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Augusto Sagnotti
Scuola Normale Superiore
U. Padova, 12-1-2011
• Dario FRANCIA (Czech Acad. Sci, Prague)• Mirian TSULAIA (U. Liverpool)• Jihad MOURAD (U. Paris VII)• Andrea CAMPOLEONI (AEI, Potsdam)• Massimo TARONNA (Scuola Normale)
• Originally: an S-matrix with(planar) duality
• Rests crucially on the presence of ∞ massive modes
• Massive modes: mostly Higher Spins (HS)
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Aμ −→ φμ1...μs4D:
D>4: Aμ −→ φμ(1)1 ...μ
(1)s1;...;μ
(N)1 ...μ
(N)sN
Symmetric
Mixed (multi-symmetric)
• [ Vacuum stability: OK with SUSY ]• Key addition: low-energy effective SUGRA
(2D data translated via RG into space-time notions)
• Deep conceptual problems are inherited from (SU)GRA.
• String Field Theory: field theory combinatorics for amplitudes. Massive modes included. Background (inde)pendendence?
Key notions hidden in massive HS excitations?
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S2 =Z √
γγab∂axμ∂bX
νGμν(X)+Z²ab∂ax
μ∂bXνBμν(X)+
Zα0√γR(2)Φ(X)+..
SD =1
2k2D
ZdDX
√−Ge−Φ
µR− 1
12H2 + 4(∂Φ)2
¶+..
• Equations: (Dirac-Fierz-Pauli) (~1930’s)
• Lagrangians: extra fields ( ≥ 1970’s)
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• Key properties of HS fields– Symmetric HS fields– Mixed symmetry
• Problems with HS interactions
• HS interactions from open strings– Limiting 3-pt functions and gauge symmetry– Conserved HS currents
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• For gravity two types of formulations:• Metric :
• Frame :
• For HS, one has similarly:• Metric-like :
• Frame-like :
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( eaμ , ωabμ ) −→ ωABμ
gμν
(φμ1...μs )
Here I will concentrate on the first type of formulation, that is more directly related to String Theory, although the second has led to a remarkable setup for fully non-linear HS interactions, the Vasiliev system.
Fronsdal (1978): natural extension of s=1,2 cases (BUT : with CONSTRAINTS)
Can simplify notation (and algebra) hiding space-time indices:
(“primes” = traces)
71st Fronsdal constraint
8
Bianchi identity :
Natural to try and forego these “trace” constraints:
• BRST (non minimal) (Buchbinder, Pashnev, Tsulaia, …, 1998-)
• Minimal compensator form (Francia, AS, Mourad, 2002 -)
Unconstrained LagrangianS=3(Schwinger)
2nd Fronsdal constraint
[2-derivative : ](Francia, 2007)
9
What are we gaining ?
s = 2:
s > 2: Hierarchy of connections and curvatures
“Irreducible” NON LOCAL form of the equations :
(de Wit and Freedman, 1980)
After some iterations: NON-LOCAL gauge invariant equation for ONLY
10
BRST equations for “contracted” Virasoro:
’ ∞
First open bosonic Regge trajectory TRIPLETSPropagate: s,s-2,s-4, …
On-shell truncation :
(A. Bengtsson, 1986; Henneaux, Teitelboim, 1987)(Pashnev, Tsulaia, 1998; Francia, AS, 2002; Bonelli, 2003; AS, Tsulaia, 2003)
(Kato and Ogawa, 1982; Witten; Neveu, West et al, 1985,,…)
Off-shell truncation : ( Buchbinder, Krykhtin, Reshetnyak 2007 )
(Francia, AS, 2002)
Frame formulation : (Sorokin, Vasiliev, 2008)
Independent fields for D > 5 Index “families” Non-Abelian gl(N) algebra mixing them
Labastida constraints (1987):
NOT all (double) traces vanish Higher traces in Lagrangians ! Constrained bosonic Lagrangians and fermionic field equations Key tool: self-adjointness of kinetic operator
Recently:
Unconstrained bosonic LagrangiansWeyl-like symmetries (sporadic cases determined by gl(N) algebra) (Un)constrained fermionic Lagrangians Key tool: Bianchi identities
(Campoleoni, Francia, Mourad, AS, 2008, and 2009)
11
Problems :
Usual coupling with gravity “naked” Weyl tensors (Aragone, Deser,1979)
Weinberg’s 1964 argument , Coleman – Mandula Velo-Zwanziger inconsistenciesWeinberg – Witten (see Porrati, 2008)
………..
(Vasiliev, 1990, 2003)(Sezgin, Sundell, 2001)
But:
(Light-cone or covariant) 3-vertices
[higher derivatives]
Scattering via current exchanges
Contact terms can resolve Velo-Zwanziger
Deformed low-derivative with ≠ Infinitely many fields 12
Berends, Burgers, van Dam, 1982)(Bengtsson2, Brink, 1983)
(Boulanger et al, 2001 -)(Metsaev, 2005,2007)
(Buchbinder,Fotopoulos, Irges, Petkou, Tsulaia, 2006)(Boulanger, Leclerc, Sundell, 2008)
(Zinoviev, 2008)(Manvelyan, Mkrtchyan, Ruhl, 2009)
(Bekaert, Mourad, Joung, 2009)
(Porrati, Rahman, 2009)(Prrati, Rahman, AS, 2010)
Vasiliev eqs:
13
• Residues of current exchanges reflect the degrees of freedom
• s=1 :
• All s :
(Francia, Mourad, AS, 2007, 2008)
Unique non-local Lagrangian:(e.g. for s=3)
(van Dam, Veltman; Zakharov, 1970)
VDVZ discontinuity : comparing of D and (D+1) massless exchanges for s≥2
All s: can describe a massive field à la Scherk-Schwarz from (D+1) dimensions : [ e.g. for s=2 : hMN (hmn cos(my) , Am sin(my), cos(my) ) ]
(A)dS extension [radial reduction] (Fronsdal, 1979; Biswas, Siegel, 2002; Higuchi, 1987 ; Porrati, 2001)
•s : Discontinuity smooth interpolation in (mL) 2 (s:
•[Liouville’s Theorem !]VDVZ discontinuity!Poles at (even) “partly massless” states
(Deser, Nepomechie, Waldron, 1983 - )14
15
• Closer look at old difficulties (for definiteness s-s-2 case)
• Aragone-Deser: NO “standard”gravity coupling
for massless HS around flat space;
• Fradkin-Vasiliev : higher-derivative terms around (A)dS;• Boulanger et al: ALMOST UNIQUE highest non-Abelian CUBIC
coupling (seed) for massless HS around flat space;
• DEFORMING highest vertex to (A)dS one can recover consistent
“standard”couplings WITH tails, singular limit as 0
NOETHER COUPLINGS:
(Boulanger, Leclerc, Sundell, 2008)
e.g. 3-3-2 :
φμ1...μs Jμ1...μs
16
Gauge fixed Polyakov path integral Koba-Nielsen amplitudes
Vertex operators asymptotic states
Sopenj1···jn =ZRn−3
dy4 · · · dyn |y12y13y23|
× h Vj1(y1)Vj2(y2)Vj3(y3) · · · Vjn(yn) iTr(Λa1 · · ·Λan) + (1↔ 2)
yij = yi − yj
Chan-Paton factors
(L0 − 1) |φi = 0 L1 |φi = 0 L2 |φi = 0
Virasoro Fierz-Pauli conditions:
KEY NOVELTY: here massive HS, but …
17
Z ∼ exp
⎡⎣−12
nXi6=j
α0 p i · p j ln |yij |−√2a0
ξi · p jyij
+1
2
ξi · ξjy2ij
⎤⎦
Starting point: 2D field theory generating function
Z[J ] = i(2π)dδ(d)(J0)C exp³− 1
2
Zd2σd2σ0 J(σ) · J(σ0)G(σ,σ0)
´
For symmetric open-string states
( 1st Regge trajectory)
Z(ξ(n)i ) ∼ exp
³Xξ(n)i Anmij (yl) ξ
(m)j + ξ
(n)i · Bni (yl; p l) + α0 p i · p j ln |yij |
´
φ i(p i, ξ i) =1
n!φiμ1···μn ξ
μ1i . . . ξ μni
“symbols”
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−p 21 =s1 − 1α0
− p 22 =s2 − 1α0
− p 23 =s3 − 1α0
Zphys ∼ exp
(rα0
2
µξ 1 · p 23
¿y23y12y13
À+ ξ 2 · p 31
¿y13y12y23
À+ ξ 3 · p 12
¿y12y13y23
À¶+ (ξ 1 · ξ 2 + ξ 1 · ξ 3 + ξ 2 · ξ 3)
)
• L0 constraint: mass
• L1 constraint: transversality
• L2 constraint: traceleness
Can impose the Virasoro constraints directly in the generating function
signs
“On-shell” couplings star- product with symbols of vertex operators
A± = φ 1
Ãp 1,
∂
∂ξ±r
α0
2p 31
!φ 2
Ãp 2, ξ +
∂
∂ξ±r
α0
2p 23
!φ 3
Ãp 3, ξ ±
rα0
2p 12
! ¯¯ξ=0
Fierz-Pauli
• 0-0-s:
• 1-1-s:
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(Berends, Burgers and Van Dam, 1986)
(Bekaert, Joung, Mourad. 2009)
A±0−0−s =
ñr
α 0
2
!sφ 1 φ 2 φ 3 · p s12
J ±(x, ξ) = Φ
Ãx ± i
rα 0
2ξ
!Φ
Ãx ∓ i
rα 0
2ξ
!(conserved!)
A±1−1−s =
ñr
α 0
2
!s−2s(s− 1)A1μA2 ν φμν... p s−212
+
ñr
α 0
2
!s hA1 · A2 φ · p s12 + sA1 · p 23A2 ν φ ν... p s−112
+ sA2 · p 31A1 ν φ ν... p s−112
i+
ñr
α 0
2
!s+2A1 · p 23A2 · p 31 φ · p s12 ,
• OLD LORE: String Theory “broken phase of something”
• AMPLITUDES: can spot extra “stuff” that drops out in the “massless” limit, where one ought to recover genuine Noether couplings based on conserved currents.
A gauge invariant pattern does show up!
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A± = exp
(rα0
2
h(∂ξ 1 · ∂ξ 2)(∂ξ 3 · p 12) + (∂ξ 2 · ∂ξ 3)(∂ξ 1 · p 23) + (∂ξ 3 · ∂ξ 1)(∂ξ 2 · p 31)
i)
× φ 1
Ãp 1; ξ 1 +
rα0
2p 23
!φ 2
Ãp 2; ξ 2 +
rα0
2p 31
!φ 3
Ãp 3; ξ 3 +
rα0
2p 12
! ¯¯ξ i=0
G operator: simplest non-trivial operator leading to (non-Abelian)gauge symmetry. Deformed Moyal-like star-product.
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The limiting couplings are induced by Noether currents:
J (x ; ξ) = exp
Ã−ir
α 0
2ξα£∂ζ1 · ∂ζ2 ∂ α
12 − 2 ∂ αζ1 ∂ζ2 · ∂ 1 + 2 ∂ α
ζ2 ∂ζ1 · ∂ 2¤!
× φ 1
Ãx − i
rα 0
2ξ , ζ1 − i
√2α 0 ∂ 2
!φ 2
Ãx + i
rα 0
2ξ , ζ2 + i
√2α 0 ∂ 1
! ¯¯ζi =0
Conserved up to massless Klein-Gordon, divergences and traces, but completion turns is completely fixed! [Can extend to Fronsdal and compensator cases.]
Gauge invariant 3-point amplitudes:
J · φ
A [0]± = exp
(±r
α 0
2
h(∂ξ 1 · ∂ξ 2)(∂ξ 3 · p 12) + (∂ξ 2 · ∂ξ 3)(∂ξ 1 · p 23) + (∂ξ 3 · ∂ξ 1)(∂ξ 2 · p 31)
i)
×φ 1
Ãp 1 , ξ 1 ±
rα 0
2p 23
!φ 2
Ãp 2 , ξ 2 ±
rα 0
2p 31
!φ 3
Ãp 3 , ξ 3 ±
rα 0
2p 12
! ¯¯ξ i =0
.
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A natural guess for the limiting behavior of FFB couplings in superstrings:
Determines corresponding Bose and Fermi conserved HS currents:
J[0]±FF (x ; ξ) = exp
Ã∓ i
rα 0
2ξα£∂ζ1 · ∂ζ2 ∂ α
12 − 2 ∂ αζ1 ∂ζ2 · ∂ 1 + 2 ∂ α
ζ2 ∂ζ1 · ∂ 2¤!
× Ψ 1
Ãx ∓ i
rα 0
2ξ , ζ1 ∓ i
√2α 0 ∂ 2
! h1+ /ξ
iΨ 2
Ãx ± i
rα 0
2ξ , ζ2 ± i
√2α 0 ∂ 1
! ¯¯ζi =0
.
A [0]±F = exp(±G) ψ 1
Ãp 1 , ξ 1 ±
rα 0
2p 23
![1+ /∂ ξ 3 ]ψ 2
Ãp 2 , ξ 2 ±
rα 0
2p 31
!
× φ 3
Ãp 3 , ξ 3 ±
rα 0
2p 12
! ¯¯ξ i =0
,
J[0]±BF (x ; ξ) = exp
Ã∓ i
rα 0
2ξα£∂ζ1 · ∂ζ2 ∂ α
12 − 2 ∂ αζ1 ∂ζ2 · ∂ 1 + 2 ∂ α
ζ2 ∂ζ1 · ∂ 2¤!
×h1+ /∂ζ 2
iΨ 1
Ãx ∓ i
rα 0
2ξ , ζ1 ∓ i
√2α 0 ∂ 2
!Φ 2
Ãx ± i
rα 0
2ξ , ζ2 ± i
√2α 0 ∂ 1
! ¯¯ζi =0
.
(Related work: Mavelyan, Mkrtchyan, Ruhl, 2010; Schlotterer, 2010)
• Free HS Fields:– [ Frame-like vs metric-like formulations ]– Constraints, compensators and curvatures– (String Theory (α΄∞) : triplets)
• Interacting HS Fields:– Cubic interactions and (conserved) currents– Interesting lessons from String Theory
• (Old) Frontier: 4-point amplitudes– Massless flat limits vs locality
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