L 8 algebras for extended geometry from Borcherds superalgebras - - PowerPoint PPT Presentation

L8 algebras for extended geometry from Borcherds superalgebras

– Part 1 – Jakob Palmkvist Based on 1804.04377 and 1711.07694 with Martin Cederwall and 1507.08828

Consider eleven-dimensional (or type IIB) supergravity in a Kaluza-Klein split with D “ 11 ´ r external dimensions. In exceptional geometry the internal tangent space is extended to a module R1 of the U-duality algebra Er in D dimensions. The fields depend on corresponding coordinates xM in addition to the external ones. By imposing a section condition BxM b BNy “ 0, where the derivatives are projected on the dual of a representation R2 ‘ r R2, the number of internal coordinates is effectively reduced to r (or r ´ 1) in an Er covariant way.

Under generalised diffeomorphisms, unifying ordinary diffeomorphisms and gauge transformations, vector fields transform with the generalised Lie derivative: δUV M “ LUV M “ U NBNV M ´ V NBNU M ` Y MN PQBNU P V Q where Y MN PQ is an Er invariant tensor with the upper pair of R1 indices in R2 ‘ r R2.

[Coimbra, Strickland-Constable, Waldram: 1112.3989] [Berman, Cederwall, Kleinschmidt, Thompson: 1208.5884]

The extended coordinate representation R1 is dual to the highest weight representation RpΛ1q, where the highest weight Λ1 is the fundamental weight associated to node 1 in the Dynkin diagram of Er.

1 2 r ´ 4 r ´ 3 r ´ 2 r ´ 1 r

The representations R2 and r R2 in the section condition are given by R2 “ R1 _ R1 a Rp2Λ1q and r R2 “ R1 ^ R1 a RpΛ2q.

Modules for these representations appear when we extend Er by adding one more node to the Dynkin diagram. This results in the Kac-Moody algebra Er`1 and the Borcherds superalgebra BpErq.

Level decompositions of BpErq and Er`1: BpErq “ ¨ ¨ ¨ ‘ R2 ‘ R1 ‘ padj ‘ 1q ‘ R1 ‘ R2 ‘ ¨ ¨ ¨ Er`1 “ ¨ ¨ ¨ ‘ r R2 ‘ R1 ‘ padj ‘ 1q ‘ R1 ‘ r R2 ‘ ¨ ¨ ¨ Example, r “ 7: BpE7q “ ¨ ¨ ¨ ‘ 133 ‘ 56 ‘ p133 ‘ 1q ‘ 56 ‘ 133 ‘ ¨ ¨ ¨ E8 “ 1 ‘ 56 ‘ p133 ‘ 1q ‘ 56 ‘ 1

This can be generalised to any integrable highest weight representation Rpλq of any Kac-Moody algebra gr of rank r. The Dynkin labels of λ specify how the additional node should be connected to those in the Dynkin diagram of gr (or more precisely, the additional off-diagonal entries in the extended Cartan matrix). Level decompositions of Bpgrq and gr`1: Bpgrq “ ¨ ¨ ¨ ‘ R2 ‘ R1 ‘ padj ‘ 1q ‘ R1 ‘ R2 ‘ ¨ ¨ ¨ gr`1 “ ¨ ¨ ¨ ‘ r R2 ‘ R1 ‘ padj ‘ 1q ‘ R1 ‘ r R2 ‘ ¨ ¨ ¨

Generalised diffeomorphisms can be defined generally for any choice of Kac-Moody algebra gr and dominant integral weight λ. They close if and only if gr is finite-dimensional, λ is a funda- mental weight Λi, and the corresponding Coxeter label ci is equal to 1. Otherwise they close only up to so called ancillary transformations.

1 2 3 2 1 2 1 2 3 4 3 2 2 2 3 4 5 6 4 2 3

[Cederwall, Palmkvist: 1711.07694]

The extended algebras Bpgrq and gr`1 can be used to express the generalised diffeomorphisms in a simple way. However, they need to be extended further and unified into Bpgr`1q. Two different Dynkin diagrams of the same algebra Bpgr`1q:

γ´1 γ0 γ1 γr´4 γr´3 γr´2 γr´1 γr β´1 β0 β1 βr´4 βr´3 βr´2 βr´1 βr

The two pZ ˆ Zq-gradings of of Bpgr`1q are related to each

• ther by m “ p and n “ p ´ q.

n m q p

The generators e, f and h associated to the outermost grey node in the second diagram sit at level p “ 0, and height q “ ˘1 and q “ 0, respectively.

If x is a nonzero element in Bpgrq at a positive level p, then re, xs is nonzero too, whereas re, re, xss “ 1 2rre, es, xs “ 0 . In general, the gr representations in the decomposition of Bpgr`1q at positive (or negative) levels p always come in doublets with respect to the Heisenberg superalgebra associated to the outermost grey node (spanned by e, f, h).

Decomposition of Bpgr`1q into g modules:

¨ ¨ ¨ p “ ´1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ ¨ ¨ ¨ n “ 0 q “ 3 r r R3 n “ 1 q “ 2 r R2 r R3 ‘ r r R3 n “ 2 q “ 1 1 R1 R2 ‘ r R2 R3 ‘ r R3 n “ 3 q “ 0 R1 1 ‘ adj ‘ 1 R1 R2 R3 ¨ ¨ ¨ ¨ ¨ ¨ R1 1

Ordinary geometry, Bpgr`1q “ slpr ` 2|1q:

p “ ´1 p “ 0 p “ 1 q “ 1 1 v q “ 0 v 1 ‘ adj ‘ 1 v q “ ´1 v 1

Double geometry, Bpgr`1q “ osppr ` 1, r ` 1|2q:

p “ ´2 p “ ´1 p “ 0 p “ 1 p “ 2 q “ 1 1 v 1 q “ 0 1 v 1 ‘ adj ‘ 1 v 1 q “ ´1 1 v 1

Exceptional geometry, gr “ sop5, 5q:

p “ ´1 p “ 0 p “ 1 p “ 2 p “ 3 p “ 4 p “ 5 q “ 2 1 16 q “ 1 1 16 10 16 45 ‘ 1 144 ‘ 16 q “ 0 16 1 ‘ 45 ‘ 1 16 10 16 45 144 q “ ´1 16 1

Exceptional geometry, gr “ E7:

p “ 0 p “ 1 p “ 2 p “ 3 p “ 4 q “ 3 1 q “ 2 1 56 1539 ‘ 133 ‘ 1 ‘ 1 q “ 1 1 56 133 ‘ 1 912 ‘ 56 8645 ‘ 133 ‘ 1539 ‘ 133 ‘ 1 q “ 0 1 ‘ 133 ‘ 1 56 133 912 8645 ‘ 133 q “ ´1 1

Back to the general case:

¨ ¨ ¨ p “ ´1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ q “ 4 ¨ ¨ ¨ q “ 3 r r R3 ¨ ¨ ¨ q “ 2 r R2 r R3 ‘ r r R3 ¨ ¨ ¨ q “ 1 1 R1 R2 ‘ r R2 R3 ‘ r R3 ¨ ¨ ¨ q “ 0 R1 1 ‘ adj ‘ 1 R1 R2 R3 ¨ ¨ ¨ q “ ´1 R1 1

Basis elements:

¨ ¨ ¨ p “ ´1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ q “ 4 ¨ ¨ ¨ q “ 3 ¨ ¨ ¨ ¨ ¨ ¨ q “ 2 r r EM, r ENs ¨ ¨ ¨ ¨ ¨ ¨ q “ 1 e r EM rEM, r ENs ¨ ¨ ¨ ¨ ¨ ¨ q “ 0 F M h , T α, k EM rEM, ENs ¨ ¨ ¨ ¨ ¨ ¨ q “ ´1 r F M f

Basis elements:

¨ ¨ ¨ p “ ´1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ q “ 4 ¨ ¨ ¨ q “ 3 ¨ ¨ ¨ ¨ ¨ ¨ q “ 2 r r EM, r ENs ¨ ¨ ¨ ¨ ¨ ¨ q “ 1 e r EM rEM, r ENs ¨ ¨ ¨ ¨ ¨ ¨ q “ 0 F M h , T α, k EM rEM, ENs ¨ ¨ ¨ ¨ ¨ ¨ q “ ´1 r F M f

We identify the internal tangent space with the odd subspace spanned by the EM and write a vector field V as V “ V MEM. It can be mapped to V 7 “ re, V s “ V M r EM at height q “ 1.

The generalised Lie derivative is now given by LUV “ rrU, r F Ns, BNV 7s ´ rrBNU 7, r F Ns, V s , and its closure follows from relations in the algebra. The section condition can be written rF M, F NsBM b BN “ r r F M, r F NsBM b BN “ 0 .

[Palmkvist: 1507.08828]

The generalised diffeomorphisms are infinitely reducible.

[Berman, Cederwall, Kleinschmidt, Thompson: 1208.5884]

Introduce an operator d : Rp Ñ Rp´1 , a ÞÑ 1 prBMa, F Ms acting on (coordinate dependent) elements on the positive levels in Bpgrq. Then d2 “ 0 thanks to the section condition. Infinite reducibility: LU “ 0 if U “ dU 1 for some U 1 P R2, dU 1 “ 0 if U 1 “ dU 2 for some U 2 P R3, . . . This derivative is covariant only for 2 ď p ď 8 ´ r.

[Cederwall, Edlund, Karlsson: 1302.6736]

The lack of covariance of d can be understood as the failure of the generalised Jacobi identity in an attempt to define a structure of an L8 algebra structure on Bpgrq.

An L8 algebra is a vector space L “ L1 ‘ L2 ‘ ¨ ¨ ¨ together with a set of k-brackets, k “ 1, 2, . . ., with degree ´1 satisfying the Z2-graded symmetry (a1 P Lℓ1 and a2 P Lℓ2) r

• r. . . , a1, . . . , a2, . . .s

s “ p´1qℓ1ℓ2r

• r. . . , a2, . . . , a1, . . .s

s , and the generalised Jacobi identity kr rr ra1s s, a2 . . . , aks s ` pk ´ 1qr rr ra1, a2s s, a3 . . . , aks s ` ¨ ¨ ¨ ¨ ¨ ¨ ` r rr ra1, a2, a3 . . . , aks ss s “ 0 (anti)symmetrised in a1, . . . , ak according to the Z2-graded symmetry above.

Unlike the ordinary Lie derivative, the generalised one is not antisymmetric, but we can antisymmetrise it: 2r rU, V s s “ LUV ´ LV U “ rrU, r F Ns, BNV 7s ´ rrBNU 7, r F Ns, V s ´ pU Ø V q . We take this 2-bracket as the starting point for the L8 algebra, together with the 1-bracket defined by r rUs s “ 0 for vector fields, that is, elements U at level p “ 1 in Bpgrq, and r ras s “ da for elements a at higher levels, r rr ras ss s “ 0 , r rr rUs s, V s s ´ r rr rV s s, Us s ` r rr rU, V s ss s “ 0 .

If r R2 “ 0, then the Jacobiator of the 2-bracket is given by 6r rr rUr1, U2s s, U3ss s “ drr rUr1, U2s s, U3ss . Thus, with the 3-bracket defined by r rU1, U2, U3s s “ ´1

3rr

rUr1, U2s s, U3ss , the generalised Jacobi identity is satisfied: 3r rr rUr1s s, U2, U3ss s ` 2r rr rUr1, U2s s, U3ss s ` r rr rUr1, U2, U3ss ss s “ 0 . However, if r R2 ‰ 0, then the Jacobiator r rr rUr1, U2s s, U3ss s cannot be written as d of something.

In general, as long as r Rk “ 0, brackets up to order k ` 1 can be defined on the positive levels of Bpgrq in a natural way such that they form the beginning of an L8 algebra. For g “ Er the first nonzero ˜ Rk appears for k “ 9 ´ r. In order to continue the L8 algebra we need to supplement R9´r “ adj with an additional R1: L1 “ R1 , . . . , L8´r “ R8´r , L9´r “ R9´r ‘ R1 . Also seen in the tensor hierarchy for exceptional field theory. For example, a section-projected 2-form in 56 needs to be introduced for E7.

[Hohm, Samtleben: 1312.4542]

In the Borcherds superalgebra picture, extending Bpgrq to Bpgr`1q, the extra representations appear naturally.

¨ ¨ ¨ p “ ´1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ ¨ ¨ ¨ q “ 3 r r R3 q “ 2 r R2 r R3 ‘ r r R3 q “ 1 1 R1 R2 ‘ r R2 R3 ‘ r R3 q “ 0 R1 1 ‘ adj ‘ 1 R1 R2 R3 ¨ ¨ ¨ ¨ ¨ ¨ R1 1 ℓ “ 1 ℓ “ 2 ℓ “ 3

The L8 algebra can be continued to all levels and all brackets.

In the Borcherds superalgebra picture, extending Bpgrq to Bpgr`1q, the extra representations appear naturally.

¨ ¨ ¨ p “ ´1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ ¨ ¨ ¨ q “ 3 r r R3 q “ 2 r R2 r R3 ‘ r r R3 q “ 1 1 R1 R2 ‘ r R2 R3 ‘ r R3 q “ 0 R1 1 ‘ adj ‘ 1 R1 R2 R3 ¨ ¨ ¨ ¨ ¨ ¨ R1 1 ℓ “ 1 ℓ “ 2 ℓ “ 3

The L8 algebra can be continued to all levels and all brackets. This will be explained in Martin’s talk . . .