Augusto Sagnotti- Higher Spins and Strings
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Augusto SAGNOTTI
Scuola Normale Superiore and INFN
Istanbul, August 2011
• Dario FRANCIA (Czech Acad. Sci, Prague)
•
Andrea CAMPOLEONI (AEI, Potsdam)• Massimo TARONNA (Scuola Normale)
• Mirian TSULAIA (U. Liverpool)• Jihad MOURAD (U. Paris VII)
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•
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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A¹ ¡! Á¹1:::¹s4D:
[ D>4: ]A¹
¡!Á¹(1)
1
:::¹(1)s1
;:::;¹(N )
1
:::¹(N )sN
[Symmetric]
[Mixed (multi-symmetric)]
(Dirac, Fierz, Pauli, 1936 - 39)
Dirac-Fierz-Pauli(DFP)
conditions:
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• [ 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 (in)dependence?
S 2 =Z p
°° ab@ ax¹@ bX
º G¹º (X )+Z ²ab@ ax
¹@ bX º B¹º (X )+
Z ®0p °R(2)©(X )+::
S D = 1
2k2D
Z dDX
p ¡Ge¡©µR¡ 1
12H 2+4(@ ©)2
¶+::
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• [ 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 (in)dependence?
S 2 =Z p
°° ab@ ax¹@ bX
º G¹º (X )+Z ²ab@ ax
¹@ bX º B¹º (X )+
Z ®0p °R(2)©(X )+::
S D = 1
2k2D
Z dDX
p ¡Ge¡©µR¡ 1
12H 2+4(@ ©)2
¶+::
Are strings really at the heart of String Theory?Lessons from (and for) (massive) HS?
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• Key properties of HS fields: – Symmetric HS fields, triplets and HS geometry;
• HS interactions: – External currents and the vDVZ discontinuity;
– Limiting string 3-pt functions and gauge symmetry;
– Conserved HS currents & exchanges;
– 4-point functions and beyond.
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Fronsdal (1978): natural extension of s=1,2 cases (BUT with CONSTRAINTS)
Can simplify notation (and algebra) hiding space-time indices:
(“primes” = traces)
1st Fronsdal constraint
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Bianchi identity :
Natural to try and forego these “trace” constraints:
• BRST (non minimal) (Buchbinder, Burdik, Pashnev, Tsulaia, , 1998-)
• Minimal compensator form (Francia, AS, Mourad, 2002 -)
Unconstrained Lagrangian
2nd Fronsdal constraint
[2-derivative : ](Buchbinder et al, 2007; Francia, 2007 )
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Bianchi identity :
Natural to try and forego these “trace” constraints:
• BRST (non minimal) (Buchbinder, Burdik, Pashnev, Tsulaia, , 1998-)
• Minimal compensator form (Francia, AS, Mourad, 2002 -)
Unconstrained LagrangianS=3(Schwinger)
2nd Fronsdal constraint
[2-derivative : ](Buchbinder et al, 2007; Francia, 2007 )
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What are we gaining ?
s = 2:
s > 2: Hierarchy of connections and curvatures (de Wit and Freedman, 1980)
After some iterations: NON-LOCAL gauge invariant equation for ϕ ONLY
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What are we gaining ?
s = 2:
s > 2: Hierarchy of connections and curvatures
NON LOCAL geometric equations :
(de Wit and Freedman, 1980)
After some iterations: NON-LOCAL gauge invariant equation for ϕ ONLY
(Francia and AS, 2002)
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BRST equations for “contracted” Virasoro:
α’ ∞
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)
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BRST equations for “contracted” Virasoro:
α’ ∞
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)
(Francia, 2010)
Geometric form: L » R¹1:::¹s;º 1:::º s 1
¤s¡1R¹1:::¹s;º 1:::º s
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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
More recently: Unconstrained bosonic Lagrangians (Un)constrained fermionic Lagrangians Key tool: Bianchi (Noether) identities
(Campoleoni, Francia, Mourad, AS, 2008, and 2009)
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Problems :
Aragone – Deser problem (with “minimal” gravity coupling)
Weinberg – Witten Coleman - Mandula Velo - Zwanziger problemWeinberg’s 1964 S-matrix argument
............................... (see Porrati, 2008)
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Problems :
Aragone – Deser problem (with “minimal” gravity coupling)
Weinberg – Witten Coleman - Mandula Velo - Zwanziger problemWeinberg’s 1964 S-matrix argument
............................... (see Porrati, 2008)
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Problems :
Aragone – Deser problem (with “minimal” gravity coupling)
Weinberg – Witten Coleman - Mandula Velo - Zwanziger problemWeinberg’s 1964 S-matrix argument
............................... (see Porrati, 2008)
(Vasiliev, 1990, 2003)(Sezgin, Sundell, 2001)
But:
(Light-cone or covariant) 3-vertices
[higher derivatives]
(Scalar) scattering via current exchanges
String contact terms resolve Velo-Zwanziger
Deformed low-derivative with Λ≠0
Infinitely many fields
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)
(Argyres, Nappi, 1989)(Porrati, Rahman, 2009)
(Porrati, Rahman, AS, 2010)
Vasiliev eqs:
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• Static sources: Coulomb-like• Residues: degrees of freedom
• e.g. s=1 :
• All s :
(Fronsdal, 1978)
… Unique NON-LOCAL Lagrangian(with proper current exchange)
E.g. for s=3:
(Francia, Mourad, AS, 2007, 2008)
NOTICE:
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(van Dam, Veltman; Zakharov, 1970)
∀s and D, m=0 :
VDVZ discontinuity: comparing D and (D+1)-dim exchanges ∀s: can describe massive fields a’ la Scherk-Schwarz from (D+1) dimensions :
[ e.g. for s=2 : h AB (hab cos(my) , Aa sin(my), ϕ cos(my) ) ]
(conserved T µν)
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T wo types of deformations:
a finite dS radius L a mass M
s=2 : (Higuchi, 1987,2002)(Porrati, 2001)(Kogan, Mouslopoulos, Papazoglou, 2001)
massless result as (ML)0massive result as (ML)∞
How to extend to all s ? Origin of the poles ?uestions:
E.g.: massive s=2 in dS in general NO gauge symmetry, BUT if
“Partial” gauge symmetry :
NOTE: the coupling J µνh µν is NOT “partially gauge invariant“Unless J µν is CONSERVED AND TRACELESS
(Deser, Nepolmechie, 1984)(Deser, Waldron, 2001)
Rational function of (ML)² :
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E.g.: massive s=2 in dS in general NO gauge symmetry, BUT if
“Partial” gauge symmetry :
NOTE: the coupling J µνh µν is NOT “partially gauge invariant“Unless J µν is CONSERVED AND TRACELESS
(Deser, Nepolmechie, 1984)(Deser, Waldron, 2001)
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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 : OK with also higher-derivative terms around (A)dS;• Metsaev (2007) light-cone• Boulanger, Leclercq, Sundell (2008): non-Abelian CUBIC “SEEDS”
HS around flat space; MORE DERIVATIVES
DEFORMING the highest vertex to (A)dS one can recover consistent “minimal-like”couplings WITH higher-derivative tails, with a singular limit as Λ 0
Naively: 2 derivatives,
BUT ACTUALLY:4 derivatives!
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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 : OK with also higher-derivative terms around (A)dS;• Metsaev (2007) light-cone• Boulanger, Leclercq, Sundell (2008): non-Abelian CUBIC “SEEDS”
HS around flat space; MORE DERIVATIVES
DEFORMING the highest vertex to (A)dS one can recover consistent “minimal-like”couplings WITH higher-derivative tails, with a singular limit as Λ 0
Naively: 2 derivatives,
BUT ACTUALLY:4 derivatives!
¤A¹ ¡ r¹r ¢ A ¡1
L2 (1 ¡ D) A¹ = 0
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Gauge fixed Polyakov path integral Koba-Nielsen amplitudes
Vertex operators asymptotic states
S openj1¢¢¢jn =
Z Rn¡3
dy4 ¢ ¢ ¢ dyn jy12y13y23j
£ h V j1(y1)V j2(y2)V j3(y3) ¢ ¢ ¢ V jn(yn) iT r(¤a1 ¢ ¢ ¢¤an)
yij = yi ¡ yj
Chan-Paton factors
(L0 ¡ 1) jªi = 0L1 jªi = 0
L2 jªi = 0
Virasoro Fierz-Pauli
HERE massive HS, but …
(AS, Taronna, 2010)
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Z » exp
24¡1
2
nXi6=j
®0 p i ¢ p j ln jyijj ¡p
2a0» i ¢ p j
yij+
1
2
» i ¢ » jy2ij
35
Z [J ] = i(2¼)d± (d)(J 0)C exp³
¡ 12
Z d2¾d2¾0 J (¾) ¢ J (¾0) G(¾; ¾0)
´
For symmetric open-string states
( 1st Regge trajectory)
Z (» (n)i ) » exp
³X» (n)i Anmij (yl) »
(m)j + »
(n)i ¢ Bni (yl; p l) + ®0 p i ¢ p j ln jyij j
´
Á i( p i; » i) =1
n!
Ái ¹1¢¢¢¹n » ¹1i : : : » ¹ni
“symbols”
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¡ p 21 = s1 ¡ 1®0 ¡ p 22 = s2 ¡ 1
®0 ¡ p 23 = s3 ¡ 1®0
Z phys » exp
(r ®0
2
µ» 1 ¢ p 23
¿y23
y12y13
À+ » 2 ¢ p 31
¿y13
y12y23
À+ » 3 ¢ p 12
¿y12
y13y23
À¶+ (» 1 ¢ » 2 + » 1 ¢ » 3 + » 2 ¢ » 3)
)
• L0 constraint: mass
• L1 constraint: transversality
• L2 constraint: vanishing trace
Virasoro constraints directly in generating function
Signs (twist)“On-shell” couplings star- product with symbols of fields
A§ = Á 1
à p 1;
@
@» §r
®0
2p 31
!Á 2
à p 2; » +
@
@» §r
®0
2p 23
!Á 3
à p 3; » §
r ®0
2p 12
! ¯
¯»=0
DFP conditions
(AS, Taronna, 2010)
KEY SIMPLIFICATION OF 3-POINT FUNCTIONS:
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• 0-0-s:
• 1-1-(s≥2):
A§0¡0¡s =
çr
® 0
2
!sÁ 1 Á 2 Á 3 ¢ p s12
J §(x; » ) = ©
Ãx § i
r ® 0
2»
!©
Ãx ¨ i
r ® 0
2»
!Conserved(massless φ)
A§1¡1¡s =
çr
® 0
2
!s¡2
s(s ¡ 1) A1¹A2 º Á¹º::: p s¡2
12
+çr
®0
2
!s hA1 ¢ A2 Á ¢ p s12 + s A1 ¢ p 23 A2 º Á º::: p s¡1
12
+ s A2 ¢ p 31 A1 º Áº::: p s¡1
12
i
+
çr
® 0
2
!s+2
A1 ¢ p 23 A2 ¢ p 31 Á ¢ p s12 ;
Wigner Function
(Berends, Burger, van Dam, 1986)
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•
OLD IDEA: String Theory broken phase of “something”
• AMPLITUDES: can spot extra “debris” that drops out in the “massless”limit, where one ought to recover couplings based on conserved currents.
A gauge invariant pattern shows up!
A§ = exp
(r ®0
2
h(@ » 1 ¢ @ » 2)(@ » 3 ¢ p 12) + (@ » 2 ¢ @ » 3)(@ » 1 ¢ p 23) + (@ » 3 ¢ @ » 1)(@ » 2 ¢ p 31)
i)
£Á 1à p 1; » 1 + r ®
0
2
p 23!Á 2Ã p 2; » 2 + r ®0
2
p 31!Á 3Ã p 3; » 3 + r ®0
2
p 12! ¯» i =0
G operator: builds a CASCADE of lower-derivative terms
(AS, Taronna, 2010)
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The limiting couplings are induced by conserved currents:
J §(x ; » ) = exp
èi
r ® 0
2» ®£
@ ³ 1 ¢ @ ³ 2 @ ®12 ¡ 2 @ ®³ 1 @ ³ 2 ¢ @ 1 + 2 @ ®³ 2 @ ³ 1 ¢ @ 2¤!
£ Á 1
Ãx ¨ i
r ® 0
2» ; ³ 1 ¨ i
p 2® 0 @ 2
!Á 2
Ãx § i
r ® 0
2» ; ³ 2 § i
p 2® 0 @ 1
! ¯¯¯³ i =0
• GENERALIZED WIGNER FUNCTIONS, conserved up to massless Klein-Gordon, divergences and traces. Unique extension to both Fronsdal andcompensator cases (with divergences and traces), conserved up to complete eqs.
• CORRESPONDING GAUGE INVARIANT 3-POINT FUNCTIONS:
J ¢Á
(AS, Taronna, 2010)
A§ = exp(r
®0
2
h(@ » 1 ¢ @ » 2 + 1)(@ » 3 ¢ p 12) + (@ » 2 ¢ @ » 3 + 1)(@ » 1 ¢ p 23) + (@ » 3 ¢ @ » 1 + 1)(@ » 2 ¢ p 31)
i)
£ Á 1 ( p 1; » 1 ) Á 2 ( p 2; » 2 ) Á 3 ( p 3; » 3 )
¯¯» i =0
Related work : Manvelyan, Mkrtchyan, Ruhl, 2010
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(AS, Taronna, 2010)
Related work : Manvelyan, Mkrtchyan, Ruhl, 2010
“Off – Shell” Vertices:
A§ = exp(r
®0
2
h(@ » 1 ¢ @ » 2 + 1)(@ » 3 ¢ p 12) + (@ » 2 ¢ @ » 3 + 1)(@ » 1 ¢ p 23) + (@ » 3 ¢ @ » 1 + 1)(@ » 2 ¢ p 31)
i)
£ Á 1 ( p 1; » 1 ) Á 2 ( p 2; » 2 ) Á 3 ( p 3; » 3 )
¯¯» i =0
Hij = (@ » i ¢ @ » j + 1) D j ¡ 12
p j ¢ @ » i @ » j ¢ @ » j
D j = p j ¢ @ » j ¡ 1
2p j ¢ » j @ » j ¢ @ » j ( de Donder operator )
V 123 = exp(
§¡) ·1 +
® 0
2³ ^
H12
^
H13 + ^
H23
^
H21 + ^
H31
^
H32´
¨µ
® 0
2
¶ 32 ³
: H12 H 31 H 23 : ¡ : H21 H 32 H 13 :´#
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A natural guess for gauge invariant FFB couplings in (type-0) superstrings:
Determines corresponding (Bose and Fermi)conserved HS currents:
J [0]§FF (x ; » ) = exp
è i
r ® 0
2» ®£
@ ³ 1 ¢ @ ³ 2 @ ®12 ¡ 2 @ ®³ 1 @ ³ 2 ¢ @ 1 + 2 @ ®³ 2 @ ³ 1 ¢ @ 2¤!
£ ¹ª 1
Ãx ¨ i
r ® 0
2» ; ³ 1 ¨ i
p 2 ® 0 @ 2
! h1 + =»
iª 2
Ãx § i
r ® 0
2» ; ³ 2 § i
p 2 ® 0 @ 1
! ¯
¯³ i=0
:
A [0]§F = exp(§G) ¹Ã 1
à p 1 ; » 1 §
r ® 0
2p 23
![1 + =@ » 3 ] Ã 2
à p 2 ; » 2 §
r ® 0
2p 31
!
£ Á 3
à p 3 ; » 3 §
r ® 0
2p 12
! ¯¯¯» i=0
;
J [0]§BF (x ; » ) = exp
è i
r ® 0
2» ®£
@ ³ 1 ¢ @ ³ 2 @ ®12 ¡ 2 @ ®³ 1 @ ³ 2 ¢ @ 1 + 2 @ ®³ 2 @ ³ 1 ¢ @ 2¤!
£h
1 + =@ ³ 2
iª 1
Ãx ¨ i
r ® 0
2» ; ³ 1 ¨ i
p 2 ® 0 @ 2
!© 2
Ãx § i
r ® 0
2» ; ³ 2 § i
p 2® 0 @ 1
! ¯¯³ i =0
:
(AS, Taronna, 2010)
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• Current-exchange formula: (Fronsdal, 1978; Francia, Mourad, AS, 2007)
• Scalar currents & “coupling function” (Berends, Burgers, van Dam, 1986)
(Bekaert, Joung, Mourad, 2009)
• Bekaert-Joung-Mourad amplitude (D=4):
Xn
1
n! 22n
(3 ¡d
2 ¡ s)n hJ [n] ; J [n]
i;
J (x; u) = ©
?
(x + iu) ©(x ¡ iu) ; a(z) =Xr
zr
r! ar
A(s) = ¡ 1
® 0s
·a
µ® 0
4(u ¡ t) +
® 0
2
p ¡ut
¶+ a
µ® 0
4(u ¡ t) ¡ ® 0
2
p ¡ut
¶¡ a0
¸£ Á 1 ( p 1) Á 2 ( p 2) Á 3 ( p 3; ) Á 4 ( p 4)
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(Taronna, 2011)
• What is the meaning of Γ ?
• Cubic vertex: tensor products of YM and scalar vertices• These quantities are gauge invariant, up to the DFP conditions
• What are the analogues of the two basic vertices for N>3 pts?
[ “Lego bricks” for S-matrix amplitudes]
A§ = exp(r
®0
2
h(@ » 1 ¢ @ » 2 + 1)@ » 3 ¢ p 12 + (@ »2 ¢ @ » 3 + 1)@ » 1 ¢ p 23 + (@ » 3 ¢ @ » 1 + 1)@ » 2 ¢ p 31
i)
£ Á 1 ( p 1; » 1 ) Á 2 ( p 2; » 2 ) Á 3 ( p 3; » 3 )
¯¯» i =0
• (Limiting) 3-pt functions:
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(Taronna, 2011)• Quartic YM vertex:
”conterterm”for linearized gauge symmetry
• Actually:gauge symmetry requires “planarly dual” pairs
• YM amplitudes:
Aa¹Abº Ac½Ad¾
µ® s¹º½¾
s+ ¯ s¹º½¾ + ®u¾¹º½
u+ ¯ u¾¹º½
¶ £tr¡
T aT bT cT d¢
+ tr¡
T dT cT bT a¢¤
+ : : :
8/3/2019 Augusto Sagnotti- Higher Spins and Strings
http://slidepdf.com/reader/full/augusto-sagnotti-higher-spins-and-strings 34/36
(Taronna, 2011)• Quartic YM vertex:
”conterterm”for linearized gauge symmetry
• Actually:gauge symmetry requires “planarly dual” pairs
• YM amplitudes:
• Gauge invariant HS amplitudes (open-string-like):
• Weinberg’s 1964 argument bypassed: lowest exchanged spin = 2s-1
• Non-local Lagrangian couplings
• Closed-string-like amplitudes: striking differences already for s=2 !
Aa¹Abº Ac½Ad¾
µ® s¹º½¾
s+ ¯ s¹º½¾ + ®u¾¹º½
u+ ¯ u¾¹º½
¶ £tr¡
T aT bT cT d¢
+ tr¡
T dT cT bT a¢¤
+ : : :
Áa¹1:::¹k Ábº 1:::º k Ác½1:::º k Ád¾1:::¾k µ® s
s+ ¯ s +
®u
u+ ¯ u¶
k
(su)k¡1
£tr ¡T aT bT cT d¢ + tr ¡T dT cT bT a¢¤
+ : : :
8/3/2019 Augusto Sagnotti- Higher Spins and Strings
http://slidepdf.com/reader/full/augusto-sagnotti-higher-spins-and-strings 35/36
• Free HS Fields: – Constraints, compensators and curvatures – (String Theory (α΄∞) : triplets)
• Interacting HS Fields: – External currents and vDVZ (dis)continuity – Cubic interactions and (conserved) currents
• (Old) Frontier Crossed: 4-point amplitudes – Massless flat limits vs locality
A. SAGNOTTI - Istanbul, 2011
8/3/2019 Augusto Sagnotti- Higher Spins and Strings
http://slidepdf.com/reader/full/augusto-sagnotti-higher-spins-and-strings 36/36
• Free HS Fields: – Constraints, compensators and curvatures – (String Theory (α΄∞) : triplets)
• Interacting HS Fields: – External currents and vDVZ (dis)continuity – Cubic interactions and (conserved) currents
• (Old) Frontier Crossed: 4-point amplitudes – Massless flat limits vs locality
A A t b l