String theory triplets and higher-spin curvatures

Transcription

1 String theory triplets and higher-spin curvatures Dario Francia Institute of Physics - Academy of Sciences of the Czech Republic SFT2010 YITP Kyoto October 19th, 2010 based on: D. F. J.Phys. Conf. Ser. 222 (2010) [hep-th] Phys. Lett. B 690 (2010) [hep-th]

2 Higher-spins & Field Theory Symmetry group of space-time fundamental particles (fields) labeled by two quantum numbers { mass: spin: m 0 s =0, 1/2, 1, 3/2, 2, 5/2, 3,... (more labels in D>5) no indications about the existence of ``preferred'' subset of values Majorana 32, Dirac 36, Fierz-Pauli 39, Wigner 39,...

3 Higher-spins & Field Theory Known interactions fill the first levels of Wigner s scheme: spin 0: Higgs boson spin 1: electroweak & strong interactions spin 2: gravitational force looks like the beginning of a sequence...

4 Higher-spins & Field Theory But: No phenomenological inputs for (elementary) higher-spins (high-spin ``particles do exist!) No-go arguments against their interactions Weinberg 64, Coleman-Mandula 67, Velo-Zwanziger 69, Aragone-Deser 79,...

5 Higher-spins & Field Theory But: No phenomenological inputs for (elementary) higher-spins (high-spin ``particles do exist!) No-go arguments against their interactions Weinberg 64, Coleman-Mandula 67, Velo-Zwanziger 69, Aragone-Deser 79,... Why this ``selection rule?

6 Higher-spins & String Theory String Theory predicts higher spins!

7 Higher-spins & String Theory String Theory predicts higher spins! spectrum of 1st quantised strings accomodates massless, spin-1 and spin-2 particles ``ST predicts Gravity together with infinitely many massive states of increasing spin: m 2 (s) 1 α s String Field Theory: successful instance of HSFT

8 Higher-spins & String Theory String Theory predicts higher spins! spectrum of 1st quantised strings accomodates massless, spin-1 and spin-2 particles ``ST predicts Gravity together with infinitely many massive states of increasing spin: m 2 (s) 1 α s String Field Theory: successful instance of HSFT long-standing conjecture:

9 Higher-spins & String Theory String Theory predicts higher spins! spectrum of 1st quantised strings accomodates massless, spin-1 and spin-2 particles ``ST predicts Gravity together with infinitely many massive states of increasing spin: m 2 (s) 1 α s String Field Theory: successful instance of HSFT long-standing conjecture:... D. Gross, E.S. Fradkin... string tension results from spontaneous breaking of higher-spin gauge symmetry?

10 Free theory: Lagrangians Let us recall the construction of massless, spin-2 Lagrangian

11 Free theory: Lagrangians Let us recall the construction of massless, spin-2 Lagrangian Graviton potential: h µν : δh µν = µ Λ ν + ν Λ µ

12 Free theory: Lagrangians Let us recall the construction of massless, spin-2 Lagrangian Graviton potential: h µν : δh µν = µ Λ ν + ν Λ µ Ricci tensor: R µν = h µν ( µ α h αν + ν α h αµ )+ µ ν h α α s.t. δ R µν =0

13 Free theory: Lagrangians Let us recall the construction of massless, spin-2 Lagrangian Graviton potential: h µν : δh µν = µ Λ ν + ν Λ µ Ricci tensor: R µν = h µν ( µ α h αν + ν α h αµ )+ µ ν h α α s.t. δ R µν =0 Einstein tensor & Lagrangian L = 1 2 h µν E µν where E µν = R µν 1 2 η µν R α α s.t. α E αµ =0 Fierz-Pauli 39 (linearised Einstein-Hilbert)

14 Free theory: Lagrangians Let us recall the construction of massless, spin-2 Lagrangian Graviton potential: fluctuation of the metric h µν : ``metric-like (Einstein-Hilbert) h µν : δh µν = µ Λ ν + ν Λ µ vielbein e a µ : ``frame-like (Cartan-Weyl) Ricci tensor: R µν = h µν ( µ α h αν + ν α h αµ )+ µ ν h α α s.t. δ R µν =0 Einstein tensor & Lagrangian L = 1 2 h µν E µν where E µν = R µν 1 2 η µν R α α s.t. α E αµ =0 Fierz-Pauli 39 (linearised Einstein-Hilbert)

15 Free theory: Lagrangians Let us recall the construction of massless, spin-2 Lagrangian Graviton potential: fluctuation of the metric h µν : ``metric-like (Einstein-Hilbert) h µν : δh µν = µ Λ ν + ν Λ µ Ricci tensor: R µν vielbein e a µ : ``frame-like (Cartan-Weyl) both options generalise to hsp (frame-like = h µν ( µ α h αν + ν α h αµ )+ µ ν h α α s.t. δ R µν =0 Vasiliev s theory) Einstein tensor & Lagrangian L = 1 2 h µν E µν where E µν = R µν 1 2 η µν R α α s.t. α E αµ =0 Fierz-Pauli 39 (linearised Einstein-Hilbert)

16 Free theory: Lagrangians spin s: gauge potential symmetric tensor of rank s ϕ µ1 µ s δϕ µ1 µ s = µ1 Λ µ2 µ s + s. t. kinetic (``Ricci ) tensor: gauge invariant completion of ϕ F µ = A µ µ α A α Maxwell F µν = h µν ( µ α h αν + ν α h αµ )+ µ ν h α α Einstein F µ ν ρ = ϕ µ ν ρ (µ α ϕ νρ) α + (µ ν ϕ α ρ) α Fronsdal 78

17 Free theory: Lagrangians spin s: gauge potential symmetric tensor of rank s ϕ µ1 µ s δϕ µ1 µ s = µ1 Λ µ2 µ s + s. t. kinetic (``Ricci ) tensor: gauge invariant completion of ϕ F µ = A µ µ α A α Maxwell F µν = h µν ( µ α h αν + ν α h αµ )+ µ ν h α α Einstein F µ ν ρ = ϕ µ ν ρ (µ α ϕ νρ) α + (µ ν ϕ α ρ) α Fronsdal 78 F µ1... µ s Two main differences w.r.t. low spins: (candidate ``Ricci tensor) gauge invariant Λ α αµ 4... µ s 0 E F 1 2 η F α α (candidate ``Einstein tensor) requires ϕ αβ α β µ 5... µ s 0

18 Free higher spins can be given a Lagrangian description generalising Maxwell and Einstein theories written in terms of the gauge potential: A µ µ α A α =0 ϕ µ ν ρ (µ α ϕ νρ) α + (µ ν ϕ α ρ) α =0

19 Free higher spins can be given a Lagrangian description generalising Maxwell and Einstein theories written in terms of the gauge potential: A µ µ α A α =0 ϕ µ ν ρ (µ α ϕ νρ) α + (µ ν ϕ α ρ) α =0 (also for massless and massive bosons and fermions of any spin and symmetry) * Labastida; Siegel, Zwiebach; Pashnev, Tsulaia, Buchbinder, Metsaev; Alkalaev; Skvortsov, Zinoviev;... Campoleoni, D.F., Mourad and Sagnotti 08, 09 *

20 Free higher spins can be given a Lagrangian description generalising Maxwell and Einstein theories written in terms of the gauge potential: A µ µ α A α =0 ϕ µ ν ρ (µ α ϕ νρ) α + (µ ν ϕ α ρ) α =0 (also for massless and massive bosons and fermions of any spin and symmetry) * Labastida; Siegel, Zwiebach; Pashnev, Tsulaia, Buchbinder, Metsaev; Alkalaev; Skvortsov, Zinoviev;... Campoleoni, D.F., Mourad and Sagnotti 08, 09 * what about the same theories expressed in terms of curvatures? ν F νµ =0 η αβ R µ α, ν β =0?

21 Model Simplest gauge theory: Maxwell gauge potential/connexion: A µ : δa µ = µ Λ field strength/curvature: F µν : δf µν =0 L = 1 4 F µν F µν µ F µν =0 Generalisation to hsp?

22 Curvatures de Wit - Freedman 80 Damour-Deser 87 Spin 3: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα R (3) µµµ, ρρρ = 3 µ ϕ ρρρ µ ρ ϕ µ, ρρ µ 2 ρ ϕ µµ, ρ 3 ρ ϕ µµµ s.t. δr (3) 0 Spin s: R (s) µ s,ν s = s ( 1) k ( s k ) s k µ k νϕ µk,ν s k k=0

23 Curvatures de Wit - Freedman 80 Damour-Deser 87 Spin 3: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα R (3) µµµ, ρρρ = 3 µ ϕ ρρρ µ ρ ϕ µ, ρρ µ 2 ρ ϕ µµ, ρ 3 ρ ϕ µµµ s.t. δr (3) 0 Higher derivatives! Spin s: R (s) µ s,ν s = s ( 1) k ( s k ) s k µ k νϕ µk,ν s k k=0

24 Curvatures de Wit - Freedman 80 Damour-Deser 87 Spin 3: δϕ αβγ = α Λ βγ + β Λ αγ + γ Λ βα R (3) µµµ, ρρρ = 3 µ ϕ ρρρ µ ρ ϕ µ, ρρ µ 2 ρ ϕ µµ, ρ 3 ρ ϕ µµµ s.t. δr (3) 0 Higher derivatives! Spin s: R (s) µ s,ν s = s ( 1) k ( s k ) s k µ k νϕ µk,ν s k k=0 no constraints

25 Plan I. Geometric Lagrangians: direct construction & problems II. Solution for irreducible spin s III. Solution for reducible spin s & tensionless strings

26 I. Geometric Lagrangians: direct construction & problems

27 Non-local theories D.F. - A.Sagnotti 02, 03, D.F. - J. Mourad A.Sagnotti 07, D.F. 07, 08 Giving a chance to hsp curvatures, inspired by lower spins: Spin 1: µ F µν =0 Spin 2: η αβ R αβ,µν =0 Spin 3: 1 η αβ γ R αβγ,µνρ =0 all non-localities are pure gauge; after a partial gaugefixing one recovers local (Fronsdal) form

28 Non-local theories II Indeed, infinitely many candidate Ricci tensors: 1 Notation: 1 η αβ γ R αβγ, µνρ R Spin 3: A ϕ (a) = 1 R + a 2 2 R s.t., a A ϕ (a) = 0 F =3 3 α ϕ

29 Non-local theories II Indeed, infinitely many candidate Ricci tensors: 1 Notation: 1 η αβ γ R αβγ, µνρ R Spin 3: A ϕ (a) = 1 R + a 2 2 R s.t., a A ϕ (a) = 0 F =3 3 α ϕ Fronsdal tensor: F =0 correct local eqn

30 Non-local theories II Indeed, infinitely many candidate Ricci tensors: 1 Notation: 1 η αβ γ R αβγ, µνρ R Spin 3: A ϕ (a) = 1 R + a 2 2 R s.t., a A ϕ (a) = 0 F =3 3 α ϕ Fronsdal tensor: F =0 correct local eqn non-local Stueckelberg field

31 Non-local theories II Indeed, infinitely many candidate Ricci tensors: 1 Notation: 1 η αβ γ R αβγ, µνρ R Spin 3: A ϕ (a) = 1 R + a 2 2 R s.t., a A ϕ (a) = 0 F =3 3 α ϕ Fronsdal tensor: F =0 correct local eqn non-local Stueckelberg field gauging away the compensator α ϕ we obtain Fronsdal = 0

32 Non-local theories III To summarise: ``Geometric Lagrangians (i.e. Lagrangians built out of curvatures) do exist; they are non-local; non-localities can be gauged away from the e.o.m. gauge-invariance but uniqueness

33 Propagators & uniqueness Consistency check: the propagator k 2 0 J (x) J (y) only physical polarisations (i.e. transverse and traceless tensors) should mediate the current exchange

34 Propagators & uniqueness Consistency check: the propagator k 2 0 J (x) J (y) only physical polarisations (i.e. transverse and traceless tensors) should mediate the current exchange Out of infinitely many non-local Lagrangians whose free equations reduce to F =3 3 α ϕ almost all of them give the wrong current-exchange but one

35 Propagators & uniqueness Consistency check: the propagator k 2 0 J (x) J (y) only physical polarisations (i.e. transverse and traceless tensors) should mediate the current exchange Out of infinitely many non-local Lagrangians whose free equations reduce to F =3 3 α ϕ almost all of them give the wrong current-exchange but one { Uniqueness restored, but

36 Propagators & uniqueness Consistency check: the propagator k 2 0 J (x) J (y) only physical polarisations (i.e. transverse and traceless tensors) should mediate the current exchange Out of infinitely many non-local Lagrangians whose free equations reduce to F =3 3 α ϕ almost all of them give the wrong current-exchange but one Uniqueness restored, but { only a posteriori

37 Propagators & uniqueness Consistency check: the propagator k 2 0 J (x) J (y) only physical polarisations (i.e. transverse and traceless tensors) should mediate the current exchange Out of infinitely many non-local Lagrangians whose free equations reduce to F =3 3 α ϕ almost all of them give the wrong current-exchange but one Uniqueness restored, but { only a posteriori the solution is not particularly simple...

38 General solution for irreducible spin s the full Lagrangian equation leading to the correct propagator for spin s looks E ϕ = A ϕ 1 2 η A ϕ + η 2 B ϕ =0 where

39 General solution for irreducible spin s the full Lagrangian equation leading to the correct propagator for spin s looks E ϕ = A ϕ 1 2 η A ϕ + η 2 B ϕ =0 where A ϕ = F 3 3 γ ϕ = n+1 k=0 ( 1) k+1 (2 k 1) { n +2 n 1 k 1 j= 1 n + j n j +1 } 2k k F [k] n+1,

40 General solution for irreducible spin s the full Lagrangian equation leading to the correct propagator for spin s looks E ϕ = A ϕ 1 2 η A ϕ + η 2 B ϕ =0 where A ϕ = F 3 3 γ ϕ = n+1 k=0 ( 1) k+1 (2 k 1) { n +2 n 1 k 1 j= 1 n + j n j +1 } 2k k F [k] n+1, B ϕ = 1 2 n+1 k=2 { 1 a k 2 k 3 n + k (n k)(n k + 1) 2(k 2) k 2 F (n+1) [k] + 2 k k (n+1) [k+2] F 2n +4k +1 2(2k 1) (n k + 1) } 2(k 1) k 1 (n+1) [k+1] F

41 General solution for irreducible spin s the full Lagrangian equation leading to the correct propagator for spin s looks E ϕ = A ϕ 1 2 η A ϕ + η 2 B ϕ =0 where A ϕ = F 3 3 γ ϕ = n+1 k=0 ( 1) k+1 (2 k 1) { n +2 n 1 k 1 j= 1 n + j n j +1 } 2k k F [k] n+1, B ϕ = 1 2 n+1 k=2 { 1 a k 2 k 3 n + k (n k)(n k + 1) 2(k 2) k 2 F (n+1) [k] + 2 k k (n+1) [k+2] F 2n +4k +1 2(2k 1) (n k + 1) } 2(k 1) k 1 (n+1) [k+1] F F n+1 = { 1 R [n+1] n s = 2 (n + 1), 1 R [n] n s =2n +1,

42 II. Solution for irreducible spin s

43 I. Irreducible spin s - minimal local Lagrangians Trace constraint on gauge parameter needed for gauge invariance of the Fronsdal tensor: δ F =3 3 Λ simplest possibility to forgo it: introduce an auxiliary, ``compensator field s.t. δα =Λ and define the unconstrained extension of F α A = F 3 3 α

44 I. Irreducible spin s - minimal local Lagrangians Trace constraint on gauge parameter needed for gauge invariance of the Fronsdal tensor: δ F =3 3 Λ simplest possibility to forgo it: introduce an auxiliary, ``compensator field s.t. δα =Λ and define the unconstrained extension of F α A = F 3 3 α β With an additional Lagrange multiplier fully unconstrained Lagrangians, for any spin (and symmetry)

45 I. Irreducible spin s - minimal local Lagrangians Trace constraint on gauge parameter needed for gauge invariance of the Fronsdal tensor: δ F =3 3 Λ simplest possibility to forgo it: introduce an auxiliary, ``compensator field s.t. δα =Λ and define the unconstrained extension of F α A = F 3 3 α β With an additional Lagrange multiplier fully unconstrained Lagrangians, for any spin (and symmetry) D.F. - A.Sagnotti 02, 03, D.F. - J. Mourad A.Sagnotti 07, D.F. 07, 08 D.F., A. Campoleoni, J. Mourad, A. Sagnotti 08, 09

46 Any relations among local Lagrangians and higher-spin curvatures? Fronsdal: intrinsically constrained; not possible to rebuild curvatures

47 Any relations among local Lagrangians and higher-spin curvatures? Fronsdal: intrinsically constrained; not possible to rebuild curvatures Minimal Lagrangians { ϕ same properties as in R µ1 µ S,ν 1 ν s Λ there are additional fields but they do not mix with the physical d.o.f.

48 Any relations among local Lagrangians and higher-spin curvatures? Fronsdal: intrinsically constrained; not possible to rebuild curvatures Minimal Lagrangians { ϕ same properties as in R µ1 µ S,ν 1 ν s Λ there are additional fields but they do not mix with the physical d.o.f. let us integrate over the auxiliary fields

49 Irreducible fields: the example of spin-3 Recall: basic kinetic tensor is A = F 3 3 α ; unconstrained Lagrangian: L = 1 2 ϕ {F 1 2 η F } α 2 α 3 2 α F

50 Irreducible fields: the example of spin-3 Recall: basic kinetic tensor is A = F 3 3 α ; unconstrained Lagrangian: L = 1 2 ϕ {F 1 2 η F } α 2 α 3 2 α F Integrating over α L eff (ϕ) = 1 2 ϕ {F 1 2 η F } 1 4 F 1 2 F

51 Irreducible fields: the example of spin-3 Recall: basic kinetic tensor is A = F 3 3 α ; unconstrained Lagrangian: L = 1 2 ϕ {F 1 2 η F } α 2 α 3 2 α F Integrating over α L eff (ϕ) = 1 2 ϕ {F 1 2 η F } 1 4 F 1 2 F coinciding, up to total derivatives, with L = 1 2 ϕ {A ϕ 1 2 η A ϕ} where A ϕ = 1 R R is the ``correct candidate Ricci tensor

52 Irreducible fields: the example of spin-3 Recall: basic kinetic tensor is A = F 3 3 α ; unconstrained Lagrangian: L = 1 2 ϕ {F 1 2 η F } α 2 α 3 2 α F Integrating over α L eff (ϕ) = 1 2 ϕ {F 1 2 η F } 1 4 F 1 2 F coinciding, up to total derivatives, with L = 1 2 ϕ {A ϕ 1 2 η A ϕ} where A ϕ = 1 R R is the ``correct candidate Ricci tensor

53 III. Solution for reducible spin s & tensionless strings

54 String Theory & Triplets Open bosonic string oscillators [α µ k,αν l ] = kδ k+l,0 η µν Virasoro generators and their rescaling limit: L k = l= α µ k l α µl, { L k 0 = 1 α L k L 0 = 1 α L 0 l k = p µ α µ k α l 0 = p µ p µ [α µ 0 = 2 α p µ ] (``tensionless limit) [l k,l l ]=kδ k+l, 0 l 0 Algebra with no central charge identically nilpotent BRST charge Q same charge from tensionless limit of open string BRST charge, after rescaling of ghosts

55 III. String-inspired Triplets ϕ µ1 µ s for ``diagonal blocks associated to symmetric, rank-s tensors, (states generated by powers of α µ 1 ) the corresponding Lagrangian is L triplet = 1 2 ϕ ϕ 1 2 sc2 ( ) s 2 D D + s ϕc +2 ( ) s 2 D C

56 III. String-inspired Triplets ϕ µ1 µ s for ``diagonal blocks associated to symmetric, rank-s tensors, (states generated by powers of α µ 1 ) the corresponding Lagrangian is L triplet = 1 2 ϕ ϕ 1 2 sc2 ( ) s 2 D D + s ϕc +2 ( ) s 2 D C equations of motion ϕ = C C = ϕ D D = C ϕ spin s C spin s 1 D spin s 2 gauge transformations δϕ = Λ δc = Λ δd = Λ

57 III. String-inspired Triplets ϕ µ1 µ s for ``diagonal blocks associated to symmetric, rank-s tensors, (states generated by powers of α µ 1 ) the corresponding Lagrangian is L triplet = 1 2 ϕ ϕ 1 2 sc2 ( ) s 2 D D + s ϕc +2 ( ) s 2 D C equations of motion ϕ = C C = ϕ D D = C ϕ spin s C spin s 1 D spin s 2 gauge transformations δϕ = Λ δc = Λ δd = Λ The triplet propagates spin s, s-2, s-4,..., 1/0

58 A. Bengtsson; Ouvry-Stern 86 M. Henneaux-C. Teitelboim 88 D.F., A. Sagnotti 02 A. Sagnotti, M. Tsulaia 03 Buchbinder et al 07 A. Fotopoulos, M. Tsulaia 08 D. Sorokin, M. Vasiliev 08 III. String-inspired Triplets ϕ µ1 µ s for ``diagonal blocks associated to symmetric, rank-s tensors, (states generated by powers of α µ 1 ) the corresponding Lagrangian is L triplet = 1 2 ϕ ϕ 1 2 sc2 ( ) s 2 D D + s ϕc +2 ( ) s 2 D C equations of motion ϕ = C C = ϕ D D = C ϕ spin s C spin s 1 D spin s 2 gauge transformations δϕ = Λ δc = Λ δd = Λ The triplet propagates spin s, s-2, s-4,..., 1/0

59 Triplets: integrating auxiliary fields Integrating over the fields C and D we find L eff (ϕ) = 1 2 ϕ ( ) ϕ ( ) s 2 ϕ ( ) 1 ϕ where the inverse of the operator O = on the space of tensors of rank k is O 1 (k) = 1 {1+ k m=1 ( 1) m m! 2 m m l=1 (1 + l 2 ) m m m}

60 Triplets: geometric Lagragians Spin 3: L eff (ϕ) = 1 2 {ϕ ϕ + 3 ϕ ϕ +3 ϕ 1 ϕ + ϕ 1 2 ϕ} Spin s:

61 Triplets: geometric Lagragians Spin 3: L eff (ϕ) = 1 2 {ϕ ϕ + 3 ϕ ϕ +3 ϕ 1 ϕ + ϕ 1 2 ϕ} Spin s: L eff (ϕ) = 1 2 s m=0 ( ) s m m ϕ 1 m 1 m ϕ

62 Triplets: geometric Lagragians Spin 3: L eff (ϕ) = 1 2 {ϕ ϕ + 3 ϕ ϕ +3 ϕ 1 ϕ + ϕ 1 2 ϕ} = 1 2 ϕ { ϕ ϕ + 2 ϕ 3 ϕ}, Spin s: L eff (ϕ) = 1 2 s m=0 ( ) s m m ϕ 1 m 1 m ϕ

63 Triplets: geometric Lagragians Spin 3: L eff (ϕ) = 1 2 {ϕ ϕ + 3 ϕ ϕ +3 ϕ 1 ϕ + ϕ 1 2 ϕ} = 1 2 ϕ { ϕ ϕ + 2 ϕ 3 ϕ}, Spin s: L eff (ϕ) = 1 2 s m=0 ( ) s m = 1 2 ϕ { ϕ + s m ϕ 1 m 1 m ϕ m=1 ( 1) m m m 1 m ϕ}

64 Triplets: geometric Lagragians Spin 3: L eff (ϕ) = 1 2 {ϕ ϕ + 3 ϕ ϕ +3 ϕ 1 ϕ + ϕ 1 2 ϕ} Spin s: = 1 2 L eff (ϕ) = 1 2 ϕ { ϕ ϕ + 2 = 1 2 ϕ R (3) s m=0 ( ) s m = 1 2 ϕ { ϕ + s ϕ 3 m ϕ 1 m 1 m ϕ m=1 ( 1) m m m 1 m ϕ} ϕ},

65 Triplets: geometric Lagragians Spin 3: L eff (ϕ) = 1 2 {ϕ ϕ + 3 ϕ ϕ +3 ϕ 1 ϕ + ϕ 1 2 ϕ} Spin s: = 1 2 L eff (ϕ) = 1 2 ϕ { ϕ ϕ + 2 = 1 2 ϕ R (3) s m=0 ( ) s m = 1 2 ϕ { ϕ + s = 1 2 ϕ 1 s 1 s R (s) ϕ 3 m ϕ 1 m 1 m ϕ m=1 ( 1) m m m 1 m ϕ} ϕ},

66 Triplets: geometric Lagragians Spin s: L eff (ϕ) = ( 1) s 2(s + 1) R (s) µ 1 µ s,ν 1 ν s 1 s 1 R (s) µ 1 µ s,ν 1 ν s Lagrangians squares of curvatures

67 Triplets: fermions Formally, Dirac equation from Klein-Gordon: ϕ =0 ψ =0 ψ =0 not true for higher spins: S 1 2 S = i F (ψ)

68 Triplets: fermions Formally, Dirac equation from Klein-Gordon: ϕ =0 ψ =0 ψ =0 not true for higher spins: S 1 2 S = i F (ψ) For the e.o.m. obtained from the fermionic triplets we find instead 1 s 1 s R (s) (ϕ) = 0 s s R (s) (ψ) = 0

69 Triplets: fermions Formally, Dirac equation from Klein-Gordon: ϕ =0 ψ =0 ψ =0 not true for higher spins: S 1 2 S = i F (ψ) For the e.o.m. obtained from the fermionic triplets we find instead 1 s 1 s R (s) (ϕ) = 0 s s R (s) (ψ) = 0 indeed, also for spin-(s + 1/2) fermions we have L eff (ψ, ψ) = ( 1) s s +1 i R (s) µ s,ν s s R (s) µ s,ν s

70 Outlook The proposal is to define ``geometric hsp Lagrangians as effective theories after elimination of non-physical fields; rationale for the appearance of non-local operators; Hsp curvatures can describe propagation of several Fronsdal fields, combined so as to reconstruct an unconstrained gauge potential;

71 Outlook The proposal is to define ``geometric hsp Lagrangians as effective theories after elimination of non-physical fields; rationale for the appearance of non-local operators; Hsp curvatures can describe propagation of several Fronsdal fields, combined so as to reconstruct an unconstrained gauge potential; For irreducible fields (integrating from minimal unconstrained Lagrangians): a priori derivation of generalised Fierz-Pauli (i.e. linearised Einstein-Hilbert) Lagrangians previously proposed; for reducible fields (integrating from tensionless string triplets): we obtain generalisations of Maxwell s theory;

72 Outlook The proposal is to define ``geometric hsp Lagrangians as effective theories after elimination of non-physical fields; rationale for the appearance of non-local operators; Hsp curvatures can describe propagation of several Fronsdal fields, combined so as to reconstruct an unconstrained gauge potential; For irreducible fields (integrating from minimal unconstrained Lagrangians): a priori derivation of generalised Fierz-Pauli (i.e. linearised Einstein-Hilbert) Lagrangians previously proposed; for reducible fields (integrating from tensionless string triplets): we obtain generalisations of Maxwell s theory; reducible case much simpler than the irreducible one; Lagrangians squares of curvatures.

73 Outlook The proposal is to define ``geometric hsp Lagrangians as effective theories after elimination of non-physical fields; rationale for the appearance of non-local operators; Hsp curvatures can describe propagation of several Fronsdal fields, combined so as to reconstruct an unconstrained gauge potential; For irreducible fields (integrating from minimal unconstrained Lagrangians): a priori derivation of generalised Fierz-Pauli (i.e. linearised Einstein-Hilbert) Lagrangians previously proposed; for reducible fields (integrating from tensionless string triplets): we obtain generalisations of Maxwell s theory; reducible case much simpler than the irreducible one; Lagrangians squares of curvatures. is there a lesson in store for hsp interactions?

74 ..

75 Massive theories..

76 Massive theories Irreducible case: deform geometric Lagrangian with generalised Fierz-Pauli mass terms L m=0 = 1 2 ϕ E ϕ L m = 1 2 ϕ {E ϕ m 2 (ϕ ηϕ η 2 ϕ )} it works because {E ϕ m 2 (ϕ ηϕ η 2 ϕ )} =0 ϕ ϕ =0

77 Massive theories Reducible case: deform geometric Lagrangian with ``generalised Proca mass term L (m) = ( 1)s 2(s + 1) R (s) µ 1 µ s,ν 1 ν s 1 s 1 R (s) µ 1 µ s,ν 1 ν s 1 2 m 2 ϕ 2

78 Massive theories Reducible case: deform geometric Lagrangian with ``generalised Proca mass term L (m) = ( 1)s 2(s + 1) R (s) µ 1 µ s,ν 1 ν s 1 s 1 R (s) µ 1 µ s,ν 1 ν s 1 2 m 2 ϕ 2 Indeed, from the eom: 1 s 1 s R m 2 ϕ =0 ϕ =0 one obtains immediately and thus ϕ m 2 ϕ =0

79 Massive theories Reducible case: deform geometric Lagrangian with ``generalised Proca mass term L (m) = ( 1)s 2(s + 1) R (s) µ 1 µ s,ν 1 ν s 1 s 1 R (s) µ 1 µ s,ν 1 ν s 1 2 m 2 ϕ 2 Indeed, from the eom: 1 s 1 s R m 2 ϕ =0 ϕ =0 one obtains immediately and thus ϕ m 2 ϕ =0 But check vs a local derivation!