An Introduction to Higher-Spin Fields

Transcription

1 An Introduction to Higher-Spin Fields Augusto SAGNOTTI Scuola Normale Superiore, Pisa Some Some reviews: N. N. Bouatta, G. G. Compere, A.S., A.S., hep-th/ D. D. Francia and and A.S., A.S., hep-th/ A.S., A.S., Sezgin, Sundell, hep-th/ X. X. Bekaert, S. S. Cnockaert, C. C. Iazeolla, M.A. M.A. Vasiliev, hep-th/ A.S., A.S., P. P. Sundell, D. D. Sorokin, M.A. M.A. Vasiliev, Phys. Phys. Reports, (?) (?) Eotvos Superstring Workshop, Budapest, Sept. 2007

2 Some Motivations for HS Key Key (old) problem in (classical) Field Theory: Only s=0,1/2,1,3/2,2 Key role in String Theory: (Non) Planar duality of tree amplitudes Modular invariance and soft U.V. Open-closed duality Budapest, Sept

3 For instance (Non-) ) planar duality rests on infinitely many poles [Actual t (or s) dependence implies a growing sequence of spins] Similarly for modular invariance: Budapest, Sept

4 What is Spin here? D=4 : Up to dualities, all cases exhausted by fully symmetric (spinor)tensors: D > 4 : Arbitrary Young tableaux : spin somehow the number of columns. Less developed, clumsy in general, many general lessons can be drawn d from the previous special set of fields. (See X. Bekaert and N. Boulanger, hp-th/ and seminar of A. Campoleoni) Budapest, Sept

5 Summary, I Lecture I: Free Higher-Spin fields Lecture II: Fierz-Pauli condition: the s=2 case in detail Singh-Hagen and Fronsdal formulations: trace constraints Kaluza-Klein Klein masses Fang-Fronsdal Fronsdal formulation for Fermi fields Free Higher-Spin fields Constrained vs unconstrained formulations Non-local bosonic formulation and Higher-Spin Geometry Non-local fermionic construction Budapest, Sept

6 Lecture III: Relation with String Theory Summary, II BRST formulation for the bosonic string Low-tension limit and triplets Compensator equations for Higher Spins Lecture IV: Compensator equations and minimal Lagrangians An alternative off-shell formulation External currents Lessons for the non-local formulation VDVZ discontinuity The Vasiliev construction (and the compensator) Problems with higher-spin interactions The Vasiliev construction and the compensator Budapest, Sept

7 Lecture I Free Higher-Spin fields Fierz-Pauli condition: the s=2 case in detail Singh-Hagen and Fronsdal formulations, trace constraints Kaluza-Klein Klein masses Fang-Fronsdal Fronsdal formulation for Fermi fields Budapest, Sept

8 Fierz-Pauli conditions: Bose (Fierz, Pauli, 1939) Spin-s s boson of mass m : Correct degrees of freedom: Combine with trace: Only traceless spatial components Budapest, Sept

9 Fierz-Pauli conditions: Fermi (Fierz, Pauli, 1939) Spin-s s fermion of mass m: Correct degrees of freedom, with: Combine with γ-trace: only γ-traceless spatial components Budapest, Sept

10 Massive case: low spins Spin 1: (Singh, Hagen, 1974) Gives Proca equation: For s=2 try (Φ( traceless): Does not not give the the correct Fierz-Pauli conditions Budapest, Sept

11 Massive case: spin 2 Can still take the divergence: Take α=2 (eliminate ) Take If we also impose: Add a Lagrange multiplier (with its kinetic and mass terms): Budapest, Sept

12 Massive case: spin 2 2x2 homogeneous system: Determinant is algebraic if : Budapest, Sept

13 Massless case: spin 2 Not surprisingly: gauge symmetry as M 0M (Fronsdal, 1978) The equations become: Gauge invariant under: Budapest, Sept

14 Massless case: spin 2 In terms of a traceful spin 2: Fronsdal eq: Fronsdal action: (Einstein, actually, in in this case!)!) This gives: Budapest, Sept

15 Kaluza-Klein Klein mass Masses can be (re)generated by K-K K K reductions (complex notation): Spin 1: Upon D+1 D reduction : Stueckelberg symmetry: For spin 2: Budapest, Sept

16 Fronsdal equation: spin 3 (Fronsdal, 1978) Not a Lagrangian equation Bianchi identity: The Lagrangian actually yields: Budapest, Sept

17 Implicit Notation For all spins, one can eliminate all indices Need only some unfamiliar combinatoric rules (Francia, AS, 2002) Budapest, Sept

18 Fronsdal equations: spin s (Fronsdal, 1978) We have seen that gauge invariance requires: We can also derive the Bianchi identity: ϕ '' Budapest, Sept

19 Fronsdal equations: spin s Fronsdal construction: Constraints: Lagrangians: Budapest, Sept

20 Massless case: spin 3/2 Rarita-Schwinger equation: familiar from supergravity gauge invariant under: Budapest, Sept

21 Massless case: spin 5/2 (Fradkin,1941; Fang,Fronsdal, 1978) Try to repeat for spin-5/2: NOT gauge invariant : BUT: can again constrain the the gauge parameter! Budapest, Sept

22 Massless case: spin s=n+1/2 Can extend to all ½-integer spins: Bianchi identity: Additional constraint: Budapest, Sept

23 Bianchi identity for spin s+1/2 Now derive the Bianchi identity: Budapest, Sept

24 Kaluza-Klein Klein mass: Bose Can extend the K-K K K construction to spin-s s case (Stueckelberg gauge symmetries) [ (s-3) 3)- parameter missing due to trace condition] [(s-4) 4)-field missing due to double trace condition] Gauge fixing the Stueckelberg symmetries one is left with: In In terms of of traceless tensors Singh-Hagen fields Budapest, Sept

25 Kaluza-Klein Klein mass: Fermi Can again extend the K-K K K construction to the spin-(n+1/2) case : (Stueckelberg gauge symmetries) [ (s-2) 2)- parameter missing due to γ-trace condition] [(s-3) 3)-field missing due to triple γ-trace condition] Gauge fixing the Stueckelberg symmetries one is left with: In In terms of of γ-traceless tensors Singh-Hagen fields Budapest, Sept

26 Lecture II Free Higher-Spin fields Constrained vs unconstrained formulations Non-local bosonic formulation and Higher-Spin Geometry Non-local fermionic construction Budapest, Sept

27 Fronsdal equations, revisited Gauge invariance for massless symmetric tensors: (Fronsdal, 1978) δϕ = Λ Λ μ... μ μ μ... μ μ μ... μ 1 s 1 2 s s 1 s 1 F μ 1... μs ϕμ1... μs μ ϕ 1 μ2... μs μ1 μ ϕ 2 μ3... μs (. +...) + ( ' +...) = 0 Originally from massive Singh-Hagen equations Unusual constraints: (Singh and Hagen, 1974) Unusual constraints: Λ ' = 0, ϕ '' = 0 Budapest, Sept

28 Bianchi identities Why the unusual constraints: 1. Gauge variation of F δ = 3( Λ ' +...) F μ... μ μ μ μ μ... μ 1 s s 2. Gauge invariance of the Lagrangian As in the spin-2 2 case, F not integrable Bianchi identity: 1 3 Fμ 2... μ s μ F 2 μ3... μ s μ 2 μ 3 μϕ 4 μ5... μ + s 2 2 ( ' +...) = ( ''...) Budapest, Sept

29 The Fronsdal actions equations: Fμ 0 R μ = μν = Naive equations: s Do not follow directly from a Lagrangian But: when combined with traces they do 1 1 Rμν ημν R= 0 Fμ 1... μ ( η F' ) s μ μ μ μ + s 2 2 Budapest, Sept

30 Constrained gauge invariance 1 δl= δϕμ 1... μs Fμ 1... μ η F s μ1μ2 μ3... μ + s 2 ( '...) If in the variation of L one inserts 1 s 1 2 Budapest, Sept δϕ = Λ + μ... μ μ μ... μ... 1 ( ) 1 δl = sλ ( ) μ2... μ F 2... F ' F ' s μ μ s μ μ μ + η s μ μ μ μ + s Bianchiidentity: ϕ" Λ' Are the constraints really necessary? (Francia, AS, 2002) s

31 The spin-3 3 case μ ν ρ δfμνρ 3 μ ν ρ ' δ = Λ F' 3 ' 2 = μ ν ρλ A fully gauge invariant (non-local) equation: F μνρ = 0 F μνρ μ ν ρ F ' 0 2 = Reduces to local Fronsdal form upon partial gauge fixing Budapest, Sept

32 Spin 3: other non-local eqs Other equivalent forms: 1 ( F ' + F ' + F ' ) = 0 3 Fμνρ μ ν ρ ν ρ μ ρ μ ν 1 F ( F ) μνρ μ νρ + ν Fρ μ + ρ Fμ ν = 3 0 Lesson: full gauge invariance with non-local terms Budapest, Sept

33 Spin 3: non-local action One can define an Einstein-like tensor: μ ν ρ 1 3 Gμνρ Fμνρ F' η F' F'... 2 μν ρ + 2 Now: 0 G νρ Gauge invariance of L with NO constraints Budapest, Sept

34 Spin 3: non-local action One can simply associate to the previous non-local equation the non-local action ( ) ( ) ( ) 2 3 ϕ ϕ ϕ' + ( ϕ ' ) ϕ' ϕ + 3 ϕ' ϕ ϕ L = μ αβγ + βγ μ α α α ϕ 2 fully invariant under δϕ αβγ = α Λ βγ + β Λ γα + γ Λαβ Budapest, Sept

35 Kinetic operators for integer spin Index-free notation: Now define: F ϕ ϕ + ϕ' = 0 (1) 2 eg.. ( p+ q)! p! q! p q p+ q F = F + F ' F ( n+ 1)(2n+ 1) n+ 1 2 ( n 1) ( n) ( n) ( n) Then: 2n+ 1 ( n) [ n] = (2 + 1) Λ n 1 δ F n Budapest, Sept

36 Kinetic operators for integer spin Defining: 1 μ1 μ s Φ ( x, ξ ) = ξ... ξ ϕ s! ξ = ξ μ1... μ s k (1) 1 + ξ ξ ξ F ( Φ ) = 0 ( k + 1)(2k + 1) k + 1 is the generic kinetic operator for higher spins when combined with traces can be reduced to 3 F = H ( δ H = 3 Λ') Budapest, Sept

37 Kinetic operators for integer spin The F (n) : Are gauge invariant for n > [(s-1)/2] Satisfy the Bianchi identities 1 F F ' = 2n ( n) ( n) n 2n+ 1 n 1 ϕ [ n+ 1] For n> [(s-1)/2] allow Einstein-like operators G ( 1) ( n p)! n 1 p ( n) p ( n)[ p] = η F p p= 0 2 n! Budapest, Sept

38 Geometric equations Christoffel connection: δ h = ε + ε μν μ ν ν μ Generalizes to ALL symmetric tensors δ Γ μ = νρ ν ρ (De Wit and Freedman, 1980) ε μ Γ R Γ μ; νν μ... μ ; ν... ν s 1 1 R μ μ ; νν μ... μ ; ν... ν s 1 Budapest, Sept s s

39 Connections ( s 1 ϕ ) s δ s 1 s 1 s = Λ s s 1 s In general:, : 1 ( 1) Γ = m + 1 k = 0 m k m k ( m) m k k ϕ Derivatives w.r.t. two sets of sym. indices Budapest, Sept

40 Connections In Einstein gravity: metric (vielbein) postulate α α ρ μν ρ μν ρμ αν ρν μα g g Γ g Γ g = 0 Linearizing: g η + h μν μν μν For spin 3 (linearized): h =Γ +Γ ρ μ μν ν; ρμ ; ρν φ =Γ +Γ +Γ σ τ μνρ νρ; στμ ρμ ; στν μν ; στρ Budapest, Sept

41 Geometric equations 1. Odd spins (s=2n+1): 1 = 0 [ n] ; v1... v s = F μν μ R μ n μ 0 2. Even spins (s=2n): R μν = 0 1 R [ n]; v1... v s = n 1 0 Budapest, Sept

42 Fermions (Fang and Fronsdal, 1978) S i( ψ ψ) = 0 δs= 2i 2 ε Bianchi identity: 1 1 S S' S = i ψ ' Constraints: ψ ' = 0, ε = 0 Budapest, Sept

43 Fermions S Notice: 1 s+ 2 1 S 2 1 s+ 2 = i F ( s) (see Francia s s seminar) γ μνρ Example: spin 3/2 (Rarita-Schwinger) ( ) = 0 ψ ν ρ = 0 Sμ i ψ μ ψ μ S μ 1 2 μ S i = ημν μ v ψ Budapest, Sept ν

44 Fermions One can again iterate: The relation to bosons generalizes to: Gauge transformations: Bianchi identities: Budapest, Sept

45 Lecture III Relation with String Theory BRST formulation for the bosonic string Low-tension limit and triplets Compensator equations for Higher Spins Compensator equations and minimal Lagrangians An alternative off-shell formulation Budapest, Sept

46 Bosonic string: BRST The starting point is the Virasoro algebra: In the tensionless limit, one is left with: Virasoro contracts (no c. charge): Budapest, Sept

47 Low-tension limit Similar simplifications for BRST charge: Making zero-modes manifest: Budapest, Sept

48 String Field equation Higher-spin massive modes: massless for 1/α 0 Free dynamics can be encoded in: Q ψ δψ 2 = 0 ( Q = 0) = Q Λ (Kato and Ogawa, 1982) (Witten, 1985) (Neveu, West et al, 1985) NO trace constraints on j or L Budapest, Sept

49 Symmetric triplets Emerge from (A. Bengtsson, 1986) (Henneaux, Teitelboim, 1987) (Pashnev, Tsulaia, 1998) (Francia, AS, 2002) The triplets are: Budapest, Sept

50 Symmetric triplets Can also eliminate C: Gauge theories of Physical state conditions: Propagate spins s,s-2,,, 0 or 1 Budapest, Sept

51 (A)dS symmetric triplets Can build directly, deforming flat-space triplets, or via BRST Directly: insist on relation between C and others BRST: gauge non-linear constraint algebra Basic commutator: Budapest, Sept

52 (A)dS symmetric triplets Can also deform directly the equations without C: It is convenient to define Bianchi identity and first equation then become: Budapest, Sept

53 (A)dS symmetric triplets The deformed equations can be derived from Alternatively: modify momenta in contracted Virasoro gauge algebra non linear (but M,N diagonal on triplets) Budapest, Sept

54 Compensator Equations In the triplet: spin-(s (s-3) compensator: The first becomes: The second becomes: Combining them: Finally (also from Bianchi): Budapest, Sept

55 Compensator Equations Summarizing: Describe a spin-s gauge field with: NO trace constraints on on the the gauge parameter NO trace constraints on on the the gauge field First can can be be reduced to to minimal non-local form BUT: NOT Lagrangian equations Budapest, Sept

56 (A)dS Compensator Eqs Flat-space compensator equations can be extended to (A)dS: Gauge invariant under The first can be turned into the second via (A)dS Bianchi Budapest, Sept

57 Compensator Equations It is possible to obtain a Lagrangian form of the compensator equations, using BRST techniques Essentially in Pashnev and Tsulaia (1997) Formulation involves number of fields ~ s Interesting BRST subtleties Here we discuss explicitly spin s=3 Fields: Budapest, Sept

58 Compensator Equations Gauge transformations: Field equations: Gauge fixing: Other fields: zero by field equations Budapest, Sept

59 Fermionic Triplets Counterparts of bosonic triplets GSO: not in 10D susy strings Yes: mixed sym generalizations Enter directly type-0 0 models (Francia and AS, 2002) Propagate s=n+1/2 and all all lower ½-integer spins Budapest, Sept

60 Fermionic Compensators Recall: Spin-(s (s-2) compensator: Gauge transformations: First compensator equation second via Bianchi Budapest, Sept

61 Fermionic Compensators We could extend the fermionic compensator eqs to (A)dS backgrounds [We could not extend the fermionic triplets] First compensator equation second via (A)dS Bianchi identity: Budapest, Sept

62 Off-Shell truncation of triplets Off-shell reduction of triplets : ( BuchbindB uchbinder, Krykhtin, Reshetnyak 2007 ) start from a triplet (s,s-2, 2, ) add two (gauge invariant) Lagrange multipliers Lagrangian : λ and μ : set to zero by the field equations Budapest, Sept

63 Minimal local Lagrangians (Francia, AS, 2005; Francia, Mourad and AS, 2007) Minimal local Lagrangians with unconstrained gauge symmetry: The Lagrangians are: Can be nicely extended to (A)dS backgrounds Budapest, Sept

64 Lecture IV External currents Lessons for the non-local formulation VDVZ discontinuity The Vasiliev construction (and the compensator) Problems with higher-spin interactions The Vasiliev constructions and free-differential algebras Strong vs weak projection: recovery of the compensator Budapest, Sept

65 Minimal local Lagrangians (Francia, AS, 2005; Francia, Mourad and AS, 2007) Minimal local Lagrangians with unconstrained gauge symmetry: The Lagrangians are: Can be nicely extended to (A)dS backgrounds Budapest, Sept

66 External currents Residues of current exchanges reflect the degrees of freedom For s=1 : For all s : Budapest, Sept

67 External currents : local case K doubly traceless using double trace constraint B: determines multiplier β for double trace constraint The exchange involves, correctly, a traceless conserved current Budapest, Sept

68 External currents : non-local case How about the non-local version of the theory? Apparently: different choices for the field equation, EQUIVALENT without currents S=3 : Bianchi identity: : changes after every iteration Budapest, Sept

69 External currents : non-local case Naively: Solution: modify the non-local Lagrangian equation Incorrect current exchange! For instance : Budapest, Sept

70 VD-V-Z Z Discontinuity for HS (van Dam, Veltman; Zakharov, 1970) For all s and D, m=0 : VDVZ discontinuity follows in general comparing D and (D+1) massless exchanges First present for s=2 via D-dependence D of ρ(d,s) For all s: can describe irreducibly a massive field a a la Scherk-Schwarz Schwarz from (D+1) dimensions : [ e.g. for s=2 : h MN (h μν cos(my), A μ sin(my), ϕ cos(my) ) ] (A)dS extension, first discussed, for s=2, by Higuchi and Porrati Discontinuity smooth interpolation in (ml) 2 Budapest, Sept

71 HS Interactions Problems with interacting higher spins : Inconsistent equations (derivatives imply further conditions) Coupling with gravity leaves naked Weyl tensors Coleman - Mandula (Aragone, Deser,1979) Way out: Infinitely many interacting fields Non-vanishing cosmological constant Λ Vasiliev equations: paradigmatic example (Berends, Burgers, van Dam, 1982) (Bengtsson 2, Brink, 1983) (Fradkin and Vasiliev, 1980 s) (Vasiliev, 1990, 2003) (Sezgin, Sundell, 2001) Budapest, Sept

72 Aragone-Deser problem Flat space Lagrangians: usually gauge invariant as a result of cancellations between terms that differ by commutators Moving to curved backgrounds introduces in general naked Riemann tensors Miracle of supergravity: such terms combine into Einstein tensors, which allows cancellations Independent higher-spin propagation possible in conformally flat backgrounds (Aragone, Deser, 1979) (A)dS: mass-like terms,, but free. Central role in Vasiliev equations Budapest, Sept

73 Vasiliev s setting: HS Interactions 1. Extend the frame formulation of gravity : For spin-s: 2. - dim. HS-algebra via oscillators (coordinates and momenta): Budapest, Sept

74 HS Interactions 3. [Weyl ordered (symmetric) polynomials in (x,p) or - products ] 4. A one-form A in adjoint of HS algebra : (all ω s s : HS vielbeins and connections) 5. A zero-form Φ in the twisted adjoint : One writes the spin-2 2 equation in the form: Riemann = Weyl whose trace gives the familiar equation Ricci=0 Weyl (+ derivatives) : fields with Young-tableau structure Scalar : ϕ Budapest, Sept

75 HS Interactions TWO forms of Vasilev equations : 1. 4-dim spinors (2-component formalism) 2. D-dim vectors Let us try to see what these equations mean (focusing on the case with vector oscillators) Budapest, Sept

76 HS Interactions Background independent (non-lagrangian)! A: s fields for every (even) rank s ( adjoint of HS algebra) (Generalized vielbeins and connections) [Chan-Paton extension to all (even and odd) ranks] Φ: fields for every rank s ( twisted adjoint of HS algebra) (Generalized Weyl and their covariant derivatives) Φ κ: converts twisted adjoint Φ to adjoint Consistent (almost by inspection) : Bianchi for F implies second d eq! Budapest, Sept

77 HS Interactions Gauge field A in (x,z) space: Internal equations: power series in Φ by successive iterations Budapest, Sept

78 Internal equations: HS Interactions with Lowest order in Φ Riemann = Weyl + Budapest, Sept

79 HS Interactions k k is a crucial ingredient : Linearized Φ equation : Unfolding : Uniform description of HS interactions Budapest, Sept

80 HS Interactions Cartan Integrable System : (Sullivan, 1977; D Auria, Fre, 1982) e.g. Chern-Simons theory manifestly consistent eqs gauge covariance manifest diff covariance non-lagrangian NEW INGREDIENT : 0-form Φ Budapest, Sept

81 HS Interactions Some missing ingredients : Y ia, Z ia to build HS algebra extending SO(2,D) must select Sp(2,R) singlets : Sp(2,R) generators (bilinears in Y, Z) K i j : Sp(2,R) generators (bilinears in Y, Z) REMOVE TRACES to obtain dynamical equations Weak projection : remove traces symmetrically from A and Φ (Vasiliev, 2003) Strong : leave traces in A (AS,Sezgin,Sundell, 2004) Budapest, Sept

82 HS Interactions Free flat limit (w. Strong projection) : s = 2 : s = 3 : a first equation,, analogous of the vielbein postulate giving ω(e) a second equation,, defining a second-order order kinetic operator a third equation giving the constraint Budapest, Sept

83 HS Interactions (AS, Sezgin, Sundell, 2004) (Dubois-Violette, Henneaux, 1999) Budapest, Sept

84 Conclusions Why Higher Spins? Field Theory String Theory (an instance with spontaneous breaking ) In these lectures: Free (unconstrained) Higher-Spin fields Relation with Vasiliev construction Some Some reviews: N. N. Bouatta, G. G. Compere, A.S., A.S., hep-th/ D. D. Francia and and A.S., A.S., hep-th/ A.S., A.S., Sezgin, Sundell, hep-th/ X. X. Bekaert, S. S. Cnockaert, C. C. Iazeolla, M.A. M.A. Vasiliev, hep-th/ A.S., A.S., P. P. Sundell, D. D. Sorokin, M.A. M.A. Vasiliev, Phys. Phys. Reports, (?) (?) Budapest, Sept