||||
8 | $\ast$ | |||||||||
7 | M2${}_{top}$ | |||||||||
6 | F1${}_{little}$, S1${}_{sd}$ | S3 | ||||||||
5 | $\ast$ | |||||||||
4 | * | * | ||||||||
3 | * |
(The first columns follow the exceptional spinors table.)
The corresponding exceptional super L-∞ algebra cocycles (schematically, without prefactors):
$\stackrel{d}{=}$ | $p =$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|
11 | $\Psi^2 E^2$ on sIso(10,1) | $\Psi^2 E^5 + \Psi^2 E^2 C_3$ on m2brane | ||||||||
10 | $\Psi^2 E^1$ on sIso(9,1) | $B_2^2 + B_2 \Psi^2 + \Psi^2 E^2$ on StringIIA | $\cdots$ on StringIIB | $B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4$ on StringIIA | $\Psi^2 E^5$ on sIso(9,1) | $B_2^4 + \cdots + \Psi^2 E^6$ on StringIIA | $\cdots$ on StringIIB | $B_2^5 + \cdots + \Psi^2 E^8$ in StringIIA | $\cdots$ on StringIIB | |
9 | $\Psi^2 E^4$ on sIso(8,1) | |||||||||
8 | $\Psi^2 E^3$ on sIso(7,1) | |||||||||
7 | $\Psi^2 E^2$ on sIso(6,1) | |||||||||
6 | $\Psi^2 E^1$ on sIso(5,1) | $\Psi^2 E^3$ on sIso(5,1) | ||||||||
5 | $\Psi^2 E^2$ on sIso(4,1) | |||||||||
4 | $\Psi^2 E^1$ on sIso(3,1) | $\Psi^2 E^2$ on sIso(3,1) | ||||||||
3 | $\Psi^2 E^1$ on sIso(2,1) |
Proposition
We may construct the prequantum (p+1)-bundle
for all super p-brane sigma-models via a kind of Lie integration. (FSS13)
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 $X$ 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 $X$ itself there is a WZW term
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 $\array{ {\widehat{\mathbb{R}}}^{d-1,1\vert N} \\ \downarrow & \searrow^{\mathrlap{\stackrel{\mathbf{L}_{WZW}}}L_\infty$-algebra, really), in the sense discussed at Lie differentiation.
Consider then the quantomorphism n-group of this formal WZW term (FRS 13a), defined by sitting in the homotopy fiber sequence
where the rightmost term is the differential concretification of the mapping stack $[\mathbf{B}^{p+1}(\mathbb{R}/\Gamma)_{conn}$, 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):
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:
Now the claim is that the obstruction to globally anomlay free super $p$-brane WZW models with respect to a given supergravity Cartan backround is a “super $p$-brane”-structure, hence a lift
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 $p Brane(d\vert N)$ it should be plausbile that this is precisely the data needed to carry the WZW turn around with Cartan’s “rolling without slipping”.
Riccardo D'Auria, Pietro Fré Geometric supergravity in $D = 11$ and its hidden supergroup, Nuclear Physics B201 (1982) 101-140 (Geometric Supergravity in D=11 and its hidden supergroup)
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
Domenico Fiorenza, Chris Rogers, Urs Schreiber Higher geometric prequantum theory
Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)
Domenico Fiorenza, Hisham Sati, Urs Schreiber The brane bouquet, International Journal of Geometric Methods in Modern Physics, Vol. 12 (2015) 1550018 (arXiv:1308.5264)