U-duality is a kind of duality in string theory.
The KK-compactifications of 11-dimensional supergravity to lower dimensional gauged supergravity theories have global/local gauge groups given by split real forms of the $E$-series of the exceptional Lie groups.
Here the compact exceptional Lie groups form a series E8,E7, E6
which is usefully thought of to continue as
(Notice that $E_4$, $E_5$ and $E_6$ are also the traditional choices for phenomenologically realistic grand unified theories, see there for more.)
The split real forms of this are traditionally written
and one sets
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 and more generally certain subgroups of it are “gauged” (have gauge fields) in gauged supergravity version.
So for instance maximal 3d supergravity has global (and in fact also local, see there) gauge group given by (the split real form of) E8.
This is no longer verbatim 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.
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” E9, E10, E11 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.
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)
More generally, there is a “magic pyramid” of super-Einstein-Yang-Mills theories and their U-duality groups.
U-duality may be understood as being the combination of T-duality for the compactification torus and S-duality of type IIB superstring theory. see (West 12, section 17.5.4).
The hidden E7-symmetry of the KK-compactification of 11-dimensional supergravity on a 7-dimensional fiber to maximal $N = 8$ 4d supergravity was first noticed in
Eugene Cremmer, Bernard Julia, The $N = 8$ supergravity theory. I. The Lagrangian, Phys. Lett. 80B (1978) 48
Eugene Cremmer, Bernard Julia, The $SO(8)$ Supergravity, Nucl. Phys. B 159 (1979) 141 (spire)
and more generally in
The further U-duality groups for compactifications of 11d SuGra including E6, E7, E8 were identified in
Eugene Cremmer, Supergravities in 5 dimensions, in Hawking, Rocek (eds.) Superspace and Supergravity, Cambridge University Press 1981
Bernard Julia, Group Disintegrations, in Hawking, Rocek (eds.) Superspace and Supergravity, Cambridge University Press 1981
Review includes (Obers-Pioline 98, section 4.2), see also
The concept and terminology of U-duality in string theory/M-theory originates in
A textbook account is in
Systematization of U-duality via the relation between supersymmetry and division algebras and the Freudenthal magic square is due to
Leron Borsten, Michael Duff, L. J. Hughes, S. Nagy, A magic square from Yang-Mills squared (arXiv:1301.4176)
A. Anastasiou, Leron Borsten, Michael Duff, L. J. Hughes, S. Nagy, A magic pyramid of supergravities, arXiv:1312.6523
Quick surveys include
Reviews focusing on gauged supergravity and the non-discrete duality groups include
with slides in
Reviews with more M-theory lore include
N.A. Obers B. Pioline, U-duality and M-Theory, Phys.Rept. 318 (1999) 113-225 (arXiv:hep-th/9809039)
Shun’ya Mizoguchi, Germar Schroeder, On Discrete U-duality in M-theory, Class.Quant.Grav. 17 (2000) 835-870 (arXiv:hep-th/9909150)
Diederik Roest, M-theory and Gauged Supergravities, Fortsch.Phys.53:119-230,2005 (arXiv:hep-th/0408175)
Discussion in line with the F-theory perspective on the $SL(2,\mathbb{Z})$-S-duality – namely “F'-theory” – is in
Discussion of 11-dimensional supergravity in a form that exhibits the higher U-duality groups already before KK-compactification, via a kind of exceptional generalized geometry,is in
The case of $SL(3,\mathbb{Z}) \times SL(2,\mathbb{Z})$ in 8d supergravity is discussed in
The case of $SL(5,\mathbb{Z})$ in 7d supergravity from M-theory is discussed in
The $E_{7(7)}$-symmetry was first discussed in
The case of $E_{8(8)}$ is discussed 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 case of E9 is discussed in
The case of E10 is discussed for bosonic degrees of freedom 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 for fermionic degrees of freedom in supersymmetric quantum cosmology in
More recent review includes
The case of of E11 is discussed in
Peter West, $E_{11}$ and M-theory, Class. Quant. Grav. 18, 4443 (2001) (arXiv:hep-th/0104081).
Peter West, A brief review of E theory (arXiv:1609.06863)
A careful discussion of the topology of the Kac-Moody 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)
General discussion of the Kac-Moody groups arising in this context is for instance in
String theory partition functions as automorphic forms for U-duality groups are discussed in