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Exotica
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 $\mathfrak{iso}(n,1)$, supergravity is defined by an extension of this to an action functional on the space connections with values in the super Poincare Lie algebra $\mathfrak{siso}(n,1)$. 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 $o(n,1) \hookrightarrow \mathfrak{siso}(n,1)$:
if we write $\mathfrak{siso}(n,1)$ as a semidirect product of the translation Lie algebra $\mathbb{R}^{(n,1)}$, the special orthogonal Lie algebra $\mathfrak{so}(n,1)$ and a spin group representation $\Gamma$, then locally a connection is a Lie algebra valued 1-form
that decomposes into three components, $A = (E, \Omega, \Psi)$:
a $\mathbb{R}^{n,1}$-valued 1-form $E$ – the vielbein
(this encodes the pseudo-Riemannian metric and hence the field of gravity);
a $\mathfrak{so}(n,1)$-valued 1-form $\Omega$ – called the spin connection;
a $\Gamma$-valued 1-form $\Psi$ – called the gravitino field.
Typically in fact the field content of supergravity is larger, in that a field $A$ is really an ∞-Lie algebra-valued differential form with values in an ∞-Lie algebra such as the supergravity Lie 3-algebra (DAuriaFreCastellani) $\mathfrak{sugra}(10,1)$. Specifically such a field
has one more component
The gauge transformations on the space of such connections that are parameterized by the elements of $\Gamma$ are called supersymmetries.
The condition of gauge invariance of an action functional on $\mathfrak{siso}$-connections is considerably more restrictive than for one on $\mathfrak{iso}$-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 $\mathfrak{iso}(n,1)$. Including the fermionic fields, this becomes connections with values in the super Poincare Lie algebra $\mathfrak{siso}(10,1)$.
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.
A solution to the bosonic Einstein equations of ordinary gravity – some Riemannian manifold – has a global symmetry if it has a Killing vector.
Accordingly, a configuration that solves the supergravity Euler-Lagrange equations is a global supersymmetry if it has a Killing spinor: a covariantly constant spinor.
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 $M^4 \times Y^6$ for $M^4$ 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 $Y^6$.
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 $Y^6$ 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 $N = 1$ supersymmetry that does so.)
More generally, in the presence of other background gauge fields, the Calabi-Yau condition here is deformed. One also speaks of generalized Calabi-Yau spaces. (For instance (GMPT05)).
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 $\sigma$-models being well defined (the WZW-model term being well defined, hence $\kappa$-symmetry being in effect). See at Green-Schwarz action – References – Supergravity equations of motion for pointers.
The compact exceptional Lie groups form a series
which is usefully thought of to continue as
Supergravity theories are controled by the corresponding split real forms
For instance the scalar fields in the field supermultiplet of $3 \leq d \leq 11$-dimensional supergravity have moduli spaces parameterized by the homogeneous spaces
for
where $K_n$ is the maximal compact subgroup of $E_{n(n)}$:
Therefore $E_{n(n)}$ 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).
It has been argued that this pattern should continue in some way further to the remaining values $0 \leq d \lt 3$, with “Kac-Moody groups” corresponding to the Kac-Moody algebras
Continuing in the other direction to $d = 10$ ($n = 1$) connects to the T-duality group $O(d,d,\mathbb{Z})$ of type II string theory.
See the references (below).
supergravity gauge group (split real form) | T-duality group (via toroidal KK-compactification) | U-duality | maximal gauged supergravity | ||
---|---|---|---|---|---|
$SL(2,\mathbb{R})$ | 1 | $SL(2,\mathbb{Z})$ S-duality | 10d type IIB supergravity | ||
SL$(2,\mathbb{R}) \times$ O(1,1) | $\mathbb{Z}_2$ | $SL(2,\mathbb{Z}) \times \mathbb{Z}_2$ | 9d supergravity | ||
SU(3)$\times$ SU(2) | SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$ | $O(2,2;\mathbb{Z})$ | $SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$ | 8d supergravity | |
Spin(10) | $SL(5,\mathbb{R})$ | $O(3,3;\mathbb{Z})$ | $SL(5,\mathbb{Z})$ | 7d supergravity | |
SU(5) | $Spin(5,5)$ | $O(4,4;\mathbb{Z})$ | $O(5,5,\mathbb{Z})$ | 6d supergravity | |
E6 | $E_{6(6)}$ | $O(5,5;\mathbb{Z})$ | $E_{6(6)}(\mathbb{Z})$ | 5d supergravity | |
E7 | $E_{7(7)}$ | $O(6,6;\mathbb{Z})$ | $E_{7(7)}(\mathbb{Z})$ | 4d supergravity | |
E8 | $E_{8(8)}$ | $O(7,7;\mathbb{Z})$ | $E_{8(8)}(\mathbb{Z})$ | 3d supergravity | |
E9 | $E_{9(9)}$ | $O(8,8;\mathbb{Z})$ | $E_{9(9)}(\mathbb{Z})$ | 2d supergravity | E8-equivariant elliptic cohomology |
E10 | $E_{10(10)}$ | $O(9,9;\mathbb{Z})$ | $E_{10(10)}(\mathbb{Z})$ | ||
E11 | $E_{11(11)}$ | $O(10,10;\mathbb{Z})$ | $E_{11(11)}(\mathbb{Z})$ |
For the moment see the remarks/references on supergravity at exceptional geometry and exceptional generalized geometry.
For supergravity Lagrangians “of ordinary type” it turns out that
is the highest dimension possible. All lower dimensional theories in this class appear as KK-compactifications of this theory or are deformations of such:
10-dimensional type II supergravity, heterotic supergravity
In dimension $(1+0)$ supergravity coupled to sigma-model fields is the spinning particle.
In dimension $(1+1)$ supergravity coupled to sigma-model fields is the spinning string/NSR superstring.
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
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 $E_{7(7)}$-symmetry was first discussed in
and $E_{8(8)}$ in
Hermann Nicolai, $D = 11$ Supergravity with Local $SO(16)$ Invariance , Phys. Lett. B 187, 316 (1987).
K. Koepsell, Hermann Nicolai, Henning Samtleben, An exceptional geometry for $d = 11$ supergravity?, Class. Quant. Grav. 17, 3689 (2000) (arXiv:hep-th/0006034).
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
Arjan Keurentjes, The topology of U-duality (sub-)groups (arXiv:hep-th/0309106)
Arjan Keurentjes, U-duality (sub-)groups and their topology (arXiv:hep-th/0312134)
A discussion in the context of generalized complex geometry / exceptional generalized complex geometry is in
Paulo Pires Pacheco, Daniel Waldram, M-theory, exceptional generalised geometry and superpotentials (arXiv:0804.1362)
Nicholas Houston, Supergravity and Generalized Geometry Thesis (2010) (pdf)
The case of “$E_{10}$” is discussed in
Thibault Damour, Marc Henneaux, Hermann Nicolai, $E(10)$ and a ‘small tension expansion’ of M theory, Phys. Rev. Lett. 89, 221601 (2002) (arXiv:hep-th/0207267);
Axel Kleinschmidt, Hermann Nicolai, $E(10)$ and $SO(9,9)$ invariant supergravity, JHEP 0407, 041 (2004) (arXiv:hep-th/0407101)
and that of “$E_{11}$” 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 $N = 1$ global supersymmetry left and their relation to Calabi-Yau compactifications are for instance in