Duality in Double and Exceptional Field Theory

Home , Dual graviton

Olaf Hohm

Simons Center, Stony Brook University

‚ E. A. Bergshoeff, O.H., V. A. Penas, F. Riccioni, 1603.07380

‚ O.H., H. Samtleben: 1308.1673, 1406.3348, 1410.8145

Workshop Dualities in Supergravities, Strings and Branes, Texas A & M University, April 2016 1 Plan of the talk:

‚ Motivation: dual ﬁelds, branes and exotic branes

‚ Review: dualization of p-forms and (linearized) gravity

‚ Exotic dualization: Kalb-Ramond 2-form to pD´2, 2q Young tableaux

‚ Dualization of (linearized) DFT

‚ Dual ﬁelds in EFT

‚ Conclusions and Outlook

2 Motivation: dual ﬁelds, branes and exotic branes

˜ ‚ p-form gauge potential Ap dual to pD ´ p ´ 2q-form AD´p´2

˜ ˜ Fp`1 “ dAp Ñ FD´p´1 ”‹Fp`1 ” dAD´p´2

‚ Dualization essential for brane word-volume theories

E.g. Kalb-Ramond 2-form B2 Ñ B6

NS5 brane: SNS5 “ B6 żΣ6

‚ Under T-duality Opd, dq B2 Ø metric Opd, dq in DFT manifest (B2 and g uniﬁed) Ñ dual graviton ‚ dual graviton needed for world-volume dynamics of Kaluza-Klein monopole [Eyras, Janssen, Lozano (1998)] Generally, various mixed-Young tableaux ﬁelds expected by dualities ñ exotic branes [Bergshoeff, Riccioni et al. (2010–), de Boer & Shigemori (2012)] 3 Reminder: off-shell dualization of p-forms

Action for p-form gauge ﬁeld, Fp`1 “ dAp,

1 D a ...a SrAs “ ´ d x Fa ...a F 1 p`1 2pp ` 1q! 1 p`1 ż equivalent to ﬁrst-order (master) action

1 D 1 a ...a a ...a SrA,˜ F s “ d x ´ Fa ...a F 1 p`1 ´ A˜ 1 p`2Ba Fa ...a pp ` 1q! 2 1 p`1 1 2 p`2 ż ´ ¯ with gauge symmetry

a ...a aa ...a δA˜ 1 p`2 “BaΛ 1 p`2 , δF “ 0

˜ ‚ EOM for A ñ Bianchi ñ Fp`1 “ dAp

a1...ap`1 ˜ba1...ap`1 ‚ EOM for F ñ F “BbA ñ integrating out F yields dual theory for A˜ (note: no Levi-Civita symbol Ñ multivector rather than form) 4 Dualization of linearized Einstein gravity

Linearized Einstein-Hilbert action for vielbein ﬂuctuation ha|b

D b ac 1 acb 1 abc SEHrhs “ d x fab f c ´ 2fabcf ´ 4fabcf ż ” c c ı linearized coefﬁcients of anholonomy fab “ 2Brahbs| First-order (master) action [West (2001), Boulanger, Cnockaert & Henneaux (2003)]

D b ac 1 acb 1 abc abc d Srf, Ds “ d x fab f c ´ 2fabcf ´ 4fabcf ` 3D d Bafbc ż abc ´ d ¯ ‚ EOM for D d ñ Bianchi Brafbcs “ 0 ñ linearized gravity c c d c ‚ EOM for fab determines fab “ 6 BdD rb as ` ¨ ¨ ¨ yields dual theory for

D˜ ” 1 Dcde P b a1...aD´3|b 6 a1...aD´3cde b

mixed Young tableau pD ´ 3, 1q ﬁeld: dual graviton D˜ pure Lorentz gauge, but needed in master action ra1...aD´3|bs 5 Exotic Dualization: 2-form to pD ´ 2, 2q Young tableau ﬁeld

First, rewrite Kalb-Ramond action [Boulanger et al. (2012–)]

1 D abc 1 D a bc ab c Srbs “ ´12 d x H Habc “ ´4 d x B b Babbc ´ 2 Bab B bcb ż ż ` ˘ First-order (master) action Ñ promote Qa|bc ”Babbc to be independent

6 Decomposing dual ﬁeld (D “ 4 for simplicity)

Duality relations gauge invariant

1 efg Qb|cd “ ´3!b Γefg,cd ` 2BrcC|b|ds Γabc,de ” 3 BraBbcs,de

abij Integrability condition: act with Ba, using BraQbs|cd “ 0 aij abij abc abc 0 “ ´BaΓ ,cd`2 BaBrcC|b|ds ô R ,abc ” 3 BraΓ ,bds “ 0

ñ B consistently propagates single d.o.f., no action for p2, 2q alone 7 Exotic Dual Action for p2, 2q Field

Decomposing into symmetric and antisymmetric part, Ca|b “ sab ` aab,

1 abc,de 1 abcd ef 1 abcd ef L “ ´ 24 Γ Γabc,de ` 12 Γbcd, haef ´ 6 Γbcd, γae,f ab 1 abc ` s Gabpsq ` 6 h habc where Gab is ‘Einstein tensor’ for sab, and

1 γab,c ” 2pBasbc ` Bbsac ´ Bcsabq , habc ” 3Braabcs

ñ wrong-sign Einstein-Hilbert and Kalb-Ramond terms!

(linearized diff and KR gauge invariant with parameters ξa and ξ˜a)

However, off-diagonal action with enlarged gauge invariance ñ no contradiction with equivalence to single 2-form

8 (Linearized) Double Field Theory

Frame formulation of DFT ηab 0 H “ E A E B S ,S ” MN M N AB AB 0 η ˆ ab˙ where A local Lorentz index for OpD ´ 1, 1q ˆ OpD ´ 1, 1q M M B M Linearized action for EA “ E¯A ` hA E¯B

¯ 1 S “ d2DX e´2d SABF F ` F˘ABCF DFT A B 6 ABC ż ˆ ˙ with ‘coefﬁcients of anholonomy’

B FABC “ 3 BrAhBCs FA “B hBA ` 2 BAd

Bianchi identities

BrAFBCDs “ 0 C B FCAB ` 2 BrAFBs “ 0 A B FA “ 0 9 Master Action & Duality Relations for DFT

Master action introducing Lagrange multiplier for Bianchi identities

¯ 1 S “ dX e´2d SABF F ` F˘ABCF A B 6 ABC ż „ ABCD AB C A ` D BAFBCD ` D B FCAB ` 2 BAFB ` D B FA  Varying w.r.t. F Ñ duality relations ` ˘

ABC DABC rA BCs F˘ “ 3 BDD ` B D AB BA A 2S FB “`2 BBD ` B D ˘

Breaking OpD,Dq Ą GLpDq and OpD ´ 1, 1q2 to diagonal subgroup a A “ p , aq abcd abc ab a DABCD Ñ D D d D cd D bcd Dabcd encodes conventional duals to metric, 2-form, dilaton, exotic duals, etc., but extra ﬁelds DAB and D needed for master/dual action 10 Dual Diffeomorphisms and Dual Action for DFT

Choose basis where A “ pa, ¯aq, DFT ﬁelds: ha¯b hab h¯a¯b d

dual DFT ﬁelds: Dabcd Dabcd¯ ...D¯a¯b¯cd¯ Da¯b Dab D¯a¯b D

Dual diffeomorphisms

c ¯c δΣDa¯b “BaΣ¯b ´ B¯bΣa ` B Σca,¯b ` B Σ¯c¯b,a a ¯a δΣD “BaΣ ` B¯aΣ

δΣDab¯cd¯ “ 2 Bra Σ|¯cd¯|,bs ´ 2 Br¯c Σ|ab|,d¯s ¨ ¨ ¨ Dual DFT Action

1 ¯b¯cd,a¯ ¯e ¯b¯cd,a¯ e 1 da,¯b¯c e L “ ´ 2 Bd¯D B D¯b¯c¯e,a ´ Bd¯D B Dea,¯b¯c ´ 2 BdD B Dea,¯b¯c ` three more lines ab,¯cd¯ p2q ´ D Rab,¯cd¯pDq ´ LDFTpDa¯b,Dq where Rab,¯cd¯ ” 4 BraBr¯c Dbsd¯s linear DFT Riemann tensor 11 Dual Fields in E8p8q Exceptional Field Theory

M M E8p8q vielbein VM , coordinates Y , M “ 1,..., 248, subject to KL KL KL 0 PMN ” P1`248`3875 MN PMN BK b BL “ ` ˘ Naive generalized Lie derivative

M K M M PK L N LΛV ” Λ BKV ´ f NP f L BKΛ V Two (related) puzzles:

1) do not form an algebra [Berman, Cederwall, Kleinschmidt & Thompson (2013)] M m 2) VM Ą ϕm (m “ 1,..., 8) dual to KK vector Aµ dual graviton (Cm1...m8,n “ m1...m8ϕn) not in D “ 11

However, gauge invariant ‘Ricci scalar’ exists (M “ VVT ): 1 1 RpMq “ ´ MMN B MKL B M ` MMN B MKLB M 240 M N KL 2 M L NK 1 ` f NQ f MS MPKB M MRLB M ` ¨ ¨ ¨ 7200 P R M QK N SL 12 Resolution of puzzle: additional gauge symmetry

M K M M PK L N LM N δpΛ,ΣqV ” Λ BKV ´ f NP f L BKΛ V ´ΣLf N V

ñ closure & invariance of action provided covariantly constrained

KL KL PMN BK b ΣL “ PMN ΣK b ΣL “ 0

ñ 8 components upon solving constraint ñ δϕm “ λm , pure gauge

ñ corresponding gauge vectors BµM needed for Chern-Simons terms

General tensor hierarchy structure for any Enpnq D 3, E : D V δ V , δB Σ “ 8p8q µ “ pBµ ´ pAµ,Bµqq µM “Bµ M ` ¨ ¨ ¨ M M 1 MN D “ 4, E7p7q : Fµν “ 2BrµAνs ` ¨ ¨ ¨ ´ 2Ω BµνN

D “ 5, E6p6q : HµνρM “ 3BrµBνρsM ` ¨ ¨ ¨ ` CµνρM Compensator gauge ﬁelds needed for supersymmetry [Godazgar2, O.H., Nicolai, Samtleben (2014)] representations not part of E10 or E11 level decomposition 13 Generalized Scherk-Schwarz & Consistent KK Truncations

Ansatz in terms of UpY q P E7 K L MMN px, Y q “ UM pY q UN pY q MK Lpxq M ´1 ´1 M N Aµ px, Y q “ ρ pY qpU qN pY q Aµ pxq N ´1 ´1 N Y drops out ô consistency (twist) conditions for EM ” ρ pU qM [Aldazabal, Grana, Marques, Rosabal (2013)]

E X K E ,X K Θ α t K LEM N “ ´ MN K MN “ M p αqN ` ¨ ¨ ¨

Full consistency: non-trivial ansatz for compensating gauge ﬁeld

´2 ´1 P R α S Bµν M px, Y q “ ´2 ρ pU qS BM UP pt qR Bµν αpxq

7 5 Twist matrices constructed for AdS4 ˆS , AdS5 ˆS , hyperboloids, etc., ñ Consistency of KK truncation [de Wit & Nicolai (1986)]

14 Outlook & Summary

‚ Dual graviton and other mixed Young tableau ﬁelds in DFT ñ new features, new ﬁelds

‚ Construction of non-linear completion in completely new light, no-go theorems not applicable

‚ Fully non-linear DFT?

‚ Parts of dual graviton appear consistently and non-linearly in EFT, needed for supersymmetry and consistent KK ansatz

‚ Application for exotic branes?