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 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.
The non-spinorial part of action functionals of supergravity theories are typically given in first order formulation as functional on a space of connections with values in the Poincare Lie algebra . Including the fermionic fields, this becomes connections with values in the super Poincare Lie algebra .
This might suggest that supergravity is to be thought of as a gauge theory. There are indeed various action functionals of Chern-Simons theory-form for supergravity theories (Zanelli). These yield theories whose bosonic action functional is the Einstein-Hilbert action in certain contraction limits.
More generally (DAuriaFreCastellani) have shown that at least some versions, such as the maximal 11-dimensional supergravity, are naturally understood as higher gauge theories whose fields are ∞-Lie algebra-valued forms with values in ∞-Lie algebras such as the supergravity Lie 3-algebra. This is described in detail at D'Auria-Fre formulation of supergravity.
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
torsion constaints in supergravity?
|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|
A modern reference for the diverse flavours of supergravity theories is
Introductory lecture notes are in
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
The standard textbook monograph on supergravity and string theory using these tools 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