Is there something in between SUGRA and Strings? Olaf Hohm Double - PowerPoint PPT Presentation

Is there something in between SUGRA and Strings? Olaf Hohm ‚ Double Field Theory Siegel (1993), Hull, Zwiebach (2009), O.H., Hull, Zwiebach (2010 – ), O.H., Siegel, Zwiebach (2013 – ) ‚ Exceptional Field Theory de Wit, Nicolai (1986), O.H., Samtleben (2013 – ) ‚ Bossard, Kleinschmidt (2015) Ashoke Sen (2016) Florence, October 2016 1

Plan of the talk: ‚ Duality-covariant Geometry of DFT/ExFT ‚ Generalized Scherk-Schwarz Compactification & Consistency of Kaluza-Klein ‚ Higher-derivative α 1 Corrections ‚ Consistent theory beyond supergravity? 2

Duality-covariant Geometry of DFT/ExFT (Hidden) duality groups of SUGRA/Strings for toroidal compactification: O p d, d q , E 6 p 6 q , E 7 p 7 q , E 8 p 8 q DFT/ExFT: extended coordinates to make dualities manifest Section constraint for doubled coordinates X M “ p ˜ x i , x i q ‚ strongly constrained: B M B M A “ 2˜ B i B i A “ 0 @ A, B : ˆ 0 ˙ 1 B M A B M B “ 0 η MN “ 1 0 B i “ 0 , up to O p D, D q , but explains hidden symmetry: ‚ solved by ˜ ✟ X M “ p Y M q , M “ 1 , . . . , 2 d x µ , x µ , ✟✟✟ � ñ unbroken O p d, d q ! ˜ � � ‚ weakly constrained in full string (field) theory: level-matching B M B M A “ 0 , non-trivial string product consistent theory for massless fields plus their KK/winding modes? 3

Generalized Geometry of DFT Generalized Lie derivatives for Generalized metric H MN p g, b q P O p D, D q ` ˘ ` ˘ B M ξ P ´B P ξ M B N ξ P ´B P ξ N p L ξ H MN “ ξ P B P H MN ` H PN ` H MP “ ‰ C “ p L ξ 1 ξ 2 ´ p Gauge algebra C-bracket: ξ 1 , ξ 2 L ξ 2 ξ 1 B i “ 0 : ˜ δg “ L ξ g , δb “ d˜ ξ ` L ξ b Ñ Courant bracket in Gen. Geom. “ ‰ “ ‰ ` ˘ ξ 1 ´ 1 ξ 1 ` ˜ ξ 1 , ξ 2 ` ˜ ` L ξ 1 ˜ ξ 2 ´ L ξ 2 ˜ i ξ 1 ˜ ξ 2 ´ i ξ 2 ˜ “ ξ 2 ξ 1 , ξ 2 2 d ξ 1 ˆ 1 ˙ B ξ i ` B ij ξ j , exact term by B -shifts: ˜ ξ i Ñ ˜ h B “ P O p D, D q 0 1 ξ M 1 “ h MN ξ N @ h P O p D, D q C-bracket: ñ Any diff. & b -field gauge invariant theory compatible with Gen. Geom. ñ only DFT makes O p d, d q manifest Ñ constrains α 1 corrections! 4

Scherk-Schwarz Compactification & Consistency of Kaluza-Klein Ansatz in terms of twist matrix U p Y q in duality group & ρ p Y q H MN p x, Y q “ U MK p Y q U NL p Y q H KL p x q A µM p x, Y q “ p U ´ 1 q NM p Y q A µN p x q e ´ 2 φ p x,Y q “ ρ ´p n ´ 2 q p Y q e ´ 2 φ p x q ¨ ¨ ¨ Y -dependence factors out consistently Ñ geometric constraints on U , ρ “ X MNK U ´ 1 M U ´ 1 p L U ´ 1 N K with constant X MNK “ Θ Mα p t α q NK `¨ ¨ ¨ , Generalized Parallelizability ‚ Reduction fully consistent, including scalar potential, fermions, etc. ‚ U , ρ for spheres ñ consistency of AdS 4 ˆ S 7 [de Wit & Nicolai (1986)] , AdS 7 ˆ S 4 [Nastase, Vaman, van Nieuwenhuizen (1999)] and AdS 5 ˆ S 5 ‚ Non-geometric compactifications upon relaxing section constraint 5

Higher-derivative α 1 corrections ‚ string theory: infinite number of higher-derivative α 1 corrections; O p d, d ; R q symmetry preserved [Sen (1991)] ‚ O p d, d ; R q α 1 -deformed [Meissner (1997), O.H. & Zwiebach (2011, 2015)] ˜ ¸ g ´ 1 g ´ 1 p p ´ p b g “ g ` α 1 pB g qpB g q ` α 1 pB b qpB b q`¨ ¨ ¨ H MN “ p , p g ´ 1 g ´ p g ´ 1 p b p p b p b ‚ Deformed gauge structure in DFT, K MN ” B M ξ N ´ B N ξ M “ ‰ M ` α 1 H KL ´ γ ´ η KL q K r 2 K 2 p γ ` ¯ ξ M P B M K 1 s LP 12 “ ξ 2 , ξ 1 C ‚ Uniquely determines O p α 1 q correction up to two parameters: [O.H. & Zwiebach (2014), Nunez & Marques (2015)] " γ ` “ 1 , γ ´ “ 0 bosonic string γ ` “ 1 γ ´ “ 1 2 , heterotic string 2 ‚ γ ` “ 0 , γ ´ “ 1 (HSZ): exactly duality and gauge invariant ñ infinite number of α 1 corrections! 6

Consistent theory beyond supergravity? ‚ B M B M p φ 1 φ 2 q ‰ 0 without strong constraint ñ non-trivial product ÿ ` ˘ P 2 e i p P 1 ` P 2 q¨ X , δ p P 1 ¨ P 2 q φ 1 P 1 φ 2 φ 1 ‹ φ 2 p X q “ P 1 ,P 2 P Z 2 d ‚ Non-associative, but ‘associator’ total derivative p φ 1 ‹ φ 2 q ‹ φ 3 ´ φ 1 ‹ p φ 2 ‹ φ 3 q “ B M F M p φ 1 , φ 2 , φ 3 q ñ cubic theory consistent, beyond that: L 8 or A 8 algebra? [Zwiebach (1993), O.H., Hull & Zwiebach, unpublished] ‚ Theory for “SUGRA” fields plus their KK & winding modes? Yes: other massive modes can be integrated out [Sen (2016)] in some sense full string theory; amplitudes for mass. modes hidden ‚ One-Loop in ExFT with massive modes ñ improved UV behavior [Bossard & Kleinschmidt (2015)] 7

Summary & Outlook ‚ DFT and ExFT make duality symmetries O p d, d q , E d p d q manifest ‚ strongly constrained theory fully background independent, reformulation of SUGRA ‚ very powerful formalism: consistency of KK truncations, higher-derivative α 1 corrections ‚ but at least on T d extended coordinates physical and real; consistent theory of SUGRA fields plus KK/winding modes from strings Can it be given explicitly? ‚ Consistent truncation with KK and winding modes? Ñ analogous to 5D SUGRA consistent truncation of IIB on AdS 5 ˆ S 5 8