On higher-spin gravity in three dimensions

Transcription

1 On higher-spin gravity in three dimensions Jena, 6 November 2015 Stefan Fredenhagen Humboldt-Universität zu Berlin und Max-Planck-Institut für Gravitationsphysik

2 Higher spins Gauge theories are a success story: Spin 1: Electrodynamics, Yang-Mills... A µ Spin 2: Gravity g µ Why not go beyond? Spin 3, 4, 5,... µ 1...µ s

3 Why higher spins? 1. Generalisations of geometry: HS symmetries as generalised diffeomorphisms HS gauge g µ g 0 µ transformation Curvature? Horizons? Causal structure? Need new concepts! Example in 3d: Black hole HS gauge transformation wormhole [Ammon,Gutperle,Kraus,Perlmutter]

4 Why higher spins? 2. Quantum Gravity: Exchange of higher spins could cure UV divergence. String theory: UV problems of perturbative quantum gravity solved by exchange of infinitely many massive higher-spin excitations. = n M n Are there other ways?

5 Why higher spins? 3. String theory: String theory could arise as a broken phase of HS theories - is HS symmetry the symmetry behind string theory? Standard Higgs effect: Gauge bosons get massive by eating the Goldstone bosons. La Grande Bouffe

6 Why higher spins? 4. AdS/CFT: HS theories as class of models in which holographic dualities can be studied (no SUSY required) AdS HS gauge theory HS gauge symmetry boundary conformal field theory global HS symmetry If HS theory related to string theory: extreme stringy limit of AdS/CFT.

7 Bumpy road to success Many no-go results for d>3: flat space minimal coupling finitely many HS 90 s [Vasiliev et al.]: Consistent higher-spin gauge theories coupled to matter - infinitely many HS - non-zero cosmological constant Applications & Questions: AdS/CFT? Solutions? Uniqueness? Higgs? Quantisation?

8 Why 2+1 dimensions? Gravity in 2+1 dimensions as toy model: No propagating degrees of freedom R µ = C µ + 2 d 2 (g µ[ R ] g [ R ]µ ) 2 0 (d 1)(d 2) Rg µ[ g ] Black holes exist for < 0 Black hole entropy can be studied. AdS/CFT concepts can be tested. Asymptotic symmetries: Virasoro central charge c = 3` 2G [Bañados, Teitelboim, Zanelli] Virasoro AdS radius [Brown,Henneaux]

9 Higher spins Higher spin extensions of gravity simpler in 2+1 than in higher dimensions: no propagating degrees of freedom asymptotic symmetries: Virasoro W N, W 1 [Henneaux,Rey] [Campoleoni,S.F., Pfenninger,Theisen] generalisations of black holes [Gutperle,Kraus] [Ammon,Gutperle,Kraus,Perlmutter] AdS/CFT proposal: duality between HS gravity and minimal models [Gaberdiel, Gopakumar]

10 Outline 1. Higher-spin gravity in 3 dimensions 2. Asymptotic Symmetries 3. Holography

11 1) Gravity and higher-spins in 3d Gravity: with S[e,!] = 1 8 G Z R = d! +! ^! Tr e ^ R + 1 3`2 e ^ e ^ e Vielbeine e a with e a µ µ e b apple ab = g µ,! ab spin connections. µ! µ a = f a bc! µ bc For higher-spins: Two approaches Second order/metric-like g µ ' µ1...µ s First order/frame-like e a µ e a 1...a s 1 µ

12 Metric-like Free massless spin s particle [Fronsdal] fully symmetric tensor ' µ1...µ s ' µ1...µ s 4 = 0 (double traceless) ' µ1...µ ' µ2...µ s ) µ2 ' µ3...µ s ) = 0 (e.o.m.) ' µ1...µ s (µ1 µ2...µ s ) where µ1...µ s 3 = 0 Interactions: add term by term while retaining gauge invariance. (gauge invariance) [Bengtsson,Bengtsson,Brink] [Metsaev] [Manvelyan,Mkrtchyan,Rühl] [Sagnotti,Taronna] [Joung,Lopez,Taronna]

13 Frame-like [Vasiliev] Generalised vielbein Gauge transformations e a 1...a s 1 µ a 1...a s 1 = µ 1...µ s 1 ē a 1 µ...ē a s 1 1 such that ' µ1...µ s = ē a 1 (µ 1...ē a s 1 µ s 1 e b 1...b s 1 µ s ) a1 b 1... as 1 b s 1 background vielbein e a 1...a s 1 µ = D µ a 1...a s 1 +ē µ,b b,a 1...a s 1 with. Spin connections µ s 1 s 1 z } {! b,a 1...a s 1 µ = D µ b,a 1...a s 1 +ē µ,c bc,a 1...a s 1! bc,a 1...a s 1 µ = D µ bc,a 1...a s 1 +ē µ,d bcd,a 1...a s 1... s 1 z } { vanish in 3 dimensions!

14 Action Dualise:! a 1...a s 1 µ =! b,c(a 2...a s 1 µ f a 1) bc S[e,!] = 1 Z Tr e ^ (d! +! ^!)+ 1? 8 G 3`2 e ^ e ^ e e = (e a µ J a + e a 1a 2 µ J a1 a )dx µ! = (! a µ J a +! a 1a 2 µ J a1 a )dx µ Lie algebra structure e.g. for spin 3: [J a,j b ] = f c ab J c [J a,j bc ] = f d sl(3) a(b J c)d [J ab,j cd ] = (apple a(c f d)v f + apple b(c f d)a f )J f Metric-like quantities: g µ = e A µ e B apple AB ' µ = e A µ e B e C d ABC, [Campoleoni,S.F.,Pfenninger,Theisen]

15 Interacting higher spin theories Generalisations to sl(n): fields of spin 2 to spin n hs : fields of all spins s greater equal 2 λ Chern-Simons formulation: S = S CS [A] with S CS [Ā] S CS [A] = k 4 Z [Blencowe] [Bergshoeff,Blencowe,Stelle] [Vasiliev] Ā, =! ± 1`, A e k = ` 4G A Tr ^ da A ^ A ^ A Coupling to massive scalar known in Prokushkin-Vasiliev theory (no action).

16 2) Asymptotic symmetries To describe AdS: consider CS theory on a cylinder and specify boundary conditions S = k bound. 4 Z A dx + dx Tr + A A A + Boundary condition: A bound. =0 Gauge symmetries become global symmetries: A +[A, ] is generated by Z Z G( ) = dx i ^ dx j k Tr F ij dx i Tr A i D 2 2 S 1 constraint Global charges satisfy affine Lie algebra ĝ k.

17 Coordinates: t,, AdS asymptotics sl(2)-part: L 0,L 1,L 1 with [L m,l n ]=(m n) L m+n Gauge choice: A = b 1 ( b( ) with b( ) =e L 0 Constraint/e.o.m./boundary condition: A = b 1 ( ) a(t, ) b( ) A t = b 1 ( ) a(t, ) b( ) AdS: a AdS = L L 1 Asymptotically AdS: A A AdS bound. finite

18 AdS asymptotics Asymptotically AdS: A A AdS bound. finite Lie algebra generators W`,m in repr. of sl(2): [L n,w`,m ]=(` n m) W`,m+n Then: b 1 ( ) W`,m b( ) = m W`,m ) a = L 1 + w 1,0 L 0 + w 1, 1 L 1 + X` X m 0 w`,m W`,m Equivalent to Drinfeld-Sokolov condition (similar to the Hamiltonian reduction of WZW models) [Balog,L. Fehér,O Raifeartaigh,Forgács, Wipf

19 AdS asymptotics Asymptotically AdS: A A AdS bound. finite equivalent to Drinfeld-Sokolov condition: The asymptotic symmetries are given by the ĝ k Drinfeld-Sokolov reduction of. sl(2) Virasoro sl(n) W n hs( ) W 1 ( ) c =6k [Campoleoni, S.F.,Pfenninger, Theisen] [Brown,Henneaux] [Campoleoni,S.F.,Pfenninger,Theisen] [Henneaux,Rey] [Gaberdiel,Hartman] [Campoleoni,S.F.,Pfenninger]

20 3) Higher-spin AdS/CFT In AdS/CFT, the boundary operators correspond to sources for the bulk fields. AdS CFT Z D e S gravity & more = D e R d d x i 0 O i E CFT d i boundary of AdS d+1 i 0 O i operator corresponding to i

21 Higher-spin AdS/CFT The string theoretic AdS/CFT correspondence has a higher-spin cousin: Higher-spin gauge theories coupled to CFTs with classically matter conserved higherspin currents J µ 1...µ s d d x µ1...µ J µ 1...µ s s Z Z d d x@ µ1 µ2...µ J µ 1...µ s = d d x µ 2...µ J µ 1...µ s s µ1 Coupling bdy value of HS field CFT operator Z '

22 Higher-spin AdS/CFT Prominent case: higher-spin AdS /CFT : Vasiliev theory Free/critical bosons/fermions 4 3 [Klebanov,Polyakov] [Sezgin,Sundell] [Giombi,Yin] If HS gauge symmetry is unbroken in the quantum theory, the boundary theories are essentially trivial for d>2. In d=2, non-trivial quantum theories are known with extended symmetries: W-algebras.

23 Minimal model holography The classical W-algebras have a quantum version W 1 ( ) Ŵ 1 ( ). We can then look for families of CFTs with W-symmetries as candidates for the boundary theory. Prototype: -minimal models W n To compare to the classical higher-spin theories we look for families which admit a classical ( c!1 ) limit.

24 Minimal model holography The minimal models charges: c n,k =(n 1) =2k 1 1 W n,k n(n + 1) (n + k)(n + k + 1) (k + 1)(k +(3n + 1)/2) (k + n)(k + n + 1) Two ways to achieve c!1 : come with central <n 1 < 2k t Hooft limit: n, k!1, = n fixed n + k semi-classical limit: k! n 1 [Gaberdiel,Gopakumar] [Castro,Gopakumar,Gutperle,Raeymaekers]

25 Minimal model holography t Hooft limit: unitary Some CFT states match excitations of a scalar field coupled to HS fields many light states - interpretation? checks: partition function, 3-point functions,... [Gaberdiel,Gopakumar,Saha] [Gaberdiel,Gopakumar,Hartman,Raju] [Chang,Yin]...

26 Minimal model holography semi-classical limit: non-unitary light states match excitations of a scalar other states ~ non-perturbative solutions: conical defects: all HS charges match [Castro,Gopakumar,Gutperle,Raeymaekers] [Campoleoni,Prochazka,Raeymaekers] [Campoleoni,S.F.] further checks: partition function, correlation functions, [Perlmutter,Prochazka,Raeymaekers] [Hijano,Kraus,Perlmutter]

27 Developments HS black holes / entropy [Gutperle,Kraus] [Ammon,Gutperle,Kraus,Perlmutter] [Pérez,Tempo,Troncoso] [Castro,Hijano,Lepage-Jutier,Maloney] [Bañados,Canto,Theisen] [Ammon,Castro,Iqbal] [de Boer,Jottar] [Henneaux,Pérez,Tempo,Troncoso] SUSY generalisations [Creutzig,Hikida,Rønne] [Candu,Gaberdiel] [Henneaux,Lucena Gomez,Park,Rey] [Beccaria,Candu,Gaberdiel,Groher] [Candu,Peng,Vollenweider] Relation to string theory: relate HS-CFT dual to String-CFT dual [Gaberdiel,Gopakumar] [Gaberdiel, Peng, Zadeh] [Creutzig,Hikida] [Hikida,Rønne] Improve understanding of HS theories: Metric-like formulation/interactions [Campoleoni,S.F.,Pfenninger,Theisen] [Fujisawa,Nakayama] [S.F.,Kessel] [Kessel,Lucena Gomez,Skvortsov,Taronna]

28 HS AdS /CFT duality: dim. laboratory for HS and AdS/CFT W-symmetry of boundary CFT: Drinfeld-Sokolov reduction Minimal models candidates for CFT duals HS gauge theories: Summary fascinating extensions of gravity relation to string theory? rich possibilities for (non-susy) AdS/CFT