Algebraic Structures in Double and Exceptional Field Theory Olaf - PowerPoint PPT Presentation

Algebraic Structures in Double and Exceptional Field Theory Olaf Hohm ‚ O.H, Zwiebach, 1701.08824 ‚ O.H., Kupriyanov, L¨ ust, Traube, 1709.10004 ‚ O.H., Samtleben, 1707.06693 & to appear ‚ O.H., work in progress Centro Atomico Bariloche, Argentina, January 2018 1

Road Map ‚ Duality covariant formulation in 1) gauged supergravity (‘embedding tensor formalim’) and 2) double/exceptional field theory requires redundant or unphysical objects ñ ‘higher equivalences’ ‚ analogous features in algebraic topology and homotopy theory, where ‘ 8 -algebras’ allow one “to live with slightly false algebraic identities in a new world where they become effectively true.” [D. Sullivan] ‚ Features of physical theories usually taken for granted [ e.g.: “continuous symmetries ” Lie algebras” ] hold only ‘up to homotopy’, which quite likely provides deep pointers for (so far) elusive underlying mathematical structure of DFT/ExFT 2

Overview ‚ Strongly Homotopy (sh) or 8 -Algebras ‚ Field Theories and L 8 Algebras Ñ weakly constrained DFT? ‚ Leibniz (or Loday) Algebras and their Chern-Simons Gauge Theory ‚ Topological Phase of E 8 p 8 q ExFT as Leibniz Chern-Simons Theory ‚ General Remarks and Outlook 3

Strongly Homotopy Lie or L 8 Algebras An L 8 algebras is a graded vector space [Zwiebach (1993), Lada & Stasheff (1993)] à X “ X n , n P Z equipped with multilinear and graded antisymmetric brackets or maps x 1 , . . . , x n ÞÑ ℓ n p x 1 , . . . , x n q P X n ´ 2 ` ř i | x i | , satisfying, for each n “ 1 , 2 , 3 , . . . , the generalized Jacobi identities ÿ p´ 1 q i p j ´ 1 q ÿ ` ˘ p´ 1 q σ ǫ p σ ; x q ℓ j ℓ i p x σ p 1 q , . . . , x σ p i q q , x σ p i ` 1 q , . . . , x σ p n q “ 0 σ i ` j “ n ` 1 with the sum over all permutations of n objects with partially ordered arguments (‘unshuffles’), σ p 1 q ď ¨ ¨ ¨ ď σ p i q , σ p i ` 1 q ď ¨ ¨ ¨ ď σ p n q , and Koszul sign ǫ p σ ; x q , determined for any graded algebra with x i x j “ p´ 1 q x i x j x j x i x 1 ¨ ¨ ¨ x k “ ǫ p σ ; x q x σ p 1 q ¨ ¨ ¨ x σ p k q by 4

Explicit L 8 -relations For n “ 1 we learn that ℓ 1 ” Q is nil-potent: ℓ 1 p ℓ 1 p x qq “ 0 For n “ 2 we learn that ℓ 1 is a derivation of ℓ 2 ” r¨ , ¨s : ℓ 1 p ℓ 2 p x 1 , x 2 qq “ ℓ 2 p ℓ 1 p x 1 q , x 2 q ` p´ 1 q x 1 ℓ 2 p x 1 , ℓ 1 p x 2 qq For n “ 3 we learn that ℓ 2 ” r¨ , ¨s satisfies Jacobi only ‘up to homotopy’ 0 “ ℓ 2 p ℓ 2 p x 1 , x 2 q , x 3 q ` 2 terms ` ℓ 1 p ℓ 3 p x 1 , x 2 , x 3 qq ` ℓ 3 p ℓ 1 p x 1 q , x 2 , x 3 q ` 2 terms For n “ 4 we learn that ℓ 2 ℓ 3 ` ℓ 3 ℓ 2 is zero ‘up to homotopy’, i.e., up to the the failure of ℓ 1 to act as a derivation on ℓ 4 plus infinitely more relations 5

Constructing L 8 Algebras Given a bilinear antisymmetric 2-bracket r¨ , ¨s , is there L 8 with ℓ 2 “ r¨ , ¨s ? Yes: extend the space V by a second copy V ˚ ℓ 1 X 1 “ V ˚ Ý Ñ X 0 “ V ℓ 1 p v ˚ q “ v , ℓ 3 p u, v, w q “ ´ Jac p u, v, w q ˚ But: ‘trivial’ because v „ w iff v ´ w “ ℓ 1 p¨q , no further extension Non-trivial if Jacobiator lives in proper subspace or, more generally, in image of linear map D : U Ñ V : Jac p¨ , ¨ , ¨q “ D f p¨ , ¨ , ¨q . Theorem: ℓ 1 “ ι ℓ 1 “ D X 2 – Ker p D q Ý Ý Ý Ñ X 1 “ U Ý Ý Ý Ý Ñ X 0 carries L 8 structure, provided r Im p D q , V s Ă Im p D q , with ℓ 3 p¨ , ¨ , ¨q “ ´ f p¨ , ¨ , ¨q and generally non-trivial ℓ 4 p¨ , ¨ , ¨ , ¨q 6

Field Theories & Weakly Constrained DFT Dictionary L 8 algebra Ð Ñ field theory: ℓ 1 ℓ 1 ℓ 1 ¨ ¨ ¨ Ý Ñ X 1 Ý Ñ X 0 Ý Ñ X ´ 1 Ý Ñ X ´ 2 Ý Ñ ¨ ¨ ¨ χ ξ Ψ EOM Gauge transformations and field equations: δ ξ Ψ “ ℓ 1 p ξ q ` ℓ 2 p ξ, Ψ q ´ 1 2 ℓ 3 p ξ, Ψ , Ψ q ` ¨ ¨ ¨ 0 “ ℓ 1 p Ψ q ´ 1 2 ℓ 2 p Ψ , Ψ q ´ 1 3! ℓ 3 p Ψ , Ψ , Ψ q ` ¨ ¨ ¨ gauge algebra closes ‘up to homotopy’: trivial parameters ξ “ ℓ 1 p χ q Example: Courant algebroid/gauge structure of DFT, with ℓ 2 “ r¨ , ¨s c , defines L 8 algebra with ℓ 4 “ 0 [Roytenberg & Weinstein (1998)] Ñ generalization to weakly constrained? Indeed, in general L 8 non-trivial ℓ 2 p χ 1 , χ 2 q “ x D χ 1 , D χ 2 y p “ B M χ 1 B M χ 2 “ 0 q Ñ still very non-trivial (non-local projected product needed) [A. Sen (2016)] 7

Leibniz Algebras and their Chern-Simons Theory Leibniz (or Loday) algebra: vector space with product ˝ , satisfying x ˝ p y ˝ z q “ p x ˝ y q ˝ z ` y ˝ p x ˝ z q If ˝ antisymmetric ñ Lie algebra δ x y “ L x y ” x ˝ y Defines symmetry variations: that close: r L x , L y s z ” L x p L y z q ´ L y p L x z q “ x ˝ p y ˝ z q ´ y ˝ p x ˝ z q “ p x ˝ y q ˝ z “ L x ˝ y z (Anti-)symmetrizing in x, y : r L x , L y s z “ L r x,y s z , L t x,y u z “ 0 Thus, t , u defines ‘trivial vector’. Jacobiator is trivial: ÿ 3 rr x 1 , x 2 s , x 3 s ´ t x 1 ˝ x 2 , x 3 u “ 0 antisym ‘Trivial space’ forms ideal of bracket: r¨ , t , us “ t¨ , ¨u . Thus: Any Leibniz algebra defines L 8 algebra with ℓ 2 “ r¨ , ¨s Theorem: 8

Leibniz-valued Gauge Fields and Chern-Simons Action Leibniz-valued one-form with gauge transformations δ λ A µ “ D µ λ ” B µ λ ´ A µ ˝ λ This closes up to ‘higher gauge transformations’ (c.f. trivial parameters). Generalized Chern-Simons action ż d 3 x ǫ µνρ @ D A µ , B ν A ρ ´ 1 S CS ” 3 A ν ˝ A ρ is gauge invariant provided the inner product x , y is invariant and x x, t¨ , ¨uy “ 0 @ x ñ situation in 3D gauged SUGRA in embedding tensor formalism [de Wit, Nicolai & Samtleben (2001–2002)] ñ any Leibniz algebra with x , y as above defines Chern-Simons theory ñ general dimensions: tensor hierarchy (& corresponding L 8 algebra) 9

‘Unbroken Phase’ of E 8 p 8 q ExFT Fields: e µa , A µM , B µM , M MN , coordinates: p x µ , Y M q , M “ 1 , . . . , 248 Action: ż ´ ¯ 1 d 3 x d 248 Y e p 240 D µ M MN D µ M MN ´ V p M q ` L top p A, B q S “ R ` Consider subsector that truncates M MN , say by setting: M MN “ δ MN : E 8 p 8 q Ñ SO p 16 q However, we want unbroken phase , so we set (illegally): M MN “ 0 Perhaps justification in suitably reformulated/enlarged theory? c.f. unconstrained ‘doubled ’metric Ñ α 1 corrections [O.H., Siegel & Zwiebach (2013)] first-order formulation with degenerate frame field? [E. Witten (1988)] 10

Big Chern-Simons Theory Leibniz algebra unifies 3D Poincar´ e and (doubled) generalized diffeos: ` ˘ ξ a , λ a ; Λ M , Σ M Ξ “ ` ˘ ξ a 12 , λ 12 a ; Λ M Ξ 1 ˝ Ξ 2 “ 12 , Σ 12 M where [ R M ” f MNK B N Λ K ` Σ M , constraint: Σ M b B M “ 0 , etc.] λ 12 a “ ǫ abc λ b 1 λ c N B N λ 2 s a 2 ` 2 Λ r 1 Σ 12 M “ L 1 Σ 2 M ` Λ N 2 B M R 1 N ´ 2 α ξ a r 1 B M λ 2 s a etc . ` ˘ e a , ω a ; A M , B M For algebra element A ” inner product given by ż ´ 2 α e a ω a ` 2 A M B M ´ f KMN A M B K A N ¯ d 248 Y x A , A y “ CS action for A µ precisely top. ExFT action! pure 3D (super-)gravity ” Chern-Simons theory Generalization of: [Achucarro & Townsend (1986), Witten (1988)] 11

Consistent Kaluza-Klein to half-maximal D “ 3 SUGRA Duality: O p d ` 1 , d ` 1 q , coordinates Y M ” Y r MN s , M, N fundamental ‘Doubled vector’ Υ ” p Λ MN , Σ MN q satisfies Leibniz algebra w.r.t. ´ ¯ , L Υ 1 Σ 2 MN ` 1 L Υ 1 Λ MN 4Λ KL Υ 1 ˝ Υ 2 ” B MN K p Υ 1 q KL 2 2 Generalized Scherk-Schwarz in terms of ‘doubled’ twist matrix: ` ˘ N s , ´ 1 ρ ´ 1 U K 4 ρ ´ 1 pB KL U P ¯ M U L ¯ N ” M q U P ¯ U ¯ M ¯ r ¯ N reads for the gauge vectors: M ¯ ¯ N p x q A µ p x, Y q “ U ¯ N p Y q A µ M ¯ and is consistent provided P ¯ ¯ Q U ¯ N ˝ U ¯ L “ ´ X ¯ U ¯ M ¯ K ¯ M ¯ N, ¯ K ¯ P ¯ L Q where X is the constant embedding tensor. ñ consistency of D “ 6 , N “ p 1 , 1 q & p 2 , 0 q SUGRA on AdS 3 ˆ S 3 12

Outlook & Remarks ‚ algebraic structures beyond Lie arise naturally in string/M-theory ‚ tensor hierarchy of gauged SUGRA & ExFT suggests 8 -algebra, difficult/unnatural in terms of Lie algebra ‚ unifying algebraic structure of M-theory? Ñ Hermann and Martin’s talks affine E 9 p 9 q works analogously to E 8 p 8 q Ñ Guillaume’s talk ñ Lie algebra theory may be the “slightly wrong” framework ‚ novel products in HSZ theory [O.H., Siegel & Zwiebach (2013)] Ñ interpretation as 8 -algebra? Ñ more general story for (chiral) CFTs? Ñ Ralph’s talk ‚ global structure of doubled (extended) spaces? [O.H. & Zwiebach (2012)] 13