Exceptional Super Yang-Mills in $D=27+3$ and Worldvolume M-Theory

Bars and Sezgin have proposed a super Yang-Mills theory in $D=11+3$ space-time dimensions with an electric 3-brane that generalizes the 2-brane of M-theory. More recently, the authors found an infinite family of exceptional super Yang-Mills theories in $D=(8n+3)+3$ via the study of exceptional periodicity (EP). A particularly interesting case occurs in signature $D=s+t=27+3$, where the superalgebra supports an electric 11-brane and its 15-brane magnetic dual. The worldvolume symmetry of the 11-brane has signature $D=11+3$ and can reproduce super Yang-Mills theory in $D=11+3$. Upon reduction to $D=26+2$, the 11-brane reduces to a 10-brane with $10+2$ worldvolume signature. A single time projection gives a $10+1$ worldvolume signature and can serve as a model for $D=10+1$ M-theory as a reduction from the $D=26+1$ signature of bosonic M-theory. Extending previous results of Dijkgraaf, Verlinde and Verlinde, we also put forward the realization of spinor generators of EP algebras as total cohomologies of (the largest spatially extended) branes which centrally extend the $(1,0)$ superalgebra underlying the corresponding exceptional super Yang-Mills theory.

In this work, we ascend to D = 27 + 3 space-time dimensions, in which an electric 11-brane and its 15-brane magnetic dual arise as central extensions of the (1, 0) global supersymmetry algebra. In particular, the 11-brane gives rise to a worldvolume theory with 11 + 3 signature, thus providing a worldvolume embedding for the chiral SYM in 11 + 3 of Bars and Sezgin [3][4][5].
Following the 11-brane in the reduction from 27 + 3 −→ 26 + 2 −→ 26 + 1 leads to the reduction of the D = 11 + 3 worldvolume of the 11-brane to a D = 10 + 1 worldvolume of a 10-brane, suggesting that M-theory may be a worldvolume theory. This chain of reductions along the worldvolume of the electric 11-brane yields a natural map of the conjectured "bosonic M-theory" of Horowitz and Susskind [14] in 26 + 1 down to M-theory in 10 + 1. Moreover, the electric 10-brane in D = 26 + 2 has a 14-brane magnetic dual (both centrally extending the corresponding (1, 0) global superalgebra). Upon reduction to D = 26 + 1, this implies the existence of a "dual" (worldvolume-realized) M-theory in D = s + t = 13 + 1. The (1, 0) superalgebra in D = 27 + 3 space-time dimensions (corresponding to the level n = 3 of EP [8][9][10]) takes the form [11] 27 + 3 : Namely, the central extensions are given by a 3-brane, a 7-brane, an electric 11-brane and its dual, a magnetic 15brane. Note that the magnetic duals of the 3-brane and 7-brane, i.e. the 23-brane resp. 19-brane, do not centrally extend the algebra (2.1); however, they can be found as the largest spatially extended central charges at n = 5 resp. n = 4 levels of EP [11].
This simple reasoning yields the following consequences: • it puts forward the realization of 10 + 1 M-theory as a worldvolume theory of an electric 10-brane in a higher 26 + 1 space-time (pertaining to bosonic M-theory of Horowitz and Susskind [14]); • as such, it provides a map from the bosonic M-theory in D = 26 + 1 to M-theory in D = 10 + 1; • we observe that bosonic M-theory can be completed to a two-time theory in D = 26 + 2, in which a (1, 0) exceptional SYM can be defined, with central extensions given by [11] 26 + 2 : i.e. by a 2-brane, a 6-brane, and by an electric 10-brane and its dual, a magnetic 14-brane. This implies that there exists a theory which is dual to the worldvolume-realized 10 + 1 M-theory embedded in 26 + 2, namely the worldvolume-realized 14 + 2 theory reduced to 13 + 1 of the magnetic 13-brane embedded in 26 + 1; we dub such a 13 + 1 theory the "dual M-theory", and we leave its study for future work.
where 32768 = 2 15 is the MW semispinor in 28 + 4, while 16384 = 2 14 and (16384) = 2 14 denote the MW spinor and its conjugate in D = 27 + 3. e . By denoting with 2 N −1 and 2 N −1 the chiral semispinor representations of so 2N , as well as with ∧ i N the rank-i antisymmetric (i-form) representation of so N , we recall that, as a consequence of the well known cohomological structure of Clifford algebra Cl(N ) in N dimensions (see e.g. [15], and Refs. therein), it holds that Thus, in the case under consideration, the 2 15 -dimensional MW semispinor 32768 of so 28,4 , which branches as 16384 ⊕ 16384 under so 28,4 → so 27,3 , can be regarded as the total cohomology of a 15-brane, which in turn can be identified with the maximally spatially extended central charge of N = (1, 0) SYM (2.1) in 27 + 3 space-time dimensions [11]. This is nothing but the n = 3 case of a general fact, namely that the (chiral) spinor component 2 4n+3 of the EP algebra e (n) can be realized as the total cohomology of a (4n + 3)-brane, which in turn can be identified with the largest spatially extended central extension of the (1, 0) supersymmetry algebra in (8n + 3) + 3 space-time dimensions [11] (8n + 3) + 3 : Therefore, the spinor generators of the EP algebra e (n) 8(−24) are realized, exploiting the central extensions of the (1, 0) supersymmetry algebra in (8n + 3) + 3, in terms of brane cohomology. We stress that this realization extends the results found by Dijkgraaf, Verlinde and Verlinde [16] which, in the BPS quantization of the 5-brane, realized the 16 components of the central charge as fluxes through the odd homology cycles on the five-brane itself. Since the spinor generators are the very ones responsible for the violation of the Jacobi identity in EP algebras [8][9][10], this is a further hint that the Lie subalgebra of EP yields a purely bosonic sector. Moreover, it is worth here reminding that the n = 1 (trivial) level of EP boils down to the fact the spinor component 128 of e can be realized as the total cohomology of the 7-brane which centrally extends the (1, 0) superalgebra in 11 + 3 [3][4][5][6].
As resulting from the star-shaped algebraic structure of EP algebras [8][9][10], we note that the cubic Vinberg's T-algebra [17] at EP level n = 3 can be decomposed (in a manifestly so 25,1 -covariant way) as where 26 and 4096 = 2 12 respectively are the vector and MW semispinor irreprs. of so 25,1 . In 25 + 1 space-time dimensions, the 24 (light-cone) transverse dimensions can be projectively mapped to the Cayley-Moufang plane OP 2 , thus recovering three chart-wise Hopf fibrations (i = 1, 2, 3): 1, 2, 3). (3.8) In particular, the third orthogonal idempotent 1 of the T-algebra (3.7) enhances the signature to 26 + 1, which in turn reduces to 10 + 1 along a sixteen-dimensional Cayley-Moufang plane chart map; this provides a simple explanation, within the n = 3 level of EP, of a remarkable observation by Ramond and Sati [18][19][20][21], asserting that M-theory with hidden OP 2 fibers underlies the origin of the massless multiplet of eleven-dimensional supergravity in terms of representations of F 4 decomposed with respect to its maximal compact subgroup Spin (9). Also, we observe that, as resulting from [22], the semispinor 4096 = 2 12 can be realized as a module for the double cover of the centralizer of an element in the class 2B of the Monster Group. Wilson has given an explicit construction of the Leech lattice in terms of O 3 elements [23], which in turn can be projectively mapped to the aforementioned Cayley-Moufang plane OP 2 [24]. As resulting from [8], the spinorial "extended roots" at EP level n = 3 have norm n + 1 = 4, thus providing norm-four vectors, as expected for the Leech lattice Λ L [9]. In this framework, we put forward the conjecture that at non-perturbative level the 24 transverse directions can be realized as vectors of Λ L , which we recall to be the unique even unimodular lattice with no roots in D = 24; taken projectively, the map reduces to a discrete version of the Hopf fibrations (3.8), with 3×65520 = 196560 vectors, whereas S 7 gets discretized in terms of the 240 roots of e 8 , such that 273 × 240 = 65520, as observed by Wilson [23]. The 273-dimensional part is recovered from the T-algebra T 8,2 3 (EP level n = 2), with light-cone coordinates removed. This intriguing scenario, deserving a detailed study, is left for future investigation.

IV. CONCLUSION
Using the exceptional SYM theory in 27 + 3 space-time dimensions, whose (1, 0) non-standard global superalgebra can be centrally extended by an electric 11-brane and its 15-brane magnetic dual [11], we considered the (multi-time) worldvolume theory of the 11-brane itself as support for the (1, 0) SYM theory in 11 + 3 space-time dimensions as introduced by Bars and Sezgin some time ago [3][4][5].
Last but not least, extending the results of [16], we have put forward the intriguing brane-cohomological interpretation of spinors, in particular for the spinor generators of EP algebras. This entangles the algebraic structure of exceptional periodicity with the central extensions of exceptional super Yang Mills theories in higher dimensional space-times.