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superalgebra and (synthetic ) supergeometry
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 of 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.
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 $\mathfrak{o}(d-1,1) \hookrightarrow \mathfrak{Iso}(\mathbb{R}^{d-1,1})$. In this first order formulation of gravity a field configuration is a Cartan connection with such coefficients.
This perspective directly generalizes to supergeometry and yields the superspace formulation of theories of supegravity – super Cartan geometry.
After picking a dimension $d\in \mathbb{N}$ and writing $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ for the Poincaré Lie algebra, then a choice of “number of supersymmetries” is a choice of real spin representation $N$. Then the direct sum
regarded as a super vector space with $N$ in odd degree becomes a super Lie algebra by letting the $[even,odd]$ bracket be given by the defining action and by letting the $[odd,odd]$ bracket be given by a canonically induced bilinear and $\mathfrak{o}$-equivariant pairing – the super Poincaré Lie algebra. This still canonical contains the Lorentz Lie algebra $\mathfrak{o}(\mathbb{R}^{d-1,1})$ and the quotient
is called super Minkowski spacetime (equipped with its super translation Lie algebra structure).
From this, a super-Cartan geometry is defined in direct analogy to the Cartan formulation of Riemannian geometry
(higher-)Cartan geometry | $\mathfrak{g}$ | $\mathfrak{h}$ | $\mathfrak{g}/\mathfrak{h}$ |
---|---|---|---|
Einstein gravity | $\mathfrak{Iso}(\mathbb{R}^{d-1,1})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1}$ |
supergravity | $\mathfrak{Iso}(\mathbb{R}^{d-1,1\vert N})$ | $\mathfrak{o}(d-1,1)$ | $\mathbb{R}^{d-1,1\vert N}$ |
11-dimensional supergravity | $\mathfrak{Iso}(\widehat{\mathbb{R}}^{10,1\vert N=1})$ | $\mathfrak{o}(d-1,1)$ | $\widehat{\mathbb{R}}^{10,1\vert N=1}$ |
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 determined 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-form B-field of 10-dimensional type II supergravity and heterotic supergravity as well as the 3-form C-field of 11-dimensional supergravity.
This means that these higher dimensional supergravity theories are not in fact entirely described by super-Cartan geometry – but by super-higher Cartan geometry.
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-algebra
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 $\frac{1}{2}\bar \psi \Gamma^{a b} \wedge \psi \wedge e_a \wedge e_b \in CE^4(\mathfrak{Iso}(10,1|N=1))$ 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 algebras, hence of L-infinity algebras, in fact of Lie (p+1)-algebras for $(p+1)$ the degree of the relevant differential form field.
Specifically, me may write the above generalized CE-algebra with the extra degree-3 generator $c_3$ as the CE-algebra $CE(\mathfrak{m}2\mathfrak{brane})$
of the supergravity Lie 3-algebra $\mathfrak{m}2\mathfrak{brane}$.
Now a morphism
encodes graviton and gravitino fields as above, but in addition it encodes a 3-form
whose curvature
satisfies a modified Bianchi identity, crucial for the theory of 11-dimensional supergravity (D’Auria-Fré 82).
So this collection of differential form data is no longer a Lie algebra valued differential form, it is an L-infinity algebra valued differential form, with values in the supergravity Lie 3-algebra.
The quotient
is known as extended super Minkowski spacetime.
The Lie integration of this is a smooth 3-group $G$ which receives a map from the Lorentz group.
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.
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 | |
SU(5) | $SL(5,\mathbb{R})$ | $O(3,3;\mathbb{Z})$ | $SL(5,\mathbb{Z})$ | 7d supergravity | |
Spin(10) | $Spin(5,5)$ | $O(4,4;\mathbb{Z})$ | $O(5,5,\mathbb{Z})$ | 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 | $E_{11(11)}$ | $O(10,10;\mathbb{Z})$ | $E_{11(11)}(\mathbb{Z})$ |
(Hull-Townsend 94, table 1, table 2)
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:
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.
Discussion of evidence for supergravity from experiment/phenomenology includes the following:
in (Dalianis-Farakos 15) it is argued that the Starobinsky model of cosmic inflation, which is strongly preferred by experiment, further improves after embedding into supergravity.
string theory FAQ – Does string theory predict supersymmetry?
cosmic inflation and supersymmetry breaking via higher curvature corrections of supergravity are discussed in the context of the Starobinsky model of cosmic inflation
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
An early survey is
Textbook accounts include
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991)
Antoine Van Proeyen, Daniel Freedman, Supergravity, Cambridge University Press, 2012
Lecture notes include
P. Binetruy, G. Girardi, R. Grimm, Supergravity couplings: a geometric formulation, Phys.Rept.343:255-462,2001 (arXiv:hep-th/0005225)
Friedemann Brandt, Lectures on supergravity (arXiv:hep-th/0204035)
Bernard de Wit, Supergravity (arXiv:hep-th/0212245)
Antoine Van Proeyen, Structure of supergravity theories (arXiv:hep-th/0301005)
Joachim Gomis, Three lectures on Supergravity (pdf slides)
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
A compendium, of relevant action functionals and equations of motion is in
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
Supergravity, in the guise of 4d supergravity, was first found (constructed) in
Accounts of the early history include the following:
Sergio Ferrara, Supergravity and the quest for a unified theory (arxiv:hep-th/9405065)
R. Arnowitt, Ali Chamseddine, Pran Nath, The Development of Supergravity Grand Unification: Circa 1982-85 (arXiv:1206.3175)
David Appell, When supergravity was born, 2012 (pdf)
Peter van Nieuwenhuizen, Aspects of supergravity, 2014 (pdf)
Sergio Ferrara, A. Sagnotti, Supergravity at 40: Reflections and Perspectives, Based in part on the talk delivered by S. F. at the “Infeld Colloquium and Discrete”, in Warsaw, on December 1 2016, and on a joint CERN Courier article. Dedicated to John Schwarz on the occasion of his 75-th birthday (arXiv:1702.00743)
Stanley Deser, A brief history of Supergravity: the first three weeks (arXiv:1704.05886)
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
Discussion of implications of supergravity for phenomenology/cosmology includes