Types of quantum field thories
A quantum field theory of supergravity is similar to the theory of gravity, but where (in first order formulation) the latter is given by an action functional (the Einstein-Hilbert action functional) on the space of connections (over spacetime) with values in the Poincare Lie algebra , supergravity is defined by an extension of this to an action functional on the space of connections with values in the super Poincare Lie algebra . One says that supergravity is the theory of local (Poincaré) supersymmetry in the same sense that ordinary gravity is the theory of “local Poincaré-symmetry”. These are gauge theories for the Poincare Lie algebra and the super Poincare Lie algebra, respectively, in that the field (physics) is a Cartan connection for the inclusion :
that decomposes into three components, :
a -valued 1-form – the vielbein
a -valued 1-form – called the spin connection;
a -valued 1-form – called the gravitino field.
Typically in fact the field content of supergravity is larger, in that a field is really an ∞-Lie algebra-valued differential form with values in an ∞-Lie algebra such as the supergravity Lie 3-algebra (DAuriaFreCastellani) . Specifically such a field
has one more component
|gauge group||stabilizer subgroup||local model space||local geometry||global geometry||differential cohomology||first order formulation of gravity|
|general||Lie group/algebraic group||subgroup (monomorphism)||quotient (“coset space”)||Klein geometry||Cartan geometry||Cartan connection|
|examples||Euclidean group||rotation group||Cartesian space||Euclidean geometry||Riemannian geometry||affine connection||Euclidean gravity|
|Poincaré group||Lorentz group||Minkowski space||Lorentzian geometry||pseudo-Riemannian geometry||spin connection||Einstein gravity|
|super Poincaré group||spin group||super Minkowski spacetime||Lorentzian supergeometry||supergeometry||superconnection||supergravity|
|linear algebraic group||parabolic subgroup/Borel subgroup||flag variety||parabolic geometry|
|orthochronous Lorentz group||conformal geometry||conformal connection||conformal gravity|
|general||smooth 2-group||2-monomorphism||homotopy quotient||Klein 2-geometry||Cartan 2-geometry|
|cohesive ∞-group||∞-monomorphism (i.e. any homomorphism)||homotopy quotient of ∞-action||higher Klein geometry||higher Cartan geometry||higher Cartan connection|
|examples||extended super Minkowski spacetime||extended supergeometry||higher supergravity: type II, heterotic, 11d|
The condition of gauge invariance of an action functional on -connections is considerably more restrictive than for one on -connections. For instance there is, under mild assumptions, a unique maximally supersymmetric supergravity extension of the ordinary Einstein-Hilbert action on a 4-dimensional manifold. This in turn is obtained from the unique (under mild assumptions) maximally supersymmetric supergravity action functional on a (10,1)-dimensional spacetime by thinking of the 4-dimensional action function as being a dimensional reduction of the 11-dimensional one.
This uniqueness (under mild conditions) is one reason for interest in supergravity theories. Another important reason is that supergravity theories tend to remove some of the problems that are encountered when trying to realize gravity as a quantum field theory. Originally there had been high hopes that the maximally supersymmetric supergravity theory in 4-dimensions is fully renormalizable. This couldn’t be shown computationally – until recently: triggered by new insights recently there there has been lots of renewed activity on the renormalizability of maximal supergravity.
Ordinary Einstein gravity has a natural formulation in terms of Cartan geometry for the inclusion of the Lorentz Lie algebra into the Poincaré Lie algebra . In this first order formulation of gravity a field configuration is a Cartan connection with such coefficients.
This persepctive directly generalizes to supergeometry and yields the superspace formulation of theories of supegravity.
regarded as a super vector space with in odd degree becomes a super Lie algebra by letting the bracket to be given by the defining action and by letting the bracket be given by a canonically induced bilinear and -equivariant pairing – the super Poincaré Lie algebra. This still canonical contains the Lorentz Lie algebra and the quotient
From this, a super-Cartan geometry is defined in direct analogy to the Cartan formulation of Riemannian geometry
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 way, this being the formalization of what physicists mean when saying that they pass to “local supersymmetry”.
One subtlety to take care of is that this makes spacetime a super-spacetime locally modeled on super Minkowski spacetime. But the resulting theory is supposed to be a field theory on an ordinary spacetime locally modeled on ordinary Minkowski spacetime. This is enforced by a further constraint on the super-Cartan connection which forces it to be determing by the bosonic manifold underlying the given supermanifold. This constraint is variously known as the superspace constraints or as rheonomy .
The other subtlety to take care of is that 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 follows a key insight due to (D’Auria-Fré-Regge 80, D’Auria-Fré 82) – the D'Auria-Fre formulation of supergravity – 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 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 the degree of the relevant differential form field.
Specifically, me may write the above generalized CE-algebra with the extra degree-3 generator as the CE-algebra
of the supergravity Lie 3-algebra .
Now a morphism
is known as extended super Minkowski spacetime.
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 – higher Cartan geometry.
Here the notion of covariant derivative includes the usual Levi-Civita connection, but also in general torsion components and contributions from other background gauge fields such as a Kalb-Ramond field and the RR-fields in type II supergravity or heterotic supergravity.
Of particular interest to phenomenologists around the turn of the millennium (but maybe less so today with new experimental evidence) has been in solutions of spacetime manifolds of the form for the locally observed Minkowski spacetime (that plays a role as the background for all available particle accelerator experiments) and a small closed 6-dimensional Riemannian manifold .
In the absence of further fields besides gravity, the condition that such a configuration has precisely one Killing spinor and hence precisely one global supersymmetry turns out to be precisely that is a Calabi-Yau manifold. This is where all the interest into these manifolds in string theory comes from. (Notice though that nothing in the theory itself demands such a compactification. It is only the phenomenological assumption of the factorized spacetime compactification together with supersymmetry that does so.)
For more see
The equations of motion of those theories of supergravity which qualify as target spaces for Green-Schwarz action functional sigma models (e.g. 10d heterotic supergravity for the heterotic string and 10d type II supergravity for the type II string) are supposed to be equivalent to those -models being well defined (the WZW-model term being well defined, hence -symmetry being in effect). See at Green-Schwarz action – References – Supergravity equations of motion for pointers.
which is usefully thought of to continue as
Supergravity theories are controled by the corresponding split real forms
where is the maximal compact subgroup of :
Therefore acts as a global symmetry on the supergravity fields.
This is no longer quite true for their UV-completion by the corresponding compactifications of string theory (e.g. type II string theory for type II supergravity, etc.). Instead, on these a discrete subgroup
acts as global symmetry. This is called the U-duality group of the supergravity theory (see there for more).
See the references (below).
|supergravity gauge group (split real form)||T-duality group (via toroidal KK-compactification)||U-duality||maximal gauged supergravity|
|1||S-duality||10d type IIB supergravity|
|SL O(1,1)||9d supergravity|
|SU(3) SU(2)||SL||8d supergravity|
|E9||2d supergravity||E8-equivariant elliptic cohomology|
For supergravity Lagrangians “of ordinary type” it turns out that
|brane||in supergravity||charged under gauge field||has worldvolume theory|
|black brane||supergravity||higher gauge field||SCFT|
|D-brane||type II||RR-field||super Yang-Mills theory|
|D0-brane||BFSS matrix model|
|D4-brane||D=5 super Yang-Mills theory with Khovanov homology observables|
|D1-brane||2d CFT with BH entropy|
|D3-brane||N=4 D=4 super Yang-Mills theory|
|(D25-brane)||(bosonic string theory)|
|NS-brane||type I, II, heterotic||circle n-connection|
|NS5-brane||B6-field||little string theory|
|D-brane for topological string|
|M-brane||11D SuGra/M-theory||circle n-connection|
|M2-brane||C3-field||ABJM theory, BLG model|
|M5-brane||C6-field||6d (2,0)-superconformal QFT|
|M9-brane/O9-plane||heterotic string theory|
|topological M2-brane||topological M-theory||C3-field on G2-manifold|
|topological M5-brane||C6-field on G2-manifold|
|solitons on M5-brane||6d (2,0)-superconformal QFT|
|self-dual string||self-dual B-field|
|3-brane in 6d|
Textbook accounts include
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
Lecture notes include
Furrther surveys include
A fair bit of detail on supersymmetry and on supergravity is in
The original article that introduced the D'Auria-Fre formulation of supergravity is
Some basic facts are recalled in
The -symmetry was first discussed in
Hermann Nicolai, Supergravity with Local Invariance , Phys. Lett. B 187, 316 (1987).
The discrete quantum subgroups were discussed in
which also introduced the term “U-duality”.
Review and further discusssion is in
A careful discussion of the topology of the U-duality groups is in
Nicholas Houston, Supergravity and Generalized Geometry Thesis (2010) (pdf)
The case of “” is discussed in
and that of “” in
General discussion of the Kac-Moody groups arising in this context is for instance in
A survey of the Chern-Simons gravity-style action functionals for supergravity is in
Further physics monographs on supergravity include
I. L. Buchbinder, S. M. Kuzenko, Ideas and methods of supersymmetry and supergravity; or A walk through superspace, googB
Julius Wess, Jonathan Bagger, Supersymmetry and supergravity, 1992
Steven Weinberg, Quantum theory of fields, volume III: supersymmetry
The Cauchy problem for classical solutions of simple supergravity has been discussed in
A canonical textbook reference for the role of Calabi-Yau manifolds in compactifications of 10-dimensional supergravity is volume II, starting on page 1091 in
Discussion of solutions with global supersymmetry left and their relation to Calabi-Yau compactifications are for instance in