Flato Fronsdal theorem for higher-order singletons

Transcription

1 Flato Fronsdal theorem for higher-order singletons Thomas Basile work with Xavier Bekaert & Nicolas Boulanger [arxiv: ] LMPT (Tours, France) & UMONS (Mons, Belgium) Thessaloniki

2 INTRODUCTION

3 Higher spin holography 1 Conjectured duality between Vasiliev s bosonic higher spin theory in AdS 4 and free or critical O(N) vector model CFT 3 (scalar: [KLEBANOV-POLYAKOV 00] / fermion: [SEZGIN-SUNDELL 003])

4 Higher spin holography 1 Higher spin/vector model Free / Critical O(N) vector model: S[φ] = 1 d 3 x µ φ a µ φ a + λ N (φa φ a ) Interaction term, critical model with φ a a primary scalar field, in the fundamental representation of O(N)

5 Higher spin holography 1 Higher spin/vector model Free / Critical O(N) vector model: S[φ] = 1 d 3 x µ φ a µ φ a + λ N (φa φ a ) Interaction term, critical model with φ a a primary scalar field, in the fundamental representation of O(N)

6 Higher spin holography 1 Higher spin/vector model SCALAR: Type A Type B AdS 4 : boundary conditions Dirichlet Neumann Dirichlet Neumann CFT 3 : conformal weight =, critical = 1, Free = 1, Gross-Neveu model =, Free From minimal to non minimal model: O(N) U(N)

7 Higher spin holography 1 Conjectured duality between Vasiliev s bosonic higher spin theory in AdS 4 and free or critical O(N) vector model CFT 3 (scalar: [KLEBANOV-POLYAKOV 00] / fermion: [SEZGIN-SUNDELL 003]) Consequent evidences: matching of the 3-points function [GIOMBI - YIN 010], free energy [GIOMBI - KLEBANOV 013]

8 Higher spin holography 1 Conjectured duality between Vasiliev s bosonic higher spin theory in AdS 4 and free or critical O(N) vector model CFT 3 (scalar: [KLEBANOV-POLYAKOV 00] / fermion: [SEZGIN-SUNDELL 003]) Consequent evidences: matching of the 3-points function [GIOMBI - YIN 010], free energy [GIOMBI - KLEBANOV 013] A few years before, some precursory results were obtained, among which Flato and Fronsdal theorem

9 SINGLETONS & THE FLATO-FRONSDAL THEOREM

10 Representations of so(, d) Highest weight representations Highest weight representations of so(, d) are classified by Λ = (, l) where : is the minimal energy of the representation (AdS point of view), or equivalently its conformal weight (CFT point of view), l is r = [ d ] components vector corresponding to the spin. A representation, or module, will be noted V(, l). If the representation is irreducible, we will use D(, l)

11 Singletons & Flato-Fronsdal theorem Singletons as so(, d) modules 3 Originally discovered by Dirac in 1963 Singletons = irreducible representation of so(, d), saturating the unitarity bound Describe particle living at the conformal boundary of AdS d+1 types, scalar : D( d d+1 1, 0) and spinor : D( 1, 1 ).

12 Singletons & Flato-Fronsdal theorem Singletons as so(, d) modules 3 Originally discovered by Dirac in 1963 Singletons = irreducible representation of so(, d), saturating the unitarity bound Describe particle living at the conformal boundary of AdS d+1 types, scalar : D( d d+1 1, 0) and spinor : D( 1, 1 ). Later, singleton type representations of spin-s where discovered (only for even values of d) [ANGELOPOULOS - LAOUES 1998]

13 Singletons & Flato-Fronsdal theorem Flato-Fronsdal theorem 4 Originally formulated for so(, 3) [FLATO & FRONSDAL 1978] 1978 : Flato-Fronsdal theorem Scalar singleton, particle on AdS 4 D ( 1, 0) D ( 1, 0) = s=0 D(s + 1, s) Massless particles of integer spin, in AdS 4

14 Singletons & Flato-Fronsdal theorem Flato-Fronsdal theorem 4 Originally formulated for so(, 3) [FLATO & FRONSDAL 1978] 1978 : Flato-Fronsdal theorem Scalar singleton, particle on AdS 4 D ( 1, 0) D ( 1, 0) = s=0 D(s + 1, s) Massless particles of integer spin, in AdS 4

15 Singletons & Flato-Fronsdal theorem Flato-Fronsdal theorem 4 Originally formulated for so(, 3) [FLATO & FRONSDAL 1978] 1978 : Flato-Fronsdal theorem Scalar singleton, particle on AdS 4 D ( 1, 0) D ( 1, 0) = s=0 D(s + 1, s) Massless particles of integer spin, in AdS 4

16 Singletons & Flato-Fronsdal theorem Flato-Fronsdal theorem 4 Originally formulated for so(, 3) [FLATO & FRONSDAL 1978] 1978 : Flato-Fronsdal theorem Scalar singleton, particle on AdS 4 Bilinear in the fields on the CFT side D ( 1, 0) D ( 1, 0) = s=0 D(s + 1, s) Massless particles of integer spin, in AdS 4 Spectrum of bosonic Vasiliev s theory

17 Singletons & Flato-Fronsdal theorem Flato-Fronsdal theorem 4 Originally formulated for so(, 3) [FLATO & FRONSDAL 1978] 1978 : Flato-Fronsdal theorem D ( 1, 0) D ( 1, 0) = D(s + 1, s) s=0 D ( 1, 0) D ( 1, 1 ) = D(s + 1, s) 1/,3/,... D ( 1, ) 1 ( ) D 1, 1 = D(, 0) D(s + 1, s) s=1

18 Singletons & Flato-Fronsdal theorem Flato-Fronsdal theorem 4 Originally formulated for so(, 3) [FLATO & FRONSDAL 1978] 1978 : Flato-Fronsdal theorem D ( 1, 0) D ( 1, 0) = D(s + 1, s) s=0 D ( 1, 0) D ( 1, 1 ) = D(s + 1, s) 1/,3/,... D ( 1, ) 1 ( ) D 1, 1 = D(, 0) D(s + 1, s) s=1

19 Singletons & Flato-Fronsdal theorem Flato-Fronsdal theorem 4 Originally formulated for so(, 3) [FLATO & FRONSDAL 1978] 1978 : Flato-Fronsdal theorem D ( 1, 0) D ( 1, 0) = D(s + 1, s) s=0 D ( 1, 0) D ( 1, 1 ) = D(s + 1, s) 1/,3/,... D ( 1, ) 1 ( ) D 1, 1 = D(, 0) D(s + 1, s) s=1 Massive field, pseudo-scalar then generalized to arbitrary dimension [VASILIEV 004], and to spin-s singletons [DOLAN 006]

20 Motivation 5 Idea : generalize this theorem to a wider class of (non unitary) representation of so(, d), called higher-order singletons. Motivations : AdS d+1 /CFT d correspondence Higher spin holography : vector model at Lifschitz point higher-spin gravity [BEKAERT - GRIGORIEV 013]

21 HIGHER-ORDER SINGLETONS

22 Higher-order singletons Group theoretical definition 6 Scalar singleton of order l N 0 : V( d V( d l, 0) = span + l, 0) = span v=iɛ,0 1 i r,ɛ=± v=iɛ,0 1 i r,ɛ=± P nv v D ( d l, 0) = V( d l,0) V( d +l,0) P nv v E nα α d l, 0 α Φ E nα α d + l, 0 V( d l, 0) α Φ

23 Higher-order singletons Group theoretical definition Primary scalar field of conformal weight = d l 6 Scalar singleton of order l N 0 : V( d V( d l, 0) = span + l, 0) = span v=iɛ,0 1 i r,ɛ=± v=iɛ,0 1 i r,ɛ=± P nv v P nv v D ( d l, 0) = V( d l, 0) + l, 0) V( d E nα α d l, 0 α Φ E nα α d + l, 0 α Φ Primary scalar field of conformal weight = d + l V( d l, 0)

24 Higher-order singletons Field realization 7 AdS point of view Solution of the Klein-Gordon equation in AdS d+1 ( AdSd+1 m ) ϕ = 0 with critical mass m = l ( d, ) l N0 and boundary behaviour ϕ ρ d/ l φ ρ 0 CFT/boundary point of view Conformal scalar field, with conformal weight = d l, solution of the polywave equation l φ = 0 m = ( d)

25 Quotient On mass-shell equation (1) 8 Scalar singleton of order l : ( ) d D l, 0 = V( d l, 0) + l, 0) D( d = d l = d + l l φ = 0 as P µ P µ

26 Quotient On mass-shell equation () 9 Spinor singleton of order l : ( d + 1 D l, 1 ) = V( d+1 l, 1 ) D( d 1 + l, 1 ) l φ = 0 l 1 / ψ = / l 1 ψ = 0

27 Quotient On mass-shell equation (3) 10 Partially massless field ([Deser - Waldron 001]): D(s + d t 1, s) = V(s + d t 1, s) D(s + d 1, s t) Symmetric tensors ϕ µ1...µ s of rank s in AdS d+1 subject to gauge transformation δϕ µ1...µ s = (µ1... µt ɛ µt+1...µ s) + terms of lower order in derivatives Partially conserved currents on the boundary subject to : µ1... µt j µ 1...µ tµ t+1...µ s = 0

28 TENSOR PRODUCT DECOMPOSITION

29 Decomposition 11 Generalized Flato-Fronsdal theorem D( d + 1 D( d + 1 D( d 1, 0) = D ( s + d, s ) s=0 1, 1 ) D( d 1, 0) = D(s + d, s, s= 1, 3,... 1, 1 ) = D(d 1, 0) 1 ) r 1 D (s + d, s, 1 m ) s=1 m=0

30 Decomposition 11 Generalized Flato-Fronsdal theorem D( d + 1 For d = r + 1 D( d + 1 D( d l, 0) = l k=1 s=0 l, 1 ) D( d l 1 l, 0) = l, 1 ) = l t= l l t=1 D ( s + d k, s ) t=1 D ( d 1 t, 0 ) s= 1, 3,... D(s + d t 1, s, 1 ) r 1 D (s + d l, s, 1 m ) s=1 m=0 r 1 D (s + d t 1, s, 1 m ) s=1 m=0

31 Decomposition 11 Generalized Flato-Fronsdal theorem For d = r D( d + 1 D( d + 1 l, 1 ) = D( d l, 0) = l k=1 s=0 l, 1 ) D( d l 1 l, 0) = l t= l D ( d 1 t, 0 ) D ( s + d k, s ) t=1 s=1 s= 1, 3,... D(s + d, s, { r 1 ) D (s + d l, s, 1 m ) m=0 [ ] D(s + d l, s, 1 r, 1) D(s + d l, s, 1 r, 1) l [ ( ) ( )] D s + d t 1, s, 1 r, 1 D s + d t 1, s, 1 r, 1 t=1 } l r 4D (s + d t 1, s, 1 m ) t=1 m=0

32 HOW TO OBTAIN THIS DECOMPOSITION?

33 Decomposition method Characters 1 Characters of (semi-simple) Lie algebras C V (µ) = λ Φ m λ e (λ,µ) where V is the vector space on which g is represented, Φ the set of weights of g in V, m λ the multiplicity of λ in V, µ an arbitray weight and (, ) the scalar product on weight space. Nice properties C V W = C V C W C V/W = C V C W C V W = C V + C W

34 Decomposition method Characters 13 Character of V Λ [DOLAN, 006] C (,d) [Λ] (q, x) = q C (d) (l) ( x)p(d) (q, x) C (d) (l) ( x) being the character of the Verma module V l and P (d) (q, x) the character of the module of trivial weight 0 EXAMPLE: SCALAR SINGLETON P (d) (q, x) = χ (,d) ( [ d l,0](q, x) = qd/ q l q l) P (d) (q, x) r 1 1 (1 qx i )(1 q/x i ), if d = r q i=1 1, if d = r

35 1-LOOP CHECK

36 Casimir energy 14 The Casimir, or vacuum, energy is : where ζ E (z) = 1 Γ(z) E c = 1 ζ E( 1) + 0 dββ z 1 Z 0 (β) and the partition function Z 0 for a free CFT is given by: Z 0 (β) = χ (,d) [,l] (q = e β, 1,..., 1)

37 Casimir energy 15 EXAMPLE: SCALAR SINGLETON χ (,d) [ d l,0](q, 1) = q d/ q l q l (1 q ) d Z 0(q) For the CFT d based on two scalar singleton: Z CFT (q) = [Z 0 (q)] = q d (q l q l ) (1 q ) d E C = 0 For the dual AdS d+1 higher spin theory of partially massless field of all integer spin: Z HS (q) = qd 1 (1 q )d ( l q s k+1 d s k=1 s=0 with d s = (s + d ) (s+d 3)! (d )!s!. s=k 1 q s d s k+1 ) E C = 0

38 CONCLUSION & PERSPECTIVES

39 Conclusion & Perspectives 16 Extension of Flato Fronsdal theorem to a larger class of singleton type representations using characters of so(, d) As a byproduct, characters give access to partition functions and thus a possibility of checking AdS/CFT at 1 loop Possible extension to spin-s 1 singletons (?) Possible extension to de Sitter signature, hint in ds/cft (?)

40 THANKS FOR YOUR ATTENTION!

41 BACKUP SLIDES

42 so (d + ) characters 16 For a heighest weight module V Λ with Λ = (, l) : So the character becomes : χ V = λ λ = Λ + α Φ n α α, with n α N (1) m λ e λ χ Λ = e Λ e nαα = e Λ 1 1 e α () α Φ α Φ n α N

43 so (d + ) characters 16 In the orthonormal basis, for d = r, Φ = { e i ± e j, 0 i < j r}, with r = [ d ]. Let s isolate the o() contribution to the weights : λ 0 = n i± (3) l = l + α Φ (d) } {{ } weights of so(d) 1 i r n α α n i+ n i (4) 1 i r where n i± are the n α associated with e 0 ± e i and Φ = Φ (d) { e 0 ± e i } i 1,r

44 so (d + ) characters 16 C (r,) Λ (µ) = λ e λ = q e (l,µ) α Φ (r) = q ( 1 î r 1 î r + n α=0 (e α ) nα + nî±=0 1 (1 qxî)(1 qx 1 î ) q n î++nî x n î+ nî î ) C (r) l (µ) = q P (r) C (r) l 1 Idem d = r + 1 with e i, 0 i r add a factor 1 q in P(d)

45 Higher-order singleton Wave equation 16 Poincaré patch, with z = ρ: Klein Gordon equation then reads: ds = dρ 4ρ + 1 ρ η abdx a dx b 0 = ( AdSd+1 m ) ϕ 0 = ρ { 0 φ + ( d + ) ρ φ + ρ ρφ } ( ( d) m ) φ At the boundary: Expanding in powers of ρ: ρ = 0 ( d) = m 0 φ n 1 = n(d n)φ n

46 Generalized Verma module 16 Definition : Generalized Verma modules Let Y(, l) be an irreducible so() so(d)-module. The generalized Verma module V(, l) is defined as : V(, l) = U(g ) Y(, l) where U(g ) is the universal enveloping algebra of g, subalgebra of so(, d) generated by the lowering operators.

47 General theorem 16 D( d l, 0) D( d l, 0) = D( d+1 l, 1 ) D( d l, 0) = t= l+l k= l l l+l 1 l l 1 s= D(s + d k, s, 0), s= 1, 3,... D(s + d t 1, s, 1 ).

48 General theorem 16 In the cases d = r + 1, we have, for l l : D( d+1 l, 1 d+1 ) D( l, 1 ) = l+l [ D ( d 1 + t, 0 ) D ( d 1 t, 0 )] t= l l r 1 [ l+l s=1 m=0 t= l l +1 D (s + d t 1, s, 1 m ) ] D (s + d l l 1, s, 1 m ) D (s + d l l, s, 1 m ),

49 General theorem 16 while in the cases d = r and for l l : D( d+1 l, 1 d+1 ) D( l, 1 ) = l+l [ D ( d 1 + t, 0 ) D ( d 1 t, 0 )] t= l l r [ l+l s=1 m=0 t= l l +1 4D (s + d t 1, s, 1 m ) ] D (s + d l l 1, s, 1 m ) D (s + d l l, s, 1 m ) s=1 [ l+l t= l l +1 (D ( s + d t 1, s, 1 r, 1 ) D ( s + d t 1, s, 1 r, 1 ) ) D ( s + d l l 1, s, 1 r, 1 ) D ( s + d l l, s, 1 r, 1 ) ].