G+++-invariant formulation for the C_{q+1} maximally oxidised theories: exact intersecting brane solutions and the root-orthogonality correspondence

The very-extended Kac-Moody algebra $\mathcal{G}^{+++}$ (obtained by adding three nodes to the Dynkin diagram of a simple Lie algebra $\mathcal{G}$) is conjectured to underlie the hidden symmetry of the gravitational theory whose three-dimensional reduction exhibits the split hidden symmetry group $G$. Englert and Houart constructed, for every simple $\mathcal{G}$, an action invariant under the non-linear realisation of $\mathcal{G}^{+++}$, and proved a threefold theorem for all maximally oxidised theories with at most one dilaton: the standard intersection rules for extremal branes are equivalent to orthogonality of the associated real positive roots; the $\mathcal{G}^{+++}$ action admits an exact solution for two intersecting branes exactly when their roots are orthogonal and their sum is not a root; and these exact solutions are in one-to-one correspondence with the intersecting extremal brane solutions of the corresponding maximally oxidised theory. The exception to these theorems is the four-dimensional $C_{q+1}$ family — gravity with $q$ dilatons, an axionic sector, and vector fields — excluded because the known intersection rules assume a single dilaton; the $C_{q+1}^{+++}$-invariant action itself was constructed in the companion paper (JHEP 01 (2004) 002), whose Appendix A.3 treats this family explicitly. Later literature (2005–2007) resolved the paper's companion open question, the classification of the exotic signature phases, but the extension of the intersecting-brane theorems to $C_{q+1}$ has never been carried out. The problem asks for that extension: the multi-dilaton intersection rules and their equivalence (or not) to root orthogonality in $C_{q+1}^{+++}$, and the exact-solution correspondence.

Background

Maximally oxidised theories. A wide class of gravitational models is organised by hidden symmetry as follows. A theory of gravity in $D$ space-time dimensions coupled to differential-form gauge fields ("forms") and scalar fields ("dilatons") is called maximally oxidised if it is the highest-dimensional theory in its toroidal-reduction orbit and if its reduction to three dimensions yields scalar fields parametrising a coset $G/H$, where $G$ is a real split simple Lie group (its maximally non-compact real form, in which the root system is realised over the reals) and $H$ its maximal compact subgroup. Cremmer, Julia, Lü and Pope classified all such theories (arXiv:hep-th/9909099): they include pure gravity in any $D$ (with $G=A_{D-3}$ in Lie-algebra notation), the bosonic sector of 11-dimensional supergravity ($G=E_8$), the bosonic-string common sector ($G=D_{D-2}$), and — the family at the centre of this problem — one four-dimensional family with $G=C_{q+1}$: gravity coupled to $q$ dilatons, an axionic sigma-model sector (pseudoscalar "axion" fields obtained by dualising two-form gauge fields, parametrising a coset), and vector gauge fields together with their duals.

Very-extended Kac-Moody algebras. To any simple Lie algebra $\mathcal{G}$ one associates an infinite-dimensional Lorentzian Kac-Moody algebra $\mathcal{G}^{+++}$ (the "very extension") by adjoining three nodes to the Dynkin diagram of $\mathcal{G}$: the affine node, the overextended node, and the very extended node. The mathematics of such Lorentzian Kac-Moody algebras was developed by Gaberdiel, Olive and West, and their appearance in gravity grew out of the cosmological billiards analysis of space-like singularities (the observation that, near such a singularity, the dynamics of the scale factors reduces to a billiard motion whose reflecting walls form a Lorentzian Kac-Moody root system) and the conjecture that M-theory possesses an $E_{11}=E_8^{+++}$ hidden symmetry.

The $\mathcal{G}^{+++}$-invariant formulation. In two companion papers (JHEP 01 (2004) 002 and the source paper, JHEP 05 (2004) 059), Englert and Houart constructed, for every simple $\mathcal{G}$, an action $S_{\mathcal{G}}$ invariant under the non-linear realisation of $\mathcal{G}^{+++}$: a world-line field parametrises the coset $\mathcal{G}^{+++}/K(\mathcal{G}^{+++})$, where the denominator subalgebra is the fixed-point set of a "temporal involution" — a substitute for the Chevalley involution chosen so that the level decomposition of $\mathcal{G}^{+++}$ with respect to the $A_{D-1}$ "gravity line" subalgebra exhibits $SO(1,D-1)$ tensor structure at every level, $D$ being the dimension of the corresponding maximally oxidised theory. (A level decomposition organises the infinitely many fields of the Kac-Moody algebra into finite sets labelled by an integer level, the number of powers of the very extended generators. The Chevalley involution is the standard sign-flip automorphism of a Lie algebra; the temporal variant is the one tailored to Lorentzian signature.) The actions are constructed recursively from the level expansion and cover all very-extended algebras, including the multi-dilaton cases; Appendix A.3 of the companion paper supplies the $C_{q+1}^{+++}$-specific data. Exact solutions of these actions reproduce the extremal $p$-branes — charged, spatially extended objects saturating a charge/mass (BPS) bound — of the maximally oxidised theories.

Intersection rules and the orthogonality theorems. Extremal branes can overlap orthogonally with zero binding energy only subject to combinatorial intersection rules (derived for one-dilaton theories by Argurio, Englert and Houart): for two branes of spatial dimensions $q_A$ and $q_B$ in $D$ dimensions sharing $\bar q$ spatial directions, the allowed values of $\bar q$ are fixed by $D$ and by the dilaton couplings of the participating field strengths. The source paper proved, for all maximally oxidised theories with at most one dilaton: (Theorem 1) the intersection rules are equivalent to orthogonality $\beta\cdot\gamma=0$ of the real positive roots $\beta,\gamma$ of $\mathcal{G}^{+++}$ associated with the two branes (the roots of a Kac-Moody algebra are the weight vectors dual to its Cartan generators; a root is positive when its coefficients in the simple-root basis are all non-negative, and real when it is Weyl-conjugate to a simple root, which is the class of non-zero norm to which brane charges are assigned; each brane corresponds to such a root whose components encode its charge and dilaton coupling); (Theorem 2) the $\mathcal{G}^{+++}$-invariant action admits an exact solution describing the intersection exactly when $\beta\cdot\gamma=0$ and $\beta+\gamma$ is not a root; and (Theorem 3) the exact solutions of $S_{\mathcal{G}}$ are in one-to-one correspondence with the intersecting extremal brane solutions of the maximally oxidised theory, including purely gravitational branes. All of this excludes the $C_{q+1}$ family by construction, because the $C_{q+1}$ theories contain $q$ dilatons whereas the intersection-rule analysis assumed a single dilaton.

Exotic phases. In string and M-theory, dualities acting on a timelike circle (timelike T-duality, and the U-dualities containing it) relate theories with different space-time signatures $(T,S)$ — the numbers of timelike and space-like directions — and with flipped signs of form kinetic terms; Hull called the resulting phases $M^*$ and $M'$ in the M-theory case, and Hull and Khuri constructed brane solutions in such exotic signatures. Within the $\mathcal{G}^{+++}$ formalism the temporal involution is not invariant under all Weyl reflections (the reflections of the Dynkin diagram generating the algebra's symmetry group), as shown by Keurentjes, so exotic phases arise automatically as Weyl-related formulations; the source paper showed that whenever they exist the intersecting-brane analysis extends to them, because the generalised intersection rules depend only on the branes' total dimensions, not on the signature or kinetic-term signs. Which exotic phases arise for each $\mathcal{G}^{+++}$ was left open there; Englert, Henneaux and Houart later identified the $G_B^{++}$-invariant truncation of each $\mathcal{G}^{+++}$ theory as the sector carrying its exact brane solutions, de Buyl, Houart and Tabti then derived, for every simple family, all signatures and kinetic-term sign patterns reachable by Weyl reflections in that truncation, and Keurentjes independently encoded the same classification in level-independent, Weyl-invariant conditions implementing Poincaré duality of the form fields. Notably, $C_{q+1}^{+++}$ admits no signature-changing Weyl reflections at all, so no exotic phases are expected in that family.

The remaining gap. What no published work does — and what this problem asks — is the extension of the intersecting-brane theorems to the $C_{q+1}$ family: the multi-dilaton intersection rules and their comparison with root orthogonality in $C_{q+1}^{+++}$, and the exact-solution correspondence. The $C_{q+1}^{+++}$-invariant action itself is already in place — the companion paper's generic recursive construction covers every very-extended algebra, including the multi-dilaton cases, and its Appendix A.3 gives the $C_{q+1}^{+++}$ simple-root data and the four-dimensional $q$-dilaton action — and the further ingredients exist: the low-level field content of the $C$-series very extensions was tabulated by Kleinschmidt, Schnakenburg and West, and the associated four-dimensional theories were described by Keurentjes. But the multi-dilaton intersection rules and the theorems have never been carried out.

Problem Statement

Extend the exact intersecting brane solutions and the root-orthogonality correspondence of the $\mathcal{G}^{+++}$-invariant formulation of Englert and Houart to the $C_{q+1}$ maximally oxidised theories — the four-dimensional theories of gravity coupled to $q$ dilatons, an axionic sigma-model sector, and vector gauge fields with their duals — the family that the source paper's footnote 3 explicitly excludes from the intersecting-brane theorems because those theorems assume a single dilaton. The $C_{q+1}^{+++}$-invariant action with temporal involution is already available and is not itself the object of the question: the companion paper (arXiv:hep-th/0311255) constructs the actions recursively from a level expansion for all very-extended algebras, including the multi-dilaton cases, and its Appendix A.3 supplies the $C_{q+1}^{+++}$-specific data. Concretely, for the very-extended algebra $C_{q+1}^{+++}$, with the level decomposition taken with respect to its $A_3$ gravity line (so that level zero reproduces four-dimensional gravity and the low levels reproduce the field content of the $C_{q+1}$ theory, as tabulated by Kleinschmidt–Schnakenburg–West):

(i) Derive the intersection rules for extremal branes of the four-dimensional $C_{q+1}$ theory — the multi-dilaton generalisation of the single-dilaton rules, including the degenerate four-dimensional cases such as the instantonic value $\bar q=-1$ of the intersection parameter (which admits an interpretation in terms of Euclidean instantons) and purely gravitational branes — and prove or refute the analogue of Theorem 1: that these rules are equivalent to the orthogonality condition $\beta\cdot\gamma=0$ on the real positive roots $\beta,\gamma$ of $C_{q+1}^{+++}$ associated with the intersecting branes.

(ii) Prove or refute the analogues of Theorems 2 and 3: the $C_{q+1}^{+++}$-invariant action admits an exact solution for the intersection of two branes with roots $\beta,\gamma$ exactly when $\beta\cdot\gamma=0$ and $\beta+\gamma$ is not a root; and the exact solutions of the $C_{q+1}^{+++}$ action are in one-to-one correspondence with the intersecting extremal brane solutions of the four-dimensional $C_{q+1}$ theory. The two analogues are linked: the correspondence is stated on the exact solutions whose existence the Theorem-2 analogue asserts, so the two counterpart claims in (ii) are jointly posed rather than independent.

The common objective uniting the two parts is the completion of the root/brane dictionary and exact-solution correspondence for the one maximally oxidised family outside the scope of the known theorems.

References

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