Dynamical realisation of very-extended Kac-Moody algebras $G^{+++}$ in maximally oxidised theories: completing the brane-intersection construction

Englert, Houart and West (JHEP 0308 (2003) 025) showed that the existence of no-force ('BPS') intersecting-brane configurations, together with the requirement that branes be able to open on host branes, selects and reconstructs the maximally oxidised gravity+form theories whose three-dimensional reductions realise simply laced coset symmetries $G/H$; the brane solutions form representations of a subgroup of the Weyl and outer-automorphism group of the very extended Kac-Moody algebra $G^{+++}$. The paper left the correspondence incomplete in five linked ways: $G^{+++}$ was not established as an exact symmetry of the full dynamics; the world-volume field identification on host branes was proven only in M-theory; the construction does not cover non-simply laced $G$ (with $G_2$ anomalous); the Chern-Simons coefficients required for consistent brane opening are not fixed by the construction in non-supersymmetric theories; and the timelike/exotic embeddings on which part of the Weyl interpretation rests lack a settled physical status. The problem remains open: substantial low-level evidence supports the correspondence, yet linearised obstructions (a section condition, a missing dual-graviton component) have been identified and no completion of the non-simply laced, world-volume, or exotic threads is known; the open core is the full dynamical realisation of $G^{+++}$ together with the uncovered cases.

Background

Maximal supergravity theories develop large hidden symmetries under toroidal dimensional reduction: compactified on a torus and reduced to three spacetime dimensions, all bosonic fields can be dualised to scalars parametrising a coset space $G/H$, where $G$ is a non-compact real form of a simple Lie group and $H$ its maximal compact subgroup (for eleven-dimensional supergravity, $G=E_{8(8)}$ and $H=SO(16)$). The inverse problem - reconstructing the highest-dimensional parent theory from the three-dimensional coset - is called oxidation, and the parent is called the maximally oxidised theory. Cremmer, Julia, Lu and Pope showed that every split simple $G$ (the maximally non-compact real form of the complex Lie algebra) arises in this way from a theory of gravity, at most one dilaton, and antisymmetric form field strengths, with no supersymmetry required (arXiv:hep-th/9909099).

Englert, Houart and West (the source paper, JHEP 0308 (2003) 025, arXiv:hep-th/0307024) connected this algebraic structure to brane dynamics. In such theories, extremal branes - static charged solitons with mutually vanishing binding energy, called 'BPS' in a sense that deliberately does not refer to supersymmetry - can intersect orthogonally in no-force configurations. Intersection rules determine which theories admit such configurations, constraining the spacetime dimension, the form degrees and the dilaton couplings. A further dynamical requirement is brane opening: when the intersection of two branes fills the boundary of one of them, that brane can end on the other ('host') brane, which is consistent with charge conservation only if the action contains Chern-Simons-type terms ($A\wedge F\wedge F$ couplings, possibly involving new form fields and hence new branes, added iteratively). The paper proved, for simply laced $G$ (all roots of equal length), that these dynamical requirements reconstruct the maximally oxidised theory, and that its BPS intersecting-brane solutions form representations of a subgroup of the Weyl group (reflections in the simple roots) and of the diagram automorphisms (symmetries of the Dynkin diagram) of the very extended Kac-Moody algebra $G^{+++}$ - the infinite-dimensional Kac-Moody algebra, obtained from the Dynkin diagram of $G$ by adjoining three nodes along the 'gravity line', whose invariant bilinear form has Lorentzian signature. In this correspondence the branes of the theory are associated with the roots of $G^{+++}$, the root lengths and inner products encoding the intersection properties of the corresponding branes. The overextended algebra $G^{++}$ (two extra nodes) is the one that governs cosmological billiards - the oscillatory, Kasner-like dynamics of the fields near a spacelike cosmological singularity - (de Buyl, Henneaux, Julia and Paulot, arXiv:hep-th/0312251). For $G=E_8$, $G^{+++}$ is $E_{11}$, conjectured by West (arXiv:hep-th/0104081) to underlie M-theory; the bosonic-string endpoint corresponds to $K_{27}=D_{24}^{+++}$; pure gravity in $D$ dimensions to $A_{D-3}^{+++}$; and the general suggestion that all maximally oxidised theories carry their $G^{+++}$ symmetry is due to Englert, Houart, Taormina and West (arXiv:hep-th/0304206).

What the source paper established is a statement about BPS solutions and a discrete subgroup of the symmetry algebra - not about the full field dynamics - and its constructive logic is complete only for simply laced $G$, using supersymmetry where available. Five gaps are acknowledged in the paper's own text, and they define the open problem addressed here: the exact $G^{+++}$ symmetry of the complete dynamics is not established; the world-volume field identification used in the opening construction is proven only in M-theory and only surmised in general; the construction does not cover non-simply laced $G$, where the root-length/intersection correspondence just described develops an anomaly for $G_2$ (short-root squared length $2/3$ when long roots have squared length $2$); the Chern-Simons coefficients required by the opening construction are not fixed by it for non-supersymmetric oxidation endpoints; and the Weyl interpretation of some embeddings relies on compactifying time (T-duality along a timelike circle, where a string on a circle of radius $R$ is equivalent to one on a circle of radius $1/R$ with momentum and winding exchanged; and 'exotic' phases of M-theory of indefinite signature), whose physical status for the underlying quantum theory is unresolved. The source paper's footnote 8 records that the Chern-Simons coefficients, while not fixed by its qualitative discussion, can be fixed or at least quantised by consistency arguments that do not appeal to supersymmetry, such as quantum consistency with both electric and magnetic sources (Bekaert-Gomberoff, arXiv:hep-th/0212099).

Problem Statement

Complete the brane-dynamics-to-symmetry correspondence established for simply laced $G$ by Englert, Houart and West (JHEP 0308 (2003) 025) into a full dynamical statement, by resolving the five gaps the paper itself acknowledges:

(a) Dynamical realisation: determine whether the very extended Kac-Moody algebra $G^{+++}$ is an exact (global, field-theoretic) symmetry of the complete dynamics of the corresponding maximally oxidised theory - all fields and the full nonlinear equations of motion of the uncompactified higher-dimensional theory - rather than only of its BPS solution sector and the Weyl/outer-automorphism subgroup demonstrated there; equivalently, construct a $G^{+++}$-invariant dynamics containing the maximally oxidised theory, or establish a precise obstruction to its existence.

(b) World-volume couplings: derive generally - without appealing to supersymmetry - the world-volume field identification used by the opening construction (the identification, on the host brane, between the pull-back of the bulk gauge potential and the Hodge-dual world-volume field, proven only in the M-theory case), together with the required world-volume couplings and their gauge-invariance conditions.

(c) Non-simply laced extension: extend the intersection-rule/opening reconstruction and the root-length/intersection-rule correspondence to the non-simply laced algebras $B_n$, $C_n$ and $F_4$, and settle the anomalous $G_2$ case whose short root has squared length $2/3$ in the normalisation with long roots of squared length $2$.

(d) Chern-Simons coefficients: the opening construction requires Chern-Simons-type terms but does not fix their coefficients, whose precise values matter for the discussion of the potential $G^{+++}$ symmetry. Determine the exact coefficients (or their quantisation) of the Chern-Simons terms required by consistent brane opening for non-supersymmetric oxidation endpoints, where no supersymmetry is available to fix them.

(e) Exotic phases: settle the physical status, for the underlying quantum theory, of the embeddings and Weyl interpretations that rely on compactifying time (T-duality along a timelike direction and the exotic phases of the maximally oxidised theories).

The components are linked as one objective: (b) and (d) are the missing inputs that would make the brane construction self-contained; (c) extends its scope to the remaining simple Lie algebras; (e) secures the physical interpretation of the Weyl transformations the construction uses; and (a) is the target conjecture the completed construction is meant to support. The five components are independently refutable, but the objective is not equivalent to any one of them: it is the combined construction - a single construction, valid for all simple $G$ (or for precisely delimited classes with obstructions established for the remainder, provided the construction still delivers (b), (d) and (e) for every class it retains), that realises the maximally oxidised dynamics invariantly under $G^{+++}$ with correct Chern-Simons coefficients and derived world-volume couplings, with the timelike/exotic sector physically interpreted - or a rigorous obstruction showing that this realisation cannot be achieved, an obstruction concerning the target dynamical realisation (a) rather than a single one of (b)-(e).

References

1. F. Englert, L. Houart, P. West, 'Intersection rules, dynamics and symmetries', JHEP 0308 (2003) 025, DOI 10.1088/1126-6708/2003/08/025, https://arxiv.org/abs/hep-th/0307024 (source paper)
2. E. Cremmer, B. Julia, H. Lu, C. N. Pope, 'Higher-dimensional Origin of D=3 Coset Symmetries', arXiv:hep-th/9909099, https://arxiv.org/abs/hep-th/9909099
3. P. West, '$E_{11}$ and M Theory', Class. Quantum Grav. 18 (2001) 4443-4460, arXiv:hep-th/0104081, DOI 10.1088/0264-9381/18/21/305, https://arxiv.org/abs/hep-th/0104081
4. A. Keurentjes, 'Oxidation = group theory', arXiv:hep-th/0212050, https://arxiv.org/abs/hep-th/0212050
5. X. Bekaert, A. Gomberoff, 'Quantization of the Chern-Simons coupling constant', arXiv:hep-th/0212099, https://arxiv.org/abs/hep-th/0212099
6. F. Englert, L. Houart, A. Taormina, P. West, 'The Symmetry of M-Theories', arXiv:hep-th/0304206, https://arxiv.org/abs/hep-th/0304206
7. F. Englert, L. Houart, 'G+++ invariant formulation of gravity and M-theories: Exact BPS solutions', arXiv:hep-th/0311255, https://arxiv.org/abs/hep-th/0311255
8. A. Kleinschmidt, I. Schnakenburg, P. West, 'Very-extended Kac-Moody algebras and their interpretation at low levels', arXiv:hep-th/0309198, https://arxiv.org/abs/hep-th/0309198
9. S. de Buyl, M. Henneaux, B. Julia, L. Paulot, 'Cosmological billiards and oxidation', arXiv:hep-th/0312251, https://arxiv.org/abs/hep-th/0312251
10. F. Englert, L. Houart, 'G+++-invariant formulation of gravity and M-theories: exact intersecting brane solutions', arXiv:hep-th/0405082, https://arxiv.org/abs/hep-th/0405082
11. A. Keurentjes, 'Time-like T-duality algebra', arXiv:hep-th/0404174, https://arxiv.org/abs/hep-th/0404174
12. P. West, 'Brane dynamics, central charges and $E_{11}$', arXiv:hep-th/0412336, https://arxiv.org/abs/hep-th/0412336
13. A. Keurentjes, 'Poincare Duality and G+++ algebra’s', arXiv:hep-th/0510212, https://arxiv.org/abs/hep-th/0510212
14. F. Riccioni, D. Steele, P. West, 'Duality Symmetries and $G^{+++}$ Theories', arXiv:0706.3659, https://arxiv.org/abs/0706.3659
15. J. Gomis, D. Roest, 'Non-propagating degrees of freedom in supergravity and very extended $G_2$', arXiv:0706.0667, https://arxiv.org/abs/0706.0667
16. A. G. Tumanov, P. West, 'E11 must be a symmetry of strings and branes', arXiv:1512.01644, https://arxiv.org/abs/1512.01644
17. A. G. Tumanov, P. West, 'E11 and exceptional field theory', arXiv:1507.08912, https://arxiv.org/abs/1507.08912
18. P. West, 'A brief review of E theory', arXiv:1609.06863, https://arxiv.org/abs/1609.06863
19. G. Bossard, A. Kleinschmidt, J. Palmkvist, C. N. Pope, E. Sezgin, 'Beyond E11', arXiv:1703.01305, https://arxiv.org/abs/1703.01305
20. P. West, 'E11, Brane Dynamics and Duality Symmetries', arXiv:1801.00669, https://arxiv.org/abs/1801.00669
21. K. Glennon, P. West, 'K27 as a symmetry of closed bosonic strings and branes', arXiv:2409.08649, https://arxiv.org/abs/2409.08649
22. N. Boulanger, P. P. Cook, J. A. O'Connor, P. West, 'Unfolding $E_{11}$', arXiv:2410.21206, https://arxiv.org/abs/2410.21206