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)

Padova, 12-1-2011 2

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?

Padova, 12-1-2011 3

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)

Padova, 12-1-2011 4

• 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

Padova, 12-1-2011 5

• For gravity two types of formulations:• Metric :

• Frame :

• For HS, one has similarly:• Metric-like :

• Frame-like :

Padova, 12-1-2011 6

( 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”

18

−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:

Padova, 12-1-2011 19

(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!

Padova, 12-1-2011 20

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.

21

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

.

22

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

Padova, 12-1-2011 23