String theory

Group Theory



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 EE-series of the exceptional Lie groups.

Here the compact exceptional Lie groups form a series E8,E7, E6

E 8,E 7,E 6 E_8, E_7, E_6

which is usefully thought of to continue as

E 5:=Spin(10),E 4:=SU(5),E 3:=SU(3)×SU(2). E_5 := Spin(10), E_4 := SU(5), E_3 := SU(3) \times SU(2) \,.

(Notice that E 4E_4, E 5E_5 and E 6E_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

E 8(8),E 7(7),E 6(6) E_{8(8)}, E_{7(7)}, E_{6(6)}

and one sets

E 5(5):=Spin(5,5),E 4(4):=SL(5,),E 3(3):=SL(3,)×SL(2,). E_{5(5)} := Spin(5,5), E_{4(4)} := SL(5, \mathbb{R}), E_{3(3)} := SL(3, \mathbb{R}) \times SL(2, \mathbb{R}) \,.

For instance the scalar fields in the field supermultiplet of 3d113 \leq d \leq 11-dimensional supergravity have moduli spaces parameterized by the homogeneous spaces

E n(n)/K n E_{n(n)}/ K_n


n=11d, n = 11 - d \,,

where K nK_n is the maximal compact subgroup of E n(n)E_{n(n)}:

K 8Spin(16),K 7SU(8),K 6Sp(4) K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4)
K 5Spin(5)×Spin(5),K 4Spin(5),K 3SU(2)×SO(2). K_5 \simeq Spin(5) \times Spin(5), K_4 \simeq Spin(5), K_3 \simeq SU(2) \times SO(2) \,.

Therefore E n(n)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

E n(n)()E n(n) E_{n(n)}(\mathbb{Z}) \hookrightarrow E_{n(n)}

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 0d<30 \leq d \lt 3, with “Kac-Moody groupsE9, E10, E11 corresponding to the Kac-Moody algebras

𝔢 9,𝔢 10,𝔢 11. \mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11} \,.

Continuing in the other direction to d=10d = 10 (n=1n = 1) connects to the T-duality group O(d,d,)O(d,d,\mathbb{Z}) of type II string theory.

supergravity gauge group (split real form)T-duality group (via toroidal KK-compactification)U-dualitymaximal gauged supergravity
SL(2,)SL(2,\mathbb{R})1SL(2,)SL(2,\mathbb{Z}) S-duality10d type IIB supergravity
SL(2,)×(2,\mathbb{R}) \times O(1,1) 2\mathbb{Z}_2SL(2,)× 2SL(2,\mathbb{Z}) \times \mathbb{Z}_29d supergravity
SU(3)×\times SU(2)SL(3,)×SL(2,)(3,\mathbb{R}) \times SL(2,\mathbb{R})O(2,2;)O(2,2;\mathbb{Z})SL(3,)×SL(2,)SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})8d supergravity
SU(5)SL(5,)SL(5,\mathbb{R})O(3,3;)O(3,3;\mathbb{Z})SL(5,)SL(5,\mathbb{Z})7d supergravity
Spin(10)Spin(5,5)Spin(5,5)O(4,4;)O(4,4;\mathbb{Z})O(5,5,)O(5,5,\mathbb{Z})6d supergravity
E6E 6(6)E_{6(6)}O(5,5;)O(5,5;\mathbb{Z})E 6(6)()E_{6(6)}(\mathbb{Z})5d supergravity
E7E 7(7)E_{7(7)}O(6,6;)O(6,6;\mathbb{Z})E 7(7)()E_{7(7)}(\mathbb{Z})4d supergravity
E8E 8(8)E_{8(8)}O(7,7;)O(7,7;\mathbb{Z})E 8(8)()E_{8(8)}(\mathbb{Z})3d supergravity
E9E 9(9)E_{9(9)}O(8,8;)O(8,8;\mathbb{Z})E 9(9)()E_{9(9)}(\mathbb{Z})2d supergravityE8-equivariant elliptic cohomology
E10E 10(10)E_{10(10)}O(9,9;)O(9,9;\mathbb{Z})E 10(10)()E_{10(10)}(\mathbb{Z})
E11E 11(11)E_{11(11)}O(10,10;)O(10,10;\mathbb{Z})E 11(11)()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.


Relation to T-duality and S-duality

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=8N = 8 4d supergravity was first noticed in

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

  • Pietro Fré, Floriana Gargiulo, Ksenya Rulik, Mario Trigiante, The general pattern of Kac Moody extensions in supergravity and the issue of cosmic billiards, Nucl.Phys. B741 (2006) 42-82 (arXiv:hep-th/0507249)

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

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,)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

In various dimensions

The case of SL(3,)×SL(2,)SL(3,\mathbb{Z}) \times SL(2,\mathbb{Z}) in 8d supergravity is discussed in

The case of SL(5,)SL(5,\mathbb{Z}) in 7d supergravity from M-theory is discussed in

The E 7(7)E_{7(7)}-symmetry was first discussed in

The case of E 8(8)E_{8(8)} is discussed in

The case of E9 is discussed in

The case of E10 is discussed for bosonic degrees of freedom in

and for fermionic degrees of freedom in supersymmetric quantum cosmology in

More recent review includes

The case of of E11 is discussed in

Further details

A careful discussion of the topology of the Kac-Moody U-duality groups is in

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

Relation to automorphic forms

String theory partition functions as automorphic forms for U-duality groups are discussed in

  • Michael Green, Jorge G. Russo, Pierre Vanhove, Automorphic properties of low energy string amplitudes in various dimensions (arXiv:1001.2535)
Revised on September 23, 2016 04:15:32 by Urs Schreiber (