Lectures at Erice, Instanbul, Prague 2011

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)

• 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:

• [ 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?

S2 =Z p

°°ab@ax¹@bX

ºG¹º(X)+Z²ab@ax

¹@bXºB¹º(X)+

Z®0p°R(2)©(X)+::

SD =1

2k2D

ZdDX

p¡Ge¡©

µR¡

1

12H2 + 4(@©)2

¶+::


• [ 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?

S2 =Z p

°°ab@ax¹@bX

ºG¹º(X)+Z²ab@ax

¹@bXºB¹º(X)+

Z®0p°R(2)©(X)+::

SD =1

2k2D

ZdDX

p¡Ge¡©

µR¡

1

12H2 + 4(@©)2

¶+::

Are strings really at the heart of String Theory?

Lessons from (and for) (massive) HS?


• 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.


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

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)

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 S=3 (Schwinger)

2nd Fronsdal constraint

[2-derivative : ] (Buchbinder et al, 2007; Francia, 2007)

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


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)


BRST equations for “contracted” Virasoro:

α’ ∞

First open bosonic Regge trajectory TRIPLETS Propagate: 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)


BRST equations for “contracted” Virasoro:

α’ ∞

First open bosonic Regge trajectory TRIPLETS Propagate: 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;º1:::ºs1

¤s¡1R¹1:::¹s;º1:::ºs


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)

13 A. SAGNOTTI - Istanbul, 2011

Problems :

Aragone – Deser problem (with “minimal” gravity coupling)

Weinberg – Witten Coleman - Mandula Velo - Zwanziger problem Weinberg’s 1964 S-matrix argument ............................... (see Porrati, 2008)


Problems :



Problems :



(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:

• 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:


(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 : hAB (hab cos(my) , Aa sin(my), ϕ cos(my) ) ]

(conserved Tµν)


Two 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)0 massive result as (ML)∞

How to extend to all s ? Origin of the poles ? Questions:

E.g.: massive s=2 in dS in general NO gauge symmetry, BUT if

“Partial” gauge symmetry :

NOTE: the coupling Jµνhµν is NOT “partially gauge invariant“

Unless Jµν is CONSERVED AND TRACELESS

(Deser, Nepolmechie, 1984) (Deser, Waldron, 2001)

Rational function of (ML)² :


E.g.: massive s=2 in dS in general NO gauge symmetry, BUT if

“Partial” gauge symmetry :

NOTE: the coupling Jµνhµν is NOT “partially gauge invariant“

Unless Jµν is CONSERVED AND TRACELESS

(Deser, Nepolmechie, 1984) (Deser, Waldron, 2001)


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!


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

Gauge fixed Polyakov path integral Koba-Nielsen amplitudes

Vertex operators asymptotic states

Sopenj1¢¢¢jn =

Z

Rn¡3

dy4 ¢ ¢ ¢ dyn jy12y13y23j

£ h Vj1(y1)Vj2(y2)Vj3(y3) ¢ ¢ ¢ Vjn(yn) iTr(¤a1 ¢ ¢ ¢¤an)

yij = yi ¡ yj

Chan-Paton factors

(L0 ¡ 1) jªi = 0

L1 jªi = 0

L2 jªi = 0

Virasoro Fierz-Pauli

HERE massive HS, but …

(AS, Taronna, 2010)


Z » exp

2

4¡1

2

nX

i6=j

®0 p i ¢ p j ln jyij j ¡p

2a0»i ¢ p j

yij+

1

2

»i ¢ »jy2ij

3

5

Z[J ] = i(2¼)d±(d)(J0)C exp³¡

1

2

Zd2¾d2¾0 J(¾) ¢ J(¾0) G(¾; ¾0)

´

For symmetric open-string states

( 1st Regge trajectory)

Z(»(n)i ) » exp

³X»(n)i Anm

ij (yl) »(m)j + »

(n)i ¢Bn

i (yl; p l) + ®0 p i ¢ p j ln jyij j´

Á i(p i; » i) =1

n!Ái ¹1¢¢¢¹n » ¹1

i : : : » ¹ni

“symbols”


¡p 21 =

s1 ¡ 1

®0 ¡ p 22 =

s2 ¡ 1

®0 ¡ p 23 =

s3 ¡ 1

®0

Zphys » 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

Ãp 1;

@

@»§r

®0

2p 31

!Á 2

Ãp 2; » +

@

@»§r

®0

2p 23

!Á 3

Ãp 3; » §

r®0

2p 12

! ¯¯¯»= 0

DFP conditions

(AS, Taronna, 2010)

KEY SIMPLIFICATION OF 3-POINT FUNCTIONS:


• 0-0-s:

• 1-1-(s≥2):

A§0¡0¡s =

Ã§r

® 0

2

!s

Á 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 =

Ã§r

® 0

2

!s¡2

s(s¡ 1) A1¹ A2 º Á¹º::: p s¡212

+

Ã§r

® 0

2

!s hA1 ¢ A2 Á ¢ p s

12 + s A1 ¢ p 23 A2 º Á º::: p s¡112

+ s A2 ¢ p 31 A1 º Á º::: p s¡112

i

+

Ã§r

® 0

2

!s+2

A1 ¢ p 23 A2 ¢ p 31 Á ¢ p s12 ;

Wigner Function

(Berends, Burger, van Dam, 1986)


• 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 1

Ãp 1; » 1 +

r®0

2p 23

!Á 2

Ãp 2; » 2 +

r®0

2p 31

!Á 3

Ãp 3; » 3 +

r®0

2p 12

! ¯¯¯» i = 0

G operator: builds a CASCADE of lower-derivative terms

(AS, Taronna, 2010)


The limiting couplings are induced by conserved currents:

J §(x ; ») = exp

Ã¨i

r® 0

2»®£@³1 ¢ @³2 @ ®

12 ¡ 2 @ ®³1 @³2 ¢ @ 1 + 2 @ ®

³2 @³1 ¢ @ 2

¤!

£ Á 1

Ãx ¨ i

r® 0

2» ; ³1 ¨ i

p2® 0 @ 2

!Á 2

Ãx § i

r® 0

2» ; ³2 § i

p2® 0 @ 1

! ¯¯¯³i = 0

• GENERALIZED WIGNER FUNCTIONS, conserved up to massless Klein-Gordon, divergences and traces. Unique extension to both Fronsdal and compensator 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


(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 ¡1

2p j ¢ @» i @» j ¢ @» j

D j = p j ¢ @» j ¡1

2p j ¢ » j @» j ¢ @» j ( de Donder operator )

V123 = exp (§¡)

·1 +

® 0

2

³H 12 H 13 + H 23 H 21 + H 31 H 32

´

¨µ

® 0

2

¶ 32 ³

: H12 H 31 H 23 : ¡ : H21 H 32 H 13 :´#


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

Ã¨ i

r® 0

2»®£@³1 ¢ @³2 @ ®

12 ¡ 2 @ ®³1 @³2 ¢ @ 1 + 2 @ ®

³2 @³1 ¢ @ 2

¤!

£ ¹ª 1

Ãx ¨ i

r® 0

2» ; ³1 ¨ i

p2 ® 0 @ 2

! h1 + =»

iª 2

Ãx § i

r® 0

2» ; ³2 § i

p2 ® 0 @ 1

! ¯¯¯³i = 0

:

A [0]§F = exp(§G) ¹Ã 1

Ãp 1 ; » 1 §

r® 0

2p 23

![1 + =@ » 3

] Ã 2

Ãp 2 ; » 2 §

r® 0

2p 31

!

£ Á 3

Ãp 3 ; » 3 §

r® 0

2p 12

! ¯¯¯» i = 0

;

J[0]§BF (x ; ») = exp

Ã¨ i

r® 0

2»®£@³1 ¢ @³2 @ ®

12 ¡ 2 @ ®³1 @³2 ¢ @ 1 + 2 @ ®

³2 @³1 ¢ @ 2

¤!

£h1 + =@³ 2

iª 1

Ãx ¨ i

r® 0

2» ; ³1 ¨ i

p2 ® 0 @ 2

!© 2

Ãx § i

r® 0

2» ; ³2 § i

p2® 0 @ 1

! ¯¯¯³i = 0

:

(AS, Taronna, 2010)


• 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):

X

n

1

n! 22n(3¡ d2 ¡ s)n

hJ [n] ; J [n]i ;

J(x; u) = ©?(x + iu) ©(x¡ iu) ; a(z) =X

r

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

(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:


(Taronna, 2011)

• Quartic YM vertex: ”conterterm”for linearized gauge symmetry

• Actually: gauge symmetry requires “planarly dual” pairs

• YM amplitudes:

Aa¹ Ab

º Ac½ Ad

¾

µ® s¹º½¾

s+ ¯ s

¹º½¾ +®u¾¹º½

u+ ¯ u

¾¹º½

¶ £tr¡T aT bT cT d

¢+ tr

¡T dT cT bT a

¢¤+ : : :

(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 !

Aa¹ Ab

º Ac½ Ad

¾

µ® s¹º½¾

s+ ¯ s

¹º½¾ +®u¾¹º½

u+ ¯ u

¾¹º½

¶ £tr¡T aT bT cT d

¢+ tr

¡T dT cT bT a

¢¤+ : : :

Áa¹1:::¹k

Ábº1:::ºk

Ác½1:::ºk

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

¢¤+ : : :

• 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


• 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


Documents

Lectures at Erice, Instanbul, Prague 2011