Interacting Higher Spins on Ad SD Mirian Tsulaia

Interacting Higher Spins on Ad. S(D) Mirian Tsulaia University of Crete 1

Plan 1. Motivation 2. Free Field Equations, Free Lagrangians 3. Fields on Ad. S and Minkowski Spaces 4. Interactions Based On: A. Fotopoulos and M. T, ar. Xiv: 0705. 2939 I. Buchbinder, A. Fotopoulos, A. Petkou, M. T, Phys. Rev. D 74, 2006, 105018 Also: A. Fotopoulos, K. L. Panigrahi, M. T, Phys. Rev. D 74, 2006, 085029 A. Sagnotti, M. T, Nucl. Phys. B 682 (2004), 83. 2

1. Motivation There are two consistent theories which contain fields with arbitrary spin (all spins are required for consistency): A: (Super)string theory – Consistent both Classically and Quantum Mechanically (in D = 26, D = 10) Backgrounds: Flat for bosonic. Flat, pp-wave, Ad. S 5 X S 5 for Superstring B: Higher Spin Gauge Theory (M. Vasiliev, E. Fradkin. M. Vasiliev) – Consistent Classically. Quantum Mechanically – No S Matrix on Ad. S Backgrounds: Ad. S(D) 3

§ HS Theory is an “analog” of SUGRA, but classical consistency requires an infinite tower of massless HS fields § SUGRA’s are low energy limits of superstring theories § Is HS gauge theory any limit (an effective theory) of String Theory? ? ? § Holography: M. Bianchi, J. Morales, H. Samtleben ‘ 03, E. Sezgin and P. Sundell ‘ 02 § Is HS gauge theory (Massless Fields) a symmetric phase of String Theory (Massive Fields)? 4

A way to look at it: § In string theory masses; ms ~ s/α’ § Symmetric Phase ms → 0. α’ → ∞ (High Energy) § Note: In a high energy limit a string might curve only about a highly curved background, e. g. Ad. S with a small radius § Recently considered by N. Moeller and P. West ’ 04, also P. Horava, C. Keeler, ’ 07, previously D. Gross, P. Mende, ’ 88. § Ways to describe HS fields: M. Vasiliev, “Frame-like”; C. Fronsdal, D. Francia and A. Sagnotti, “Metric-like” 5

2. Fields on Ad. S(D) and Minkowski Space Strategy: First derive proper field equations for massless fields What is “proper”? A: Mass Shell Condition § Massless Klein-Gordon Equation for HS bosons. § Massless Klein-Gordon Equation for fields with mixed symmetry § Mixed symmetry fields are described by different Young tableaux: eg. → 6

On Ad. S(D) § Group of Isometries SO(D-1, 2) § Maximal Compact Subgroup is SO(2) X SO(D-1) § Build Representations in the Cartan-Weyl Basis 7

§ Highest weight state § Representations § E 0 = S + D-3 (for totally symmetric fields) → zero norm → zero mass § Leads to Klein Gordon Equation: 8

B: The equations must have enough gauge invariance to remove negative norm states, ghosts from the spectrum § Totally symmetric field § Mixed symmetry fields 9

How to derive: A possible way to take a BRST Charge for a bosonic string Rescale And take formally α’ → ∞ With Commutation Relations 10

§ BRST Charge is nilpotent in any Dimension Q~ = 0 § One can truncate the number if oscillators αμk, ck, bk to any finite k without affecting the nilpotency property § To get a free lagrangian for a leading Regge trajectory expand § Analogous for mixed symmetry fields § Fock vacuum § The ghost ck has ghost number 1; bk has ghost number -1 11

3. Lagrangian § Free Lagrangian § Gauge Transformation § Equation of Motion 12

Ad. S Deformation for Leading Regge Trajectory + Where γ = cb and M = ½ αμαμ 13

Coming back to α’ → ∞ § Taking the point that we have massless fields of any spin, our system (BRST charge, Lagrangian, Gauge Transformations) correctly describes these fields (totally symmetric, mixed symmetry) § Fixing gauge one can see that only physical degrees of freedom propagate 14

4. Cubic Interactions § In order to have nontrivial interactions one has to take three copies of Hilbert space considered before and deform gauge transformations § The Lagrangian § Interaction Vertex § V(αi+, ci+, bi 0) determined from Q|V>= 0; Q = Q 1 + Q 2 + Q 3 15

§ The BRST invariance of |V> ensures the gauge invariance of the action and closure of the nonabelian algebra up to first order in g. § One can solve this equation, taking V to be an arbitrary polynomial in αiμ+; bi+, ci+ with ghost number zero. § It can be done both for Ad. S (leading trajectory) and flat space-time. § The solution should not be BRST trivial: |V> ≠ Q|W>. The trivial one can be obtained from the free action by field redefinitions. 16

Finally the solution from String Field Theory vertex § In string field theory V is § Αμ 0 is proportional to (α’)1/2, but one has to also keep terms which do not contain (α’)1/2, since they are in the exponential § Put the anzatz § Where βij = cib 0 j and lij = αipj 17

Demand BRST invariance: The Closure requires: § Sii = (1, 1, 1) § To consider the whole tower together § One can do this for totally symmetric or mixed symmetry fields 18

§ We have actually done a field dependent deformation of the initial BRST charge ; § Does not work for Ad. S: there is no such solution in the Bosonic string 19

Summary: § From Bosonic string theory one can get information about interacting massless bosonic fields on a flat background § To do Ad. S along these lines one probably has to do super § Even a free BRST on Ad. S for fermions and mixed symmetry has not yet been done 20