nLab U-duality

Idea

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

$$E_8, E_7, E_6$$

which is usefully thought of to continue as

$$E_5 \coloneqq Spin(10), E_4 \coloneqq SU(5), E_3 \coloneqq SU(3) \times SU(2) \,.$$

(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

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

and one sets

$$E_{5(5)} \coloneqq Spin(5,5), E_{4(4)} \coloneqq SL(5, \mathbb{R}), E_{3(3)} \coloneqq SL(3, \mathbb{R}) \times SL(2, \mathbb{R}) \,.$$

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

$$E_{n(n)}/ K_n$$

for

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

where $K_n$ is the maximal compact subgroup of $E_{n(n)}$:

$$K_8 \simeq Spin(16), K_7 \simeq SU(8), K_6 \simeq Sp(4)$$

$$K_5 \simeq Spin(5) \times Spin(5), K_4 \simeq Spin(5), K_3 \simeq SU(2) \times SO(2) \,.$$

Therefore $E_{n(n)}$ 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

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

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 $0 \leq d \lt 3$, with “Kac-Moody groups” E9, E10, E11 corresponding to the Kac-Moody algebras

$$\mathfrak{e}_9, \mathfrak{e}_10, \mathfrak{e}_{11} \,.$$

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	D=10 type IIB supergravity
	SL$(2,\mathbb{R}) \times$ O(1,1)	$\mathbb{Z}_2$	$SL(2,\mathbb{Z})$ $\times \mathbb{Z}_2$	D=9 supergravity
SU(3)$\times$ SU(2)	SL$(3,\mathbb{R}) \times SL(2,\mathbb{R})$	$O(2,2;\mathbb{Z})$	$SL(3,\mathbb{Z})\times SL(2,\mathbb{Z})$	D=8 supergravity
SU(5)	$SL(5,\mathbb{R})$	$O(3,3;\mathbb{Z})$	$SL(5,\mathbb{Z})$	D=7 supergravity
Spin(10)	$Spin(5,5)$	$O(4,4;\mathbb{Z})$	$O(5,5,\mathbb{Z})$	D=6 supergravity
E₆	$E_{6(6)}$	$O(5,5;\mathbb{Z})$	$E_{6(6)}(\mathbb{Z})$	D=5 supergravity
E₇	$E_{7(7)}$	$O(6,6;\mathbb{Z})$	$E_{7(7)}(\mathbb{Z})$	D=4 supergravity
E₈	$E_{8(8)}$	$O(7,7;\mathbb{Z})$	$E_{8(8)}(\mathbb{Z})$	D=3 supergravity
E₉	$E_{9(9)}$	$O(8,8;\mathbb{Z})$	$E_{9(9)}(\mathbb{Z})$	D=2 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.

Properties

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).

Related concepts

• embedding tensor, tensor hierarchy

• exceptional field theory

• duality in string theory

  • T-duality

  • S-duality

  • U-duality

• M-theory on G₂-manifolds

• magic pyramid

• mysterious duality

References

General

The hidden global E7-symmetry (and local $K(E_7) \simeq SU(8)$-symmetry) of the KK-compactification of 11-dimensional supergravity on a 7-dimensional torus to D=4 N=8 supergravity was first realized

• Eugene Cremmer, Bernard Julia, The $N = 8$ supergravity theory. I. The Lagrangian, Phys. Lett. B 80 (1978) 48 [spire:131572, doi:10.1016/0370-2693(78)90303-9]

• Eugene Cremmer, Bernard Julia, The $SO(8)$ Supergravity, Nucl. Phys. B 159 (1979) 141 [doi;10.1016/0550-3213(79)90331-6, spire:140465]

• Bernard de Wit, Hermann Nicolai: D = 11 Supergravity With Local $SU(8)$ Invariance, Nucl. Phys. B 274, 363 (1986) [doi:10.1016/0550-3213(86)90290-7, spire:227409]

• Bernard de Wit, Hermann Nicolai: Local $SU(8)$ invariance in $d = 11$ supergravity, talk at Nuffield Workshop on Supersymmetry and its Applications, 0357 [spire:218601]

and the hidden global E8-symmetry (and local $K(\mathfrak{e}_8) \simeq \mathfrak{so}_{16}$-symmetry) in

• Eugene Cremmer: Supergravities in 5 dimensions, in: Superspace and Supergravity, Cambridge University Press (1981) [spire:155020, pdf, ISBN:0521239087]

• Hermann Nicolai: $d=11$ Supergravity with local $SO(16)$ invariance, Physics Letters B 187 3–4 (1987) 316-320 [doi:10.1016/0370-2693(87)91102-6]

See also:

• Bernard Julia, Group Disintegrations, in: Superspace and Supergravity, Cambridge University Press (1981) [spire:155019, ISBN:0521239087]

The concept and terminology of U-duality in string theory/M-theory originates with:

• Chris Hull, Paul Townsend: Unity of Superstring Dualities, Nucl. Phys. B 438 (1995) 109-137 [arXiv:hep-th/9410167, doi:10.1016/0550-3213(94)00559-W]

Discussion via the BFSS matrix model:

• Shmuel Elitzur, Amit Giveon, David Kutasov, Eliezer Rabinovici: Algebraic Aspects of Matrix Theory on $T^d$, Nucl. Phys. B 509 (1998) 122-144 [arXiv:hep-th/9707217, doi:10.1016/S0550-3213(97)00622-6]

Early review:

• Eugène Cremmer, Bernard Julia, H. Lu, Christopher Pope: Dualisation of Dualities, I, Nucl. Phys. B 523 (1998) 73-144 [arXiv:hep-th/9710119, doi:10.1016/S0550-3213(98)00136-9]

• Niels Obers, Boris Pioline, U-duality and M-Theory, Phys. Rept. 318 (1999) 113-225 [arXiv:hep-th/9809039, doi:10.1016/S0370-1573(99)00004-6]

• Hermann Nicolai: On M-Theory, J Astrophys Astron 20, (1999) 149–164 [arXiv:hep-th/9801090, doi:10.1007/BF02702349]

• Bernard de Wit, Hermann Nicolai: Hidden Symmetries, Central Charges and All That, Class. Quant. Grav. 18 (2001) 3095-3112 [arXiv:hep-th/0011239, doi:10.1088/0264-9381/18/16/302]

• 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. B 741 (2006) 42-82 [arXiv:hep-th/0507249]

• Paul Cook, Connections between Kac-Moody algebras and M-theory, PhD thesis, King’s College London (2007) [arXiv:0711.3498, webpage]

Monographs:

• Peter West, section 17.3 of Introduction to Strings and Branes, Cambridge University Press 2012

Systematization of U-duality via the relation between supersymmetry and division algebras and the Freudenthal magic square is due to

• Leron Borsten, Michael Duff, Mia J. Hughes, Silvia Nagy, A magic square from Yang-Mills squared, Phys. Rev. Lett. 112 (2014) 131601 [arXiv:1301.4176]

• Alexandros Anastasiou, Leron Borsten, Michael Duff, Mia J. Hughes, Silvia Nagy, A magic pyramid of supergravities, JHEP (2014) 178 [arXiv:1312.6523]

Quick surveys include

• Jacques Distler, Split real forms (blog post).

Reviews focusing on gauged supergravity and the non-discrete duality groups include

• Henning Samtleben, Lectures on Gauged Supergravity and Flux Compactifications (arXiv:0808.4076)

with slides in

• Henning Samtleben, Gauged supergravity and U-duality, 2007 (pdf)

Further discussion with an eye towards M-theory:

• 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 (2005) 119-230 [arXiv:hep-th/0408175, doi:10.1002/prop.200410192]

Discussion in line with the F-theory perspective on the $SL(2,\mathbb{Z})$-S-duality – namely “F'-theory” – is in

• Alok Kumar, Cumrun Vafa, U-Manifolds, Phys.Lett. B396 (1997) 85-90 (arXiv:hep-th/9611007)

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

• Olaf Hohm, Henning Samtleben, Exceptional Form of $D=11$ Supergravity, Phys. Rev. Lett. 111, 231601 (2013) (arXiv:1308.1673)

Review of U-duality and exceptional generalized geometry in KK-compactification of D=11 supergravity:

• Henning Samtleben, 11D Supergravity and Hidden Symmetries, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2303.12682, doi:10.1007/978-981-19-3079-9]

On U-duality (and possibly mysterious duality) via Hypothesis H as automorphisms of iterated (rational) cyclic loop spaces of the (rational) 4-sphere:

• Hisham Sati, Alexander Voronov, Mysterious Triality and Rational Homotopy Theory, Comm. Math. Phys. 400 (2023) 1915-1960 [arXiv:2111.14810, doi:10.1007/s00220-023-04643-7]

• Hisham Sati, Alexander Voronov, Mysterious Triality and M-Theory [arXiv:2212.13968]

• Hisham Sati, Alexander A. Voronov: Mysterious Triality and the Exceptional Symmetry of Loop Spaces [arXiv:2408.13337]

review in:

• Alexander Voronov (joint with Hisham Sati), Mysterious Duality, talk at Texas Tech 2021 (abstract pdf pdf, slides pdf, pdf)

• Alexander Voronov: The $E_k$ symmetry of dimensional reductions of M-theory, talk at M-Theory and Mathematics 2023, NYU Abu Dhabi (2023) [web]

$n=3$

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

• James Liu, Ruben Minasian, U-branes and $T^3$ fibrations, Nucl.Phys. B510 (1998) 538-554 (arXiv:hep-th/9707125)

$n=4$

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

• Moshe Rozali, Matrix Theory and U-Duality in Seven Dimensions, Phys.Lett. B400 (1997) 260-264 (arXiv:hep-th/9702136)

$n=7$

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

• Bernard de Wit, Hermann Nicolai, $D = 11$ Supergravity With Local $SU(8)$ Invariance, Nucl. Phys. B 274, 363 (1986)

$n=8$

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).

$n=9$

The case of E9 is discussed in

• François Englert, Laurent Houart, Axel Kleinschmidt, Hermann Nicolai, Nassiba Tabti, An E9 multiplet of BPS states, JHEP 0705:065,2007 (arXiv:hep-th/0703285)

$n=10$

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

• Thibault Damour, Axel Kleinschmidt, Hermann Nicolai: Hidden symmetries and the fermionic sector of eleven-dimensional supergravity, Phys. Lett. B 634 (2006) 319-324 [arXiv:hep-th/0512163, doi:10.1016/j.physletb.2006.01.015]

and for supersymmetric quantum cosmology in

• Thibault Damour, Quantum supersymmetric Bianchi IX Cosmology and its hidden Kac-Moody Structure, talk at Gravitation, Solitons and Symmetries (pdf)

Review includes

• Hermann Nicolai, Wonders of $E_{10}$ and $K(E_{10})$ (2008) (pdf)

• Hermann Nicolai, On Exceptional Geometry and Supergravity, talk at Gravitation, Solitons and Symmetries (pdf)

Discussion of phenomenology:

• Axel Kleinschmidt, Hermann Nicolai, Standard model fermions and $K(E_{10})$ (arXiv:1504.01586)

• Krzysztof A. Meissner, Hermann Nicolai, Standard Model Fermions and Infinite-Dimensional R-Symmetries, Phys. Rev. Lett. 121, 091601 (2018) (arXiv:1804.09606)

• Krzysztof A. Meissner, Hermann Nicolai, Planck Mass Charged Gravitino Dark Matter, Phys. Rev. D 100, 035001 (2019) (arXiv:1809.01441)

$n=11$

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)

“$n=26, 27$”

Discussion for bosonic M-theory:

• Keith Glennon, Peter West: $K_{27}$ as a symmetry of closed bosonic strings and branes [arXiv:2409.08649]

• Keith Glennon: Bosonic M-Theory From a Kac-Moody Algebra Perspective [arXiv:2501.03000]

Further details

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

• Philipp Fleig, Axel Kleinschmidt, Eisenstein series for infinite-dimensional U-duality groups (arXiv:1204.3043, slides)

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)