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

L 8 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 R 1 of the U-duality algebra E r in D dimensions. The fields depend on corresponding coordinates x M in addition to the external ones. By imposing a section condition B x M b B N y “ 0, where the derivatives are projected on the dual of a representation R 2 ‘ r R 2 , the number of internal coordinates is effectively reduced to r (or r ´ 1) in an E r covariant way.

Under generalised diffeomorphisms, unifying ordinary diffeomorphisms and gauge transformations, vector fields transform with the generalised Lie derivative: δ U V M “ L U V M “ U N B N V M ´ V N B N U M ` Y MN PQ B N U P V Q where Y MN PQ is an E r invariant tensor with the upper pair of R 1 indices in R 2 ‘ r R 2 . [Coimbra, Strickland-Constable, Waldram: 1112.3989] [Berman, Cederwall, Kleinschmidt, Thompson: 1208.5884]

The extended coordinate representation R 1 is dual to the highest weight representation R p Λ 1 q , where the highest weight Λ 1 is the fundamental weight associated to node 1 in the Dynkin diagram of E r . r 1 2 r ´ 4 r ´ 3 r ´ 2 r ´ 1 The representations R 2 and r R 2 in the section condition are given by R 2 “ R 1 _ R 1 a R p 2Λ 1 q and r R 2 “ R 1 ^ R 1 a R p Λ 2 q .

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

Level decompositions of B p E r q and E r ` 1 : B p E r q “ ¨ ¨ ¨ ‘ R 2 ‘ R 1 ‘ p adj ‘ 1 q ‘ R 1 ‘ R 2 ‘ ¨ ¨ ¨ E r ` 1 “ ¨ ¨ ¨ ‘ r R 2 ‘ R 1 ‘ p adj ‘ 1 q ‘ R 1 ‘ r R 2 ‘ ¨ ¨ ¨ Example, r “ 7: B p E 7 q “ ¨ ¨ ¨ ‘ 133 ‘ 56 ‘ p 133 ‘ 1 q ‘ 56 ‘ 133 ‘ ¨ ¨ ¨ E 8 “ 1 ‘ 56 ‘ p 133 ‘ 1 q ‘ 56 ‘ 1

This can be generalised to any integrable highest weight representation R p λ q of any Kac-Moody algebra g r of rank r . The Dynkin labels of λ specify how the additional node should be connected to those in the Dynkin diagram of g r (or more precisely, the additional off-diagonal entries in the extended Cartan matrix). Level decompositions of B p g r q and g r ` 1 : B p g r q “ ¨ ¨ ¨ ‘ R 2 ‘ R 1 ‘ p adj ‘ 1 q ‘ R 1 ‘ R 2 ‘ ¨ ¨ ¨ g r ` 1 “ ¨ ¨ ¨ ‘ r R 2 ‘ R 1 ‘ p adj ‘ 1 q ‘ R 1 ‘ r R 2 ‘ ¨ ¨ ¨

Generalised diffeomorphisms can be defined generally for any choice of Kac-Moody algebra g r and dominant integral weight λ . They close if and only if g r is finite-dimensional, λ is a funda- mental weight Λ i , and the corresponding Coxeter label c i is equal to 1. Otherwise they close only up to so called ancillary transformations. 2 2 3 1 2 3 2 1 1 2 3 4 3 2 2 3 4 5 6 4 2 [Cederwall, Palmkvist: 1711.07694]

The extended algebras B p g r q and g r ` 1 can be used to express the generalised diffeomorphisms in a simple way. However, they need to be extended further and unified into B p g r ` 1 q . Two different Dynkin diagrams of the same algebra B p g r ` 1 q : γ r γ ´ 1 γ 0 γ 1 γ r ´ 4 γ r ´ 3 γ r ´ 2 γ r ´ 1 β r β ´ 1 β 0 β 1 β r ´ 4 β r ´ 3 β r ´ 2 β r ´ 1

The two p Z ˆ Z q -gradings of of B p g r ` 1 q are related to each other 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 B p g r q at a positive level p , then r e, x s is nonzero too, whereas r e, r e, x ss “ 1 2 rr e, e s , x s “ 0 . In general, the g r representations in the decomposition of B p g r ` 1 q 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 B p g r ` 1 q into g modules: ¨ ¨ ¨ p “ ´ 1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ ¨ ¨ ¨ n “ 0 r r q “ 3 R 3 n “ 1 R 3 ‘ r r r r q “ 2 R 2 R 3 n “ 2 R 2 ‘ r R 3 ‘ r q “ 1 1 R 1 R 2 R 3 n “ 3 q “ 0 1 ‘ adj ‘ 1 ¨ ¨ ¨ R 1 R 1 R 2 R 3 ¨ ¨ ¨ R 1 1

Ordinary geometry, B p g r ` 1 q “ sl p r ` 2 | 1 q : p “ ´ 1 p “ 0 p “ 1 q “ 1 1 v q “ 0 1 ‘ adj ‘ 1 v v q “ ´ 1 v 1

Double geometry, B p g r ` 1 q “ osp p r ` 1 , r ` 1 | 2 q : p “ ´ 2 p “ ´ 1 p “ 0 p “ 1 p “ 2 q “ 1 1 v 1 q “ 0 1 ‘ adj ‘ 1 1 v v 1 q “ ´ 1 1 v 1

Exceptional geometry, g r “ so p 5 , 5 q : p “ ´ 1 p “ 0 p “ 1 p “ 2 p “ 3 p “ 4 p “ 5 q “ 2 1 16 q “ 1 45 ‘ 1 144 ‘ 16 1 16 10 16 q “ 0 1 ‘ 45 ‘ 1 16 16 10 16 45 144 q “ ´ 1 16 1

Exceptional geometry, g r “ E 7 : p “ 0 p “ 1 p “ 2 p “ 3 p “ 4 q “ 3 1 q “ 2 1539 ‘ 133 ‘ 1 ‘ 1 1 56 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 ¨ ¨ ¨ r r q “ 3 R 3 ¨ ¨ ¨ R 3 ‘ r r r r q “ 2 ¨ ¨ ¨ R 2 R 3 R 2 ‘ r R 3 ‘ r q “ 1 ¨ ¨ ¨ 1 R 1 R 2 R 3 q “ 0 R 1 1 ‘ adj ‘ 1 R 1 R 2 R 3 ¨ ¨ ¨ q “ ´ 1 R 1 1

Basis elements: ¨ ¨ ¨ p “ ´ 1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ q “ 4 ¨ ¨ ¨ q “ 3 ¨ ¨ ¨ ¨ ¨ ¨ r r E M , r q “ 2 E N s ¨ ¨ ¨ ¨ ¨ ¨ r r E M , r q “ 1 E N s ¨ ¨ ¨ ¨ ¨ ¨ e E M F M h , T α , k q “ 0 E M r E M , E N s ¨ ¨ ¨ ¨ ¨ ¨ r F M q “ ´ 1 f

Basis elements: ¨ ¨ ¨ p “ ´ 1 p “ 0 p “ 1 p “ 2 p “ 3 ¨ ¨ ¨ q “ 4 ¨ ¨ ¨ q “ 3 ¨ ¨ ¨ ¨ ¨ ¨ r r E M , r q “ 2 E N s ¨ ¨ ¨ ¨ ¨ ¨ r r E M , r q “ 1 E N s ¨ ¨ ¨ ¨ ¨ ¨ e E M F M h , T α , k q “ 0 E M r E M , E N s ¨ ¨ ¨ ¨ ¨ ¨ r F M q “ ´ 1 f We identify the internal tangent space with the odd subspace spanned by the E M and write a vector field V as V “ V M E M . It can be mapped to V 7 “ r e, V s “ V M r E M at height q “ 1.

The generalised Lie derivative is now given by L U V “ rr U, r F N s , B N V 7 s ´ rrB N U 7 , r F N s , V s , and its closure follows from relations in the algebra. The section condition can be written r F M , F N sB M b B N “ r r F M , r F N sB M b B N “ 0 . [Palmkvist: 1507.08828] The generalised diffeomorphisms are infinitely reducible. [Berman, Cederwall, Kleinschmidt, Thompson: 1208.5884]

Introduce an operator a ÞÑ 1 p rB M a, F M s d : R p Ñ R p ´ 1 , acting on (coordinate dependent) elements on the positive levels in B p g r q . Then d 2 “ 0 thanks to the section condition. Infinite reducibility: L U “ 0 if U “ dU 1 for some U 1 P R 2 , dU 1 “ 0 if U 1 “ dU 2 for some U 2 P R 3 , . . . 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 L 8 algebra structure on B p g r q .

An L 8 algebra is a vector space L “ L 1 ‘ L 2 ‘ ¨ ¨ ¨ together with a set of k -brackets, k “ 1 , 2 , . . . , with degree ´ 1 satisfying the Z 2 -graded symmetry ( a 1 P L ℓ 1 and a 2 P L ℓ 2 ) s “ p´ 1 q ℓ 1 ℓ 2 r r r . . . , a 1 , . . . , a 2 , . . . s r . . . , a 2 , . . . , a 1 , . . . s s , and the generalised Jacobi identity k r rr r a 1 s s , a 2 . . . , a k s s ` p k ´ 1 qr rr r a 1 , a 2 s s , a 3 . . . , a k s s ` ¨ ¨ ¨ ¨ ¨ ¨ ` r rr r a 1 , a 2 , a 3 . . . , a k s ss s “ 0 (anti)symmetrised in a 1 , . . . , a k according to the Z 2 -graded symmetry above.

Unlike the ordinary Lie derivative, the generalised one is not antisymmetric, but we can antisymmetrise it: 2 r r U, V s s “ L U V ´ L V U “ rr U, r F N s , B N V 7 s ´ rrB N U 7 , r F N s , V s ´ p U Ø V q . We take this 2-bracket as the starting point for the L 8 algebra, together with the 1-bracket defined by r r U s s “ 0 for vector fields, that is, elements U at level p “ 1 in B p g r q , and r r a s s “ da for elements a at higher levels, r rr r a s ss s “ 0 , r rr r U s s , V s s ´ r rr r V s s , U s s ` r rr r U, V s ss s “ 0 .

If r R 2 “ 0, then the Jacobiator of the 2-bracket is given by 6 r rr r U r 1 , U 2 s s , U 3 s s s “ d rr r U r 1 , U 2 s s , U 3 s s . Thus, with the 3-bracket defined by s “ ´ 1 r r U 1 , U 2 , U 3 s 3 rr r U r 1 , U 2 s s , U 3 s s , the generalised Jacobi identity is satisfied: 3 r rr r U r 1 s s , U 2 , U 3 s s s ` 2 r rr r U r 1 , U 2 s s , U 3 s s s ` r rr r U r 1 , U 2 , U 3 s s ss s “ 0 . However, if r R 2 ‰ 0, then the Jacobiator r rr r U r 1 , U 2 s s , U 3 s s s cannot be written as d of something.