1 String Lessons for Higher Spins Massimo Taronna TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA A A A AA A A AA A A Scuola Normale Superiore  INFN, Pisa Based on: Master Thesis(2009)[hep-th/ ](M.T. advisor: Prof. A. Sagnotti)  hep-th/ : A.Sagnotti and M.T.

2 What is String Theory? It is a scheme based on the mechanical model of a vibrating relativistic string It is a scheme based on the mechanical model of a vibrating relativistic string Although very natural it raises several questions: Background (in)dependence(?).... Although very natural it raises several questions: Background (in)dependence(?).... Key ingredient for consistency: Infinite tower of HS excitations The mechanical model hides the geometry A possible way out Probe String Theory within a regime where HS excitations are relevant The difficulty Soft UV behavior, open-closed duality, modular invariance, etc… String Theory is a consistent Higher-Spin Theory!

3 Higher Spins and Field Theory Starting Point Symmetry group of space-time (Majorana, 1932; Dirac, 1936; Fierz-Pauli, 1939; Wigner, 1939; …) Fundamental one-particle states are labeled by two quantum numbers Mass: Spin: More labels for There is at present no indication about the existence of a preferred subset of values! The Challenge Uncover the systematics of Higher Spins

4 Constructing a non-linear Gauge Theory Postulate some non-abelian algebra and construct the corresponding invariant field theory Analyze vertices in a preexisting non linear theory Classify all consistent deformations of the free theory (Nöther procedure) Low spin High spin YANG-MILLS GENERAL RELATIVITY SUGRA VASILIEV’s SYSTEM STRING (FIELD) THEORY Bengtsson,; Ouvry and Stern (1986)... M.T. (2009) A. Sagnotti and M.T. (2010) Berends, Burgers, van Dam(1985) Barnich, Hanneaux (1993)... Bekaert (2008) Bekaert, Joung, Mourad (2009) Manvelyan, Mktrchyan, Rühl (2010) Fotopoulos, Tsulaia (2010)

5 Conjectures and Open problems Does string theory result from the spontaneous breaking of HS gauge symmetry? How unique is String Theory? Using String Theory as a theoretical laboratory one can extract a lot of information about consistent interactions of HS states Massive scattering amplitudes and couplings do not suffer from the usual problems encountered in the massless case Massive scattering amplitudes and couplings do not suffer from the usual problems encountered in the massless case In the massless limit ordinary gauge invariance emerges! In the massless limit ordinary gauge invariance emerges! D. Gross; E.S. Fradkin; …

6 Plan  Open Bosonic String Tree-level S-matrix Amplitudes  String symbol calculus  Generating function for 3-point correlation functions  3-point Amplitudes  String Higher-Spin Couplings  Couplings 0-0-s  Couplings 1-1-s  General case  Higher-Spin Conserved Currents

7 (Open) Bosonic-String S-matrix The starting point is the gauge fixed version of the Polyakov path integral Vertex operators associated to asymptotic states via the state-operator isomorphism Chan-Paton factors

8 Symbol Calculus It is convenient to introduce the symbols of the usual string oscillators In terms of symbols the usual vertex operator correlation function that enters into the definition of the S-matrix becomes Symbol of the vertex operator Weyl-Wigner calculus! Generating Function

9 Generating Functions For external states of the first Regge trajectory of the open bosonic string Unphysical dependence on the y ij

10 Three-point Amplitudes L 0 constraint: mass parameterization for the first Regge trajectory The issue is now to impose the Virasoro constraints at the level of the generating function in order to select the physical information that it contains L 1 constraint: transversality One thus obtains the physical generating function for three-point amplitudes L 2 constraint: tracelessness

11 Three-point Amplitudes The amplitudes are given by the * - product of the physical generating function with the symbols of the vertex operators The result is Carrying out the * - product one arrives at Analogous simplifications for four-point amplitudes and string currents!

s Couplings In this case the current generating function is conserved and is given by coordinate space (Berends, Burgers and Van Dam, 1986) (Bekaert, Joung, Mourad, 2009) Example: the complex scalar HS currents

s Couplings This coupling can be extracted from our general result and is given by The on-shell gauge variation with respect to the spin-1 external legs vanishes identically On the contrary the gauge variation with respect to the spin-s external leg gives To recover the massless limit one should take care of such hidden subleading terms Simplest non trivial example!

14 Massless limit of the Theory The procedure just outlined can be implemented for the general coupling! The procedure just outlined can be implemented for the general s 1 -s 2 -s 3 coupling! This amplitude is gauge invariant up to the linearized massless equations and determines uniquely its off-shell completion, up to partial integrations and field redefinitions The result is: G: non trivial gauge invariant operator Coupling function: completely arbitrary in Field Theory (?)

15 Cubic vertices s1-s2-s3 couplings Number of derivatives S 1 +S 2 +S 3 S 1 +S 2 +S 3 -2 S 1 +S 2 -S 3... Minimum number of derivatives No minimal coupling of HS with gravity, but multipolar couplings are possible!

16 HS (bosonic) Conserved Currents The couplings so far obtained are induced by Noether currents Conserved up to massless Klein-Gordon equation, divergences and traces, but their completion turns out to be completely fixed, up to redefinitions! Spin one currents HS currents

17 HS (fermionic) Conserved Currents An educated guess is possible in the fermionic case Conserved up to massless Dirac equation, divergences and (gamma-)traces, but their completion turns out to be completely fixed, up to redefinitions! Spin one currents HS currents Analogous result for the currents with one fermion and one boson

Outlook  All three-point couplings (both abelian and non-abelian!)  Signals of non-abelian (unbroken) gauge symmetry at the cubic order  All four-point amplitudes (similar to the computation of three-point amplitudes)  All reference to the mechanical model completely eliminated  Educated guess for fermionic couplings and related currents  On-shell results contain (so far) all the physical information For the first Regge trajectory of the open bosonic string: BEHIND THE CORNER: Consistent HS gauge theory  Non localities can play a crucial role 18