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Home , Cayley plane, E8 lattice, Leech lattice

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1 Introduction 2

2 Octonions, E8 and the 3 2.1 TheOctonions ...... 3 2.2 E8 andtheLeechLattice ...... 3

3 Jordan Algebras and D = 5 Magic Supergravity 4 3.1 JordanAlgebras ...... 4 3.2 D =5MagicSupergravity...... 5

4 The Leech Lattice and U-Duality 6 4.1 The Leech Lattice and BPS Black Holes ...... 6 4.2 Conway’s Group Co0 and U-Duality ...... 6

5 Conclusion 7

1 Introduction

Recently, Wilson has given a three-dimensional octonionic construction of the Leech lattice and has shown its , Conway’s group Co = 2 Co , is generated by 3 3 matrices over the octonions [1, 2]. Oc- 0 · 1 × tonions have found wide application in the black hole-qudit correspondence [3]-[9][19, 34, 36, 39, 40], through the study of the quantum informational in- terpretations of supergravities arising from toroidally compactified M-theory and magic supergravity theories [10]-[23],[28]-[30][33, 37, 38]. The = 2, N D = 5 exceptional magic supergravity, in particular, has the exceptional O Jordan algebra over the octonions J3 as its charge space with U-duality group ( 26) [14, 20]. Building− on the results of Wilson, we show the generators of the auto- morphism group of the Leech lattice are elements of , expressed as unitary 3 3 matrices over the octonions O. As F4 E6( 26), the generators of the × ⊂ − automorphisms of the Leech lattice can be interpreted as U-duality transfor- mations in = 2, D = 5 exceptional magic supergravity. We map points N of the Leech lattice to elements of the exceptional Jordan algebra, where generators of the full automorphism group of the Leech lattice are rotations in the charge space of BPS black holes.

2 2 Octonions, E8 and the Leech Lattice

2.1 The Octonions Let V be a finite dimensional vector space over a field F = R, C. An algebra structure on V is a bilinear map V V V (1) × → (x,y) x y. 7→ • A composition algebra is an algebra A = (V, ), admitting an identity ele- • ment, with a non-degenerate η satisfying x,y A η(x y)= η(x)η(y). (2) ∀ ∈ • If x A such that x = 0 and η(x) = 0, η is said to be isotropic and gives ∃ ∈ 6 rise to a split composition algebra. When x A, x = 0, η(x) = 0, η is ∀ ∈ 6 6 anisotropic and yields a composition division algebra. Theorem 2.1. A finite dimensional vector space V over F = R, C can be endowed with a composition algebra structure if and only if dimF(V ) = 1, 2, 4, 8. If F = C, then for a given dimension all composition algebras are isomorphic. For F = R and dimF(V ) = 8 there are only two non-isomorphic composition algebras: the octonions O for which η is anisotropic and the split-octonions Os for which η is isotropic and of signature (4, 4). Moreover for all composition algebras, the quadratic form η is uniquely defined by the algebra structure. The composition division algebra O has an orthonormal basis e = 1, e , ..., e labeled by the projective line PL(7) = F , with { 0 1 7} {∞} ∪ 7 product given by e e = e e = e and images under et et and 1 2 − 2 1 4 7→ +1 et e t [1, 2]. The is given by η(x) = xx, where x is the conjugate 7→ 2 of x.

2.2 E8 and the Leech Lattice

The embeds in the octonion algebra O in many different ways. Following Wilson [1, 2], we take the 240 roots to be 112 octonions et eu for ± ± any distinct t, u PL(7), and 128 octonions of the form 1 ( 1 e e ) ∈ 2 ± ± 1 ±···± 7 that have an odd number of minus signs. The lattice spanned by these 240 octonions will be denoted as L, which is a scaled copy of the E8 lattice. Using L as a copy of E8 in the octonions, the octonionic Leech lattice ΛO is defined [1, 2] as the set of triples (x,y,z) O3, with norm 1 ∈ N(x,y,z)= 2 (xx + yy + zz), such that 1. x,y,z L ∈ 2. x + y,x + z,y + z Ls ∈ 3. x + y + z Ls ∈ Wilson has shown ΛO is isometric to the Leech lattice [2].

3 Theorem 2.2. (Wilson) The full automorphism group Co0 of the Leech lattice is generated by the following symmetries [1]: (i) an S3 of coordinate permutations; (ii) the map r0 : (x,y,z) (x,ye1, ze1) 1 7→ (iii) the maps x (x(1 e ))(1 + et) for t = 2, 3, 4, 5, 6, 7 7→ 2 − 1 (iv) the map (x,y,z) 1 ((y + z)s,xs y + z,xs + y z) 7→ 2 − − In the proof of the theorem, to construct a non-monomial symmetry, Wilson takes s to be a complex number s = 1 ( 1+ √ 7), satisfying ss = 2 [1]. 2 − − Note: The maps given in (iii) act on all three coordinates simultaneously.

Corollary 2.3. The full automorphism group Co0 of the Leech lattice is generated by F4 symmetries.

Proof. The S3 coordinate permutations of Wilson’s theorem are given by six matrices in the usual O(3) representation, where it is known O(3) F [24]. ⊂ 4 For the remaining symmetries, we note that F contains all unitary 3 3 4 × matrices over the octonions [24, 30, 32] and give the corresponding unitary matrices for each symmetry listed in Wilson’s theorem. The matrices are explicitly:

1 0 0   (ii) : R1 = 0 e1 0  0 0 e1  1 e 0 0 − 1 (iii) : R = 1  0 1 e 0  , 2 √2 − 1  0 0 1 e  − 1 1+ et 0 0 R = 1  0 1+ e 0  3 √2 t  0 0 1+ et  0 s s (iv) : R = 1  s 1 1  4 2 −  s 1 1  − and noting e2 = e2 = 1 and ss = 2, it is seen the matrices are unitary. 1 t − 3 Jordan Algebras and D =5 Magic Supergravity

3.1 Jordan Algebras A Jordan algebra over a field F = R, C is a vector space over F equipped with a bilinear form (Jordan product) (X,Y ) X Y satisfying X,Y : → ◦ ∀ X Y = Y X ◦ ◦ X (Y X2) = (X Y ) X2. ◦ ◦ ◦ ◦

4 Given an associative algebra , we can define the Jordan product using A ◦ the associative product in , given by X Y 1 (XY + Y X). Under A ◦ ≡ 2 the Jordan product, ( , ) becomes a special Jordan algebra. Any Jordan A ◦ algebra that is simple and not special is called an exceptional Jordan algebra or Albert Algebra. The exceptional Jordan algebras are 27-dimensional R O Os Jordan algebras over , isomorphic to either J3 or J3 . The automorphism A and reduced structure groups for these algebras are denoted as Aut(Jn ) A O and Str0(Jn ), respectively. In the case of J3 , the Jordan algebra of 3 3 O O O × Hermitian matrices over , Aut(J3 ) = F4 and Str0(J3 ) = E6( 26) [14, 20, 24]. − O The rank one elements of J3 are points of the octonionic projective 2 plane OP = F4/SO(9), also known as the Cayley plane. F4 and E6( 26) act on the Cayley plane as isometry and collineation groups, respectively− [24, 25, 26, 30].

3.2 D =5 Magic Supergravity In D = 5, the = 8 and = 2 magic supergravities are coupled to 6, 9, N N 15 and 27 vector fields with U-duality groups SL(3, R), SL(3, C), SU ∗(6), E6( 26) and E6(6), respectively [4, 13, 14]. The orbits of BPS black hole so- lutions− were classified [13, 22] by studying the underlying Jordan algebras of A degree three under the actions of their reduced structure groups, Str0(J3 ), corresponding to the U-duality groups of the = 8 and = 2 supergravi- N N ties. This is seen by associating a given black hole solution with charges qI (I = 1, ..., nV ) an element

n r1 A1 A2 I   J = q eI = A r A ri R, Ai A (3) X 1 2 3 ∈ ∈ I=1  A2 A3 r3  A A of a Jordan algebra of degree three J3 over a composition algebra , where the eI form a basis for the nV -dimensional Jordan algebra. This establishes a correspondence between Jordan algebras of degree three and the charge spaces of BPS black holes [13]. The entropy of a black hole solution can be expressed [4, 13, 23] in the form

S = π I (J) (4) p| 3 | where I3 is the cubic invariant given by the

I3 = det(J). (5) The U-duality orbits classify the black hole solutions via rank as Rank J = 3 iff I (J) = 0 S = 0, 1/8-BPS 3 6 6 Rank J = 2 iff I (J) = 0, J ♮ = 0 S = 0, 1/4-BPS (6) 3 6 Rank J = 1 iff J ♮ = 0, J = 0 S = 0, 1/2-BPS 6 5 and J ♮ is the quadratic adjoint map given by 1 J ♮ = J J = J 2 tr(J)J + (tr(J)2 tr(J 2))I. (7) × − 2 −

The U-duality group E6( 26) preserves the cubic invariant and thus the entropy of BPS black holes− in = 2, D = 5 exceptional magic supergravity. N 4 The Leech Lattice and U-Duality

4.1 The Leech Lattice and BPS Black Holes Recall that Wilson’s octonionic construction of the Leech lattice uses triples v = (x,y,z) O3, where x,y,z L, with L being a copy of the E root ∈ ∈ 8 lattice. We can thus map such triples to elements of the exceptional Jordan O algebra J3 via: O v vvT = V J (8) 7→ ∈ 3 This allows one to map points of the Leech lattice to black hole charge vectors in = 2, D = 5 exceptional magic supergravity. Denoting the set N O of all such mapped points as J3 (Λ) we recover a map from the (octonionic) Leech lattice to the exceptional Jordan algebra:

O ΛO J (Λ) (9) → 3 O We shall refer to the black hole solutions with charge vectors in J3 (Λ) as the Leech BPS black holes, which in general admit charges in a 24-dimensional charge space since they are constructed from elements of O3.

4.2 Conway’s Group Co0 and U-Duality Given a charge vector for a Leech BPS black hole, we can act on it via the F4 generators of the Co0 as:

V UVU † = V ′ (10) 7→ where UVU † is unambiguous since each U contains only a single octonion. Wilson has given an octonionic description of the 196560 (norm four) vectors that span the Leech lattice [2]. They are given by: (2λ, 0, 0) (3 240 = 720) × (λs, (λs)j, 0) (3 240 16 = 11520) ± × × ((λs)j, λk, (λj)k) (3 240 16 16 = 184320) ± ± × × × where λ is a root of the E lattice and j, k et t PL(7) and all 8 ∈ {± | ∈ } permutations of the three coordinates are allowed. Mapping the 196560 norm four vectors of the first shell of the Leech O lattice to rank one elements of J3 corresponds to mapping to 1/2-BPS

6 black hole charge vectors. Recall that geometrically such charge vectors are points of the Cayley plane OP2, so the generators of the Conway group Co0, which are 3 3 unitary matrices over the octonions (hence elements of × 2 F4) give isometries of these 1/2-BPS black hole charge vectors in OP . As F4 E6( 26), the isometries are U-duality rotations preserving the vanishing ⊂ − entropy of the corresponding 1/2-BPS black hole solutions. As Co is the double cover of Conway’s group Co , and 21+24 Co 0 1 · 1 is a maximal subgroup of the Monster, it is natural to seek a U-duality realization of the . The Monster is known to act as the group of linear transformations on a vector space of dimension 196883+1, 98280=196560/2 dimensions of which come from the set of norm four vectors of the first shell of the Leech lattice, with opposite pairs identified. [45]

5 Conclusion

Building on the results of Wilson [1, 2], we have shown the automorphism group of the Leech lattice, Co , is generated by unitary 3 3 matrices 0 × over the octonions, and are therefore F4 transformations. As F4 E6( 26), ⊂ − and E6( 26) is the U-duality group of = 2, D = 5 exceptional magic − N supergravity, we interpreted such generators of Co0 as U-duality transfor- mations. As the charge space for BPS black holes in = 2, D = 5 excep- N O tional magic supergravity is the exceptional Jordan algebra J3 , this shows O Co0 = 2 Co1 encodes U-duality transformations. Since Aut(J3 )= F4 and O · Str0(J3 ) = E6( 26), and F4 is the trace preserving part of E6( 26), the F4 − − O generators for Co0 were shown to preserve the Leech lattice norm in J3 , as expected from isometries. In other studies, the Leech lattice and the subgroup M24 of its auto- morphism group have found applications in [41, 42, 43, 44], shedding light on [45] and its generalizations. It was found [41] there is a Mathieu moonshine relating the elliptic genus of K3 to the M24. More recently, Mathieu moonshine was shown to hold for D = 4, = 2 string compactifications, which arise from het- N erotic strings on K3 T 2 or from type II strings on Calabi-Yau threefolds, × where the one-loop prepotential universally exhibits a structure encoding degeneracies of M24 [42]. In moving from D = 5 to D = 4, the octonionic = 2 magic supergrav- N ity exhibits an E7( 25) U-duality symmetry, containing a maximal subgroup SO(2, 10) SU(1,−1) found in the moduli space of complex structure de- × formations of the FHSV model [46], which is directly related to heterotic strings on K3 T 2. In [47] it was argued the D = 4, = 2 octonionic × N magic supergravity admits a stringy interpretation closely related to the FHSV model. In D = 11, Ramond noticed [48, 49] the massless supermultiplet of

7 eleven-dimensional supergravity can be generated from the decomposition of certain representation of F4 into those of its maximal compact subgroup 2 Spin(9). It was argued [49] that the Cayley plane, OP = F4/SO(9), may geometrically shed light on the fundamental degrees of freedom underlying M-theory. Building on these observations, Sati proposed [50, 51] a Kaluza- 2 Klein construction, where hidden OP bundles with structure group F4 en- code the massless fields of M-theory, resulting in a 27-dimensional candidate for bosonic M-theory [52]. Perhaps = 2, D = 5 exceptional magic super- N gravity has its full intepretation in such a 27-dimensional model. Moreover, the appearance of F4 in a model of M-theory with Cayley plane fibers would allow a Leech lattice construction, with F4 generators of Co0 acting on such fibers. It is a pressing question as to the role of the Monster group in 27-dimensions, as the 196560 norm four vectors of the Leech lattice in 24-dimensions contribute to the degrees of freedom of the on which the Monster acts.

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