The Geometry of Exceptional Super Yang-Mills Theories

Some time ago, Sezgin, Bars and Nishino have proposed super Yang-Mills theories (SYM's) in $D=11+3$ and beyond. Using the"Magic Star"projection of $\mathfrak{e}_{8(-24)}$, we show that the geometric structure of SYM's in $11+3$ and $12+4$ space-time dimensions is recovered from the affine symmetry of the space $AdS_{4}\otimes S^{8}$, with the $8$-sphere being a line in the Cayley plane. By reducing to transverse transformations, along maximal embeddings, the near horizon geometries of the M2-brane ($AdS_{4}\otimes S^{7}$) and M5-brane ($AdS_{7}\otimes S^{4}$) are recovered. Generalizing the construction to higher, generic levels of the recently introduced"Exceptional Periodicity"(EP) and exploiting the embedding of semi-simple rank-3 Jordan algebras into rank-3 T-algebras of special type, yields the spaces $AdS_{4}\otimes S^{8n}$ and $AdS_{7}\otimes S^{8n-3}$, with reduced subspaces $AdS_{4}\otimes S^{8n-1}$ and $AdS_{7}\otimes S^{8n-4}$, respectively. Within EP, this suggests generalizations of the near horizon geometry of the M2-brane and its Hodge (magnetic) duals, related to $(1,0)$ SYM's in $(8n+3)+3$ dimensions, constituting a particular class of novel higher-dimensional SYM's, which we name exceptional SYM's. Remarkably, the $n=3$ level gives $AdS_{4}\otimes S^{23}$, hinting at M2 and M21 branes as solutions of bosonic M-theory, and reduction to $AdS_{3}\otimes S^{23}$ gives support for Witten's monstrous $AdS$/CFT construction.

Through a projection of e 8(−24) along an sl 3,R subalgebra (the so-called "Magic Star" projection, cfr. Fig. 1), the hidden Jordan algebraic structure within e 8(−24) becomes manifest [10][11][12][13]. The central vertex e 6(−26) of the projection then encodes the reduced structure symmetry of the degree three exceptional Jordan algebra (aka Albert algebra) J O 3 [30], mapped to six vertices of the star projection. Using the aforementioned gradings of e 6(−26) , e 7(−25) and e 8(−24) , as well as the magic star, a periodic extension of the exceptional Lie algebras can be formulated. This periodic extension, dubbed Exceptional Periodicity (EP) [13], allows higher dimensional extensions of the exceptional Lie algebras, while also permitting arbitrarily high dimensional magic star projections. Although the resulting algebras (with generalized roots) no longer satisfy the Jacobi relation, they do contain Lie algebraic reductive parts, as well as a normalized cocycle, as seen in lattice vertex algebras [13,14].
In this study, using some gradings of the exceptional Lie algebras and their exceptionally periodic extensions, we show that the geometric structure of SYM's in 11 + 3 and 12 + 4 space-time dimensions can be recovered from the affine symmetry of the space AdS 4 ⊗ S 8 , with the 8-sphere being a line in the Cayley plane OP 2 , the space of rank-1 projectors of J O 3 [15,57]. The symmetry of S 8 , SO(9) (the lightcone little group of M-theory [56]), is a maximal and symmetric subgroup of F 4 , and as such is the stabilizer of OP 2 F 4 /SO(9) itself. A fixed point of OP 2 , a rank-1 idempotent of J O 3 , identifies one of three possible embeddings SO(9) ⊂ F 4 and spans an orthogonal direction that can serve as the 11th dimension of M-theory. By considering transverse directions, along maximal embeddings, the near horizon geometries of the M2-brane (AdS 4 ⊗ S 7 ) and M5-brane (AdS 7 ⊗ S 4 ) are recovered.
Generalizing the construction to higher levels of exceptional periodicity (parametrized by n ∈ N), where Jordan algebras of degree three are lifted to the special class of rank-3 Vinberg's T-algebras [22], maximal embeddings that respect the symmetry of the T-algebraic spin factors yield the spaces AdS 4 ⊗ S 8n and AdS 7 ⊗ S 8n−3 , with reduced subspaces AdS 4 ⊗ S 8n−1 and AdS 7 ⊗ S 8n−4 , respectively. Through exceptional periodicity, this suggests generalizations of the M2-brane and M5-brane near horizon geometries from SYM's in (8n + 3) + 3 space-time dimensions, as descending from SYM's in (8n + 4) + 4 space-time dimensions [16].
The plan of the paper is as follows. Within EP, we explicitly study levels n = 1 (the trivial level, corresponding to exceptional Lie algebras, in particular to e 8(−24) in our case), n = 2 (the first non-trivial level) and n = 3, respectively in Secs. II, III B and III D. Interestingly, at level n = 3, the resulting branchings give AdS 4 ⊗S 23 and hint at M2 and M21 branes as solutions of the (conjectured) bosonic M-theory [17]; by reduction to AdS 3 ⊗ S 23 , we recover a space that lends support for Witten's monstrous AdS/CFT construction for three-dimensional gravity [18], as the Conway group Co 0 (the symmetry of the Leech lattice [19]) is recovered from the SO(24) R-symmetry 1 of a discretized S 23 . On the other hand, our analysis at levels n = 2 and 3 can be bridged by the observation that every K3 sigma model contains a symmetry group that is a subgroup of Co 0 [20].
Such higher-dimensional SYM's can be defined at every level of EP for e 8(−24) , whose generic n-th level is considered in Sec. III C. The resulting "EP/SYM correspondence" (which will be investigated further in suggests a spectral formulation of M-theory from the class of Vinberg's special cubic T-algebras [22]. This generalizes the structure of matrix theory [23] in D = 10 + 1 to a more general class of nonassociative matrix operator algebras that periodically exhibit nonassociative geometry, up to infinite dimensions. This is as Connes [24] has done for noncommutative geometry from noncommutative C * -algebras, which falls under the more general motivic program of Grothendieck [25]. II. e 8(−24) A. N = (1, 0) SYM in 11 + 3 The Cayley plane OP 2 is the projective space of all rank-1 projectors of the exceptional Jordan algebra J O 3 [15]; by stabilizing a point of the Cayley plane, the affine symmetry E 6(−26) is reduced to SO(9, 1) [15]. This SO(9, 1) subgroup acts on a line (an S 8 ) of the Cayley plane as affine transformations. It is also the symmetry of the 10-dimensional spin factor from the Peirce decomposition (cfr. e.g. [26,27]) the fixed point (primitive idempotent of J O 3 ) corresponds to the 1 in the r.h.s. of (2.1), and it serves as a point at infinity for the 8-dimensional transverse space. The affine symmetry and MW semispinor 16 occur in the 3-grading 2 of e 6(−26) : where 16 is the conjugate MW semispinor in 9 + 1 space-time dimensions. (2.2) can be interpreted as the maximal symmetric embedding which is a consequence of the maximal Jordan-algebraic embedding : where J O 2 Γ 9,1 (cfr. e.g. (5.2) of [29]) is a 10-dimensional Lorentzian spin factor [54,55] and R is the span of the fixed idempotent 3 .
The "Magic Star" of e 8(−24) (cfr. Fig. 1 and its caption) is a projection along an sl 3,R subalgebra, with e 6(−26) projected to a central vertex [10][11][12][13]. In e 8(−24) , the analog of so 9,1 is so 12,4 ; indeed, the following maximal symmetric embedding holds : which is still a consequence of (2.4); qconf here denotes the quasi-conformal symmetry of the corresponding rank-3 Jordan algebra [31]. The branching corresponding to (2.5) gives naturally rise to the following 5-grading where 64 and 64 are the MW semispinor and its conjugate in D = s + t = 11 + 3 space-time dimensions. Part of (2.7) has already appeared in [7,8]. As mentioned above, mathematically, it identifies (the minimally non-compact, real form of) a Kantor triple system, of "extended Poincaré type", over e 8 [9]. Moreover, we recall that so 11,3 , occurring in the 0-graded reductive component of the 5-grading (2.7), is the space-time, purely bosonic, symmetry Lie algebra of the N = (1, 0) SYM studied by Sezgin, Bars and Nishino in D = 11 + 3 space-time dimensions [4][5][6] (see also Sec. III A below). Reducing to transversal rotations, the isometry group SO(9) of S 8 OP 1 (which can be regarded as a line in OP 2 ) breaks to SO(8) acting on S 7 . This corresponds to the two-step chain of maximal symmetric embeddings into SO (12,4) : (2.10) breaking down to isometries of AdS 4 by a further maximal and symmetric embedding, yields Alternatively, the following embedding also holds : where SO(3, 2) × SO(8) acts as isometries of AdS 4 ⊗ S 7 , the near horizon geometry of the M2-brane (whose world volume symmetry is described by SO(1, 2)), as a solution of 10 + 1 M-theory (or of its low-energy limit, N = 1 eleven-dimensional supergravity); see e.g. [32][33][34]. Considering the reduction S 5 → S 4 , where S 4 HP 1 can be conceived as a line in HP 2 , the isometry group SO(6) of S 5 breaks to SO(5) over S 4 . Thus, the reduction O → H, reduces OP 1 S 8 → HP 1 S 4 with point at infinity given by a fixed primitive idempotent of J H 3 . This corresponds to the following chain of maximal symmetric embeddings into SO(12, 4) : breaking down to isometries of AdS 7 by a further maximal and symmetric embedding, yields Alternatively, the following embedding also holds : where SO(6, 2) × SO(5) acts as isometries of AdS 7 ⊗ S 4 , the near horizon geometry of the M5-brane, Hodge dual to the M2-brane (whose world volume symmetry SO(1, 2) still occurs as a commuting factor) in 10 + 1 space-time dimensions (see e.g. [32][33][34]).
Note also that SO(12, 4) contains the isometries of AdS 7 and of the M5-brane worldvolume, times a dilatational factor: (2.18) or, equivalently, through two different maximal and symmetric embeddings, the isometries of AdS 7 times the conformal symmetry of the M5-brane worldvolume, or, respectively, the conformal symmetry of AdS 7 times the isometries of the M5-brane worldvolume: (2.20)
We should also recall that the symmetry property of the γ-matrices repeats itself every 8 dimensions in space-time [44], while the chirality (dottedness) of the spinors alternates every two dimensions. In particular, the properties of spinors are defined by two parameters : D = s + t mod(8) and ρ := s − t mod(8) (the mod(8) periodicity being the Bott periodicity; see e.g. [45], and Refs. therein). Thus, the fact that all superalgebras (3.1)-(3.4) are characterized by a MW (semi)spinor generator can be traced back that they all have ρ = 8 = 0 mod (8).
B. The first non-trivial level (n = 2): e 8 (−24) , and it gives naturally rise to the following 5-grading : where 1024 and 1024 are the MW semispinor and its conjugate in 19 + 3 space-time dimensions.
(3.10) is the first non-trivial generalization of the 5-grading (2.7) of e 8(− 24) , and, in light of the discussion in Sec. III A, it provides the vector and spinor structures for a novel, exceptional N = (1, 0) SYM in 19 + 3, generalizing [16] the work of Sezgin, Bars and Nishino [4][5][6]21]. It is worth stressing that 19 + 3 is the signature of a unimodular lattice appearing in the description of the space of periods for a complex K3 surface S and a Kahler class of H 1,1 (S, R) [46]. Also, the moduli space of N = (4, 4) string theories with K3 target space has a discrete symmetry group that is the integral orthogonal group of an even unimodular lattice of signature (20,4) [53]. This may permit further application of (3.10) (with normalized cocycle) in the study of vertex operator algebras for BPS states of K3 sigma models with Mathieu group M 24 symmetry [20].

M2-brane in 18 + 1
Considering the Lie group associated to the reductive (simple) part of e within EP (cfr. Fig. 2). Fixing a rank-1 idempotent of T 8,2 3 induces the following SO(17, 1)-covariant Peirce decomposition [13]: where 256 denotes the MW semispinor in signature 17 + 1, and 1 is the fixed rank-1 idempotent of T 8,2 3 . (3.12) can be regarded as a consequence of the maximal embedding [13,35] which in turn might give rise to a quasi-conformal interpretation of the definition (3.9) itself [14]. The 18-dimensional Lorentzian spin factor Γ 17,1 [30] has SO(17, 1) space-time symmetry, which is also the affine symmetry of S 16 , a sphere of the transverse degrees of freedom with a fixed (rank-1 idempotent) point at infinity. It is here worth recalling that this structure is seen in the 3-grading of the first non-trivial extension e which might enjoy a reduced structure symmetry interpretation, as well [14]. Hence, SO(20, 4) contains maximally (and symmetrically) the affine symmetries of AdS 4 ⊗ S 16 .
Considering the reduction to transversal rotations 5 S 16 → S 15 , and the isometry group SO(17) reduces to SO (16). This corresponds to the two-step chain of maximal symmetric embeddings into SO (20,4) : (16). (3.24) 5 Note that the geometric picture in terms of projective lines in higher projective spaces is lost for all non-trivial levels of EP, namely for n 2, because octonionic (projective) geometry is defined only until dimension two. This is also reflected in the fact that T 8,2 3 is not a rank-3 Jordan algebra, and the triality among its block components within SO(16)-covariant Peirce decomposition is spoiled [13]. Again, another consequence of the aforementioned fact is that EP algebras are not Lie algebras (because the Jacobi identity does not hold on their non-reductive component [13]). However, after Rosenfeld [47], the isometry SO(16) of S 15 can be regarded as the stabilizer of (O ⊗ O) P 2 = E 8 /SO (16), which is an example (with the largest exceptional global isometry) of the so-called Tits' buildings [48,49]. 6 After Rosenfeld [47], the stabilizer SO(12) of S 12 can be regarded as the stabilizer (up to a commuting SU (2) factor) of the Tits' building (H ⊗ O) P 2 = E 7 / (SO(12) × SU (2)) [48,49]. The procedure of the previous subsection can be generalized to an arbitrary level n of EP (characterized by Bott periodicity) as follows [13] (n ∈ N; recall that e 8(−24) ≡ e where 2 (8n+6)/2 is the MW semispinor in (8n + 4) + 4 space-time dimensions, while 2 (8n+6)/2−1 and 2 (8n+6)/2−1 denote the MW spinor and its conjugate in (8n + 3) + 3 space-time dimensions. (3.25) and (3.26) respectively provide the n-th generalization of (3.9) and (3.10) within EP [13]. Again, in light of the discussion in Sec. III A, this is tantalizing evidence for the possible existence of a countably infinite tower (parametrized by n ∈ N) of novel, exceptional N = (1, 0) SYM's in (8n + 3) + 3 space-time dimensions. This generalization of the work by Sezgin, Bars and Nishino [4][5][6]21], briefly considered in Sec. III A, is the object of a forthcoming paper [16].
2. M2-brane in (8n + 2) + 1 Considering the Lie group associated to the reductive (simple) part of e Once again, SO(3, 3) yields affine transformations of AdS 4 . On the other hand, SO(8n + 1, 1) can be regarded as the affine symmetry of S 8n , which is the sphere acquired from T 8,n 3 , the rank-3 T-algebra of special type [22] which provides the n-th generalization of the Albert algebra J O 3 ≡ J 8 3 ≡ T 8,1 3 within EP (cfr. Fig. 2). Fixing a rank-1 idempotent of T 8,n 3 induces the following SO(8n + 1, 1)-covariant Peirce decomposition [13]: where 2 4n denotes the MW semispinor in signature (8n + 1) + 1, and 1 is the fixed rank-1 idempotent of T 8,n 3 . (3.28) can be regarded as a consequence of the maximal embedding [13,35] which in turn might give rise to a quasi-conformal interpretation of the definition (3.25) itself [14]. The (8n + 2)dimensional Lorentzian spin factor Γ 8n+1,1 has SO(8n + 1, 1) space-time symmetry, which is also the affine symmetry of S 8n , a sphere of the transverse directions with a fixed idempotent point at infinity. It is here worth recalling that this structure is seen in the 3-grading of the n-th generalization e (n) 6(−26) of e 6(−26) within EP [35]: which might enjoy a reduced structure symmetry interpretation, as well [14]. Hence, SO(8n+4, 4) contains maximally (and symmetrically) the affine symmetries of AdS 4 ⊗ S 8n . Considering the reduction to transversal rotations 8 S 8n → S 8n−1 , and the isometry group SO(8n + 1) reduces to SO(8n). This corresponds to the two-step chain of maximal symmetric embeddings into SO(8n + 4, 4) : Breaking down to isometries of AdS 7 by a further maximal and symmetric embedding, yields: Alternatively, the following embedding also holds : where SO(6, 2) × SO(8n − 3) acts as isometries of AdS 7 ⊗ S 8n−4 , which can thus be regarded as a generalization of the near horizon geometry of the M5-brane to an M(8n − 3)-brane, which is the Hodge dual of M2. 10 In other words, the AdS 7 ⊗ S 8n−3 geometry, "dual" to AdS 4 ⊗ S 8n , has a spherical factor with exactly the dimensionality of the magnetic brane 11 (Hodge dual of the M2-brane) in (8n + 2) + 1 dimensional Lorentzian spacetime. In fact, as yielded by (3.34), the very dual magnetic AdS 7 ⊗ S 8n−3 geometry results from reducing SO(8n + 4, 4) with respect to the affine symmetry SO(8n − 2, 1) of S 8n−3 .

M2-brane in bosonic M-theory
Considering the Lie group associated to the reductive (simple) part of e where SO(3, 3) yields affine transformations of AdS 4 and SO(25, 1) can be regarded as the affine symmetry of S 24 , or alternatively as the space-time symmetry of bosonic string theory [17]. By specializing to n = 3 the previous n-parametrized treatment, SO(25, 1) can be regarded as the affine symmetry of S 24 , which is the sphere acquired from T 8,3 3 , the rank-3 T-algebra of special type [22] which provides the 3rd generalization of the Albert algebra within EP (cfr. Fig. 2). Fixing a rank-1 idempotent of T 8, 3 3 induces the following SO(25, 1)-covariant Peirce decomposition [13]: where 2 12 denotes the MW semispinor in signature 25 + 1, and 1 is the fixed rank-1 idempotent of T 8,3 3 . (3.44) can be regarded as a consequence of the maximal embedding [13,35] The 26-dimensional Lorentzian spin factor Γ 25,1 has bosonic string theory space-time symmetry, which is also the affine symmetry of S 24 , a sphere of the transverse degrees of freedom with fixed idempotent point at infinity. The span of this fixed idempotent can serve as a 27th dimension for D = 27 M-theory [17]. Note, bosonic string theory space-time symmetry is seen in the 3-grading of the 3rd generalization e  where SO(6, 2) × SO(21) acts as isometries of AdS 7 ⊗ S 20 , which can thus be regarded as a generalization of the near horizon geometry of the M5-brane to an M21-brane, Hodge dual of the M2-brane, in 26 + 1 space-time dimensions. In view of the considerations above, and of Horowitz and Susskind's conjectured M21-brane in [17], AdS 7 ⊗ S 20 would also provide support for a possible vacuum of bosonic M-theory, "dual" to AdS 4 ⊗ S 23 . Note also that SO (28,4) contains the isometries of AdS 7 and of the M21-brane worldvolume, times a dilatational factor: SO(28, 4) ⊃ SO(6, 2) × SO(21, 1) × SO(1, 1), (3.54) or, equivalently, through two different maximal and symmetric embeddings, the isometries of AdS 7 times the conformal symmetry of the M21-brane worldvolume, or, respectively, the conformal symmetry of AdS 7 times the isometries of the M21-brane worldvolume: where SO(2, 2) × SO(24) yields isometries of AdS 3 ⊗ S 23 . We note that the space AdS 3 ⊗ S 23 is especially interesting, as it lends support for the Monster AdS/CFT for three-dimensional gravity proposed by Witten [18]. If we suppose AdS 3 ⊗ S 23 is a possible vacuum of bosonic string theory 12 , where the expected R-symmetry is SO(24) (from D = 27 M-theory reduced to D = 3), we can identify the Conway group Co 0 ⊂ M (where M is the Monster group) in SO(24) as acting on a discretized S 23 with points given by vectors of the Leech lattice [19,50], in which the first shell of 196, 560 vectors have norm four [19]. This allows a finite group action as the Conway group Co 0 is the automorphism group of the Leech lattice [19], acting as isometries of the discretized 13 S 23 . The appearance of the Conway group is more than fortuitous, as every K3 sigma