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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)

 A. SAGNOTTI - Istanbul, 2011

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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anbul, 2011

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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

àp 1;

@ 

@» §r 

®0

2p 31

!Á 2

àp 2; »  +

@ 

@» §r 

®0

2p 23

!Á 3

àp 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 1àp 1; » 1 + r ®

0

2

p 23!Á 2àp 2; » 2 + r ®0

2

p 31!Á 3àp 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

àp 1 ; » 1 §

r ® 0

2p 23

![1 + =@ » 3 ] Ã 2

àp 2 ; » 2 §

r ® 0

2p 31

!

£ Á 3

àp 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)

 A. SAGNOTTI - Prague, 2011

8/3/2019 Augusto Sagnotti- Higher Spins and Strings

http://slidepdf.com/reader/full/augusto-sagnotti-higher-spins-and-strings 32/36

(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:

 A. SAGNOTTI - Istanbul, 2011

8/3/2019 Augusto Sagnotti- Higher Spins and Strings

http://slidepdf.com/reader/full/augusto-sagnotti-higher-spins-and-strings 33/36

(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