From Higher Spins to Strings Rajesh Gopakumar Harish-Chandra - PowerPoint PPT Presentation

From Higher Spins to Strings Rajesh Gopakumar Harish-Chandra Research Institute Strings 2014, Princeton, June 24th, 2014 Based on: M. R. Gaberdiel and R. G. (arXiv:1406.tmrw and also 1305.4181)

Why are We Studying Higher Spin Theories? • Free YM theory has a tower of conserved currents dual to Vasiliev H-spin gauge fields ( Sundborg, Witten ). • Signals the presence of a large unbroken symmetry phase of the string theory ( Gross, Witten, Moore, Sagnotti et.al. ). • Can the Vasiliev H-Spin symmetries help to get a handle on the extended stringy symmetry in tensionless limit? • might be a good test case since it already has Virasoro AdS 3 (and then extended to - Henneaux-Rey, Campoleoni et.al. ). W ∞ • Symmetric product CFT for D1-D5 system has been believed to be dual to tensionless limit of string theory.

The Punchline Vasiliev higher spin symmetry organises all the ( T 4 ) N +1 /S N +1 states of the orbifold symmetric product CFT = Tensionless limit of strings AdS 3 × S 3 × T 4 on .

Stringy Symmetries In particular: The chiral sector (conserved currents) can be written in terms of representations of the higher spin symmetry algebra. X Z NS ( q, y ) = n ( Λ ) χ (0; Λ ) ( q, y ) Λ ∈ U ( N ) Chiral part of Characters of N = 4 Symm. Prod. multiplicity of minimal model singlets in Λ S N +1 coset: reps. W ∞ Infinite (stringy) extension of symmetry. W ∞

Explicitly..... • The vacuum character ( ) contains the usual W ∞ Λ = 0 generators - bilinears in free fermions and bosons. • Additional chiral generators ( ) can be written Λ 6 = 0 down explicitly in terms of free fermions and bosons . N +1 − 1 / 2 ψ i β X ψ i α Λ = [2 , 0 . . . , 0] ↔ − 1 / 2 i =1 N +1 − 1 / 2 ψ j β − 1 / 2 ψ i γ − 1 / 2 ψ j δ X ψ i α Λ = [0 , 2 , 0 . . . , 0] ↔ − 1 / 2 i,j =1

N = 4 Large AdS 3 × S 3 × T 4 • String theory on has small N = 4 SUSY . • Useful to consider via a limit of H-spin holography for large coset CFTs. ( Gaberdiel-R.G. ) N = 4 • Large SCA has two SU(2) Kac-Moody algebras . N = 4 k − Thus labelled by one extra parameter: . γ = k + + k − • Small obtained as a contraction - . N = 4 k + → ∞ • Only one SU(2) KM algebra at level . k −

N = 4 Large Coset Holography 4(N+1) free fermions The CFT: su ( N + 2) (1) = su ( N + 2) k ⊕ so (4 N + 4) 1 ⊕ u (1) (1) ⊕ u (1) (1) ∼ κ ⊕ u (1) . su ( N ) (1) u ( N ) k +2 κ c = 6( k + 1)( N + 1) . Take ‘t Hooft limit N, k → ∞ k + N + 2 with fixed. (Gaberdiel-R.G.) N + 1 λ = N + k + 2 = γ Has Large ( van Proeyen et.al., Sevrin et.al. ) with N = 4 k + = ( k + 1); k − = ( N + 1)

Coset Holography (Contd.) The H-Spin Dual: • Vasiliev theory based on gauge group (Prokushkin- shs 2 [ λ ] Vasiliev) . • One higher spin gauge supermultiplet for each spin s ≥ 1 s : ( 1 , 1 ) SU(2) labels s + 1 2 : ( 2 , 2 ) R ( s ) : s + 1 : ( 3 , 1 ) ⊕ ( 1 , 3 ) s + 3 2 : ( 2 , 2 ) s + 2 : ( 1 , 1 ) . • Generates an asymptotic super algebra which matches W ∞ nontrivially with coset ( Gaberdiel-Peng, Beccaria et.al. ).

Representations W ∞ • Primaries labelled by ( Λ + ; Λ − , u ) ∈ u (1) κ (will be omitted) ∈ su ( N + 2) k ∈ su ( N ) k +2 • (0;f) “Perturbative” matter multiplets of H-Spin theory (with multi-particles) ( Chang-Yin ). (0; Λ ) k + 3 N + k + 2 → 1 − λ 2 h (0; f) = 2 (0; Λ ) ⊗ (0; Λ ∗ ) ⊂ H (diag) = H (pert) = M M ( Λ + ; Λ − ) ⊗ ( Λ ∗ + ; Λ ∗ − ) Λ + , Λ − Λ Contains “light states”

N = 4 contracts N = 4 c = 6( k + 1)( N + 1) k →∞ → c = 6( N + 1) − − − − k + N + 2 • Coset CFT reduces to a continuous orbifold . ( T 4 ) N +1 /U ( N ) • The WZW factors decompactify to give 4(N+1) free bosons which combine with the 4(N+1) free fermions, gauged by U(N) . 2 · ( N , 1 ) ⊕ 2 · ( ¯ Bosons: N , 1 ) ⊕ 4 · ( 1 , 1 ) ( N , 2 ) ⊕ ( ¯ N , 2 ) ⊕ 2 · ( 1 , 2 ) Fermions: fund. of U(N) SU (2) R Singlet of U(N)

Continuous Orbifold • Untwisted sector: U(N) singlets formed from fermions/bosons. i α e h (0; f ) = 1 − λ → 1 (0;¯ ¯ ψ i β k →∞ E.g. ; ( Note: ) f) ⊗ (0; f) ↔ ψ − − − − 2 2 • More generally, Vasiliev States M (0; Λ ) ⊗ (0; Λ ∗ ) = H (pert) H untwisted = Λ N = 2 Similar to bosonic and cases ( Gaberdiel-Suchanek, Gaberdiel-Kelm ) • Twisted Sector: Continuous twists (U(N) holonomies) leads to a continuum (incl. light states). Labelled by . ( Λ + ; Λ − ) : w/ Λ + 6 = 0

A Tale of Two Orbifolds • How do we relate to ? ( T 4 ) N +1 /U ( N ) ( T 4 ) N +1 /S N +1 • and N , ¯ S N +1 ⊂ U ( N ) N → N N Dim. Irrep. of S N +1 2 · ( N , 1 ) ⊕ 2 · ( ¯ N , 1 ) ⊕ 4 · ( 1 , 1 ) → 4 · ( N, 1 ) ⊕ 4 · (1 , 1 ) Bosons: ( N , 2 ) ⊕ ( ¯ N , 2 ) ⊕ 2 · ( 1 , 2 ) → 2 · ( N, 2 ) ⊕ 2 · (1 , 2 ) Fermions: How fermions and bosons in usual symmetric product orbifold transform � � ⇒ ( T 4 ) N +1 /U ( N ) untwisted ⊂ ( T 4 ) N +1 /S N +1 � � � � untwisted

Two Orbifolds (Contd.) • Therefore: H (pert) = (0; Λ ) ⊗ (0; Λ ∗ ) ⊂ H (Sym . Prod . ) � M � � untwisted Λ • i.e. Vasiliev states are a closed subsector of the Symmetric Product CFT = Tensionless string theory. • More generally, states of the symmetric product CFT must transform in specific representations of the chiral algebra of the continuous orbifold (the U(N) invariant i.e. currents). W ∞ y ) = |Z vac ( q, y ) | 2 + ( q, y ) | 2 + |Z (U) |Z (T) X X ( q, y ) | 2 Z NS ( q, ¯ q, y, ¯ j β β j Other untwisted sectors Twisted sectors

Stringy Chiral Algebra • The vacuum sector ( invariant currents) can therefore be S N +1 organised in terms of coset ( ) representations - from the W ∞ untwisted sector of the continuous orbifold. X Z vac ( q, y ) = n ( Λ ) χ (0; Λ ) ( q, y ) Λ ∈ U ( N ) • Each such representation comes with a multiplicity which would be given by the number of times the singlet of S N +1 appears in the U(N) representation . Λ • A vast extension of - generators not just bilinear in W ∞ fermions/bosons but also cubic, quartic etc.

Reality Check • Explicitly verify this equality to low orders - use DMVV prescription to compute 2 y 2 + 12 + 2 y − 2 � 1 2 + 2 y + 2 y − 1 � � � Z vac ( q, y ) = 1 + q q 2 y 3 + 32 y + 32 y − 1 + 2 y − 3 � 3 � + q 2 2 y 4 + 52 y 2 + 159 + 52 y − 2 + 2 y − 4 � q 2 � + 2 y 5 + 62 y 3 + 426 y + 426 y − 1 + 62 y − 3 + 2 y − 5 � 5 � + q 2 2 y 6 + 64 y 4 + 767 y 2 + 1800 + 767 y − 2 + 64 y − 4 + 2 y − 6 � q 3 � + 7 2 ) . + O ( q

It Agrees! X 2 ψ i β ψ i α Vasiliev higher spin fields Additional higher spin generators : − 1 − 1 2 i Z vac ( q, y ) = χ (0;0) ( q, y ) + χ (0;[2 , 0 ,..., 0]) ( q, y ) + χ (0;[0 , 0 ,..., 0 , 2]) ( q, y ) + χ (0;[3 , 0 ,..., 0 , 0]) ( q, y ) + χ (0;[0 , 0 , 0 ,..., 0 , 3]) ( q, y ) + χ (0;[2 , 0 ,..., 0 , 1]) ( q, y ) + χ (0;[1 , 0 , 0 ,..., 0 , 2]) ( q, y ) + 2 · χ (0;[4 , 0 ,..., 0 , 0]) ( q, y ) + 2 · χ (0;[0 , 0 , 0 ,..., 0 , 4]) ( q, y ) + χ (0;[0 , 2 , 0 ,... 0 , 0]) ( q, y ) + χ (0;[0 , 0 ,... 0 , 2 , 0]) ( q, y ) + χ (0;[3 , 0 ,..., 0 , 1]) ( q, y ) + χ (0;[1 , 0 , 0 ,..., 0 , 3]) ( q, y ) + 2 · χ (0;[2 , 0 , 0 ,..., 0 , 2]) ( q, y ) + χ (0;[1 , 2 , 0 ,..., 0]) ( q, y ) + χ (0;[0 ,..., 0 , 2 , 1]) ( q, y ) + χ (0;[2 , 1 , 0 ,..., 0 , 1]) ( q, y ) + χ (0;[1 , 0 ,..., 0 , 1 , 2]) ( q, y ) + χ (0;[0 , 2 , 0 ,..., 0 , 1]) ( q, y ) + χ (0;[1 , 0 ,..., 0 , 2 , 0]) ( q, y ) + 3 · χ (0;[3 , 0 ,..., 0 , 2]) ( q, y ) +3 · χ (0;[2 , 0 ,..., 0 , 3]) ( q, y ) + χ (0;[1 , 1 , 0 ,..., 0 , 2]) ( q, y ) + χ (0;[2 , 0 ,..., 0 , 1 , 1]) ( q, y ) + χ (0;[0 , 0 , 2 , 0 ,..., 0]) ( q, y ) + χ (0;[0 ,..., 0 , 2 , 0 , 0]) ( q, y ) + 3 · χ (0;[0 , 2 , 0 ,..., 0 , 2]) ( q, y ) +3 · χ (0;[2 , 0 ,..., 0 , 2 , 0]) ( q, y ) + χ (0;[1 , 1 , 0 ,..., 0 , 1 , 1]) ( q, y ) + O ( q 7 / 2 ) .

Reality Check (Contd.) • Can do something similar for the simplest non-trivial untwisted sector - which contains 16 of the 20 marginal ops. Z (U) X ( q, y ) = n 1 ( Λ ) χ (0; Λ ) ( q, y ) 1 Λ Contains ψ i α Multiplicity of N dim. irrep of in Λ S N +1 − 1 2 • Compute LHS (2 y + 2 y − 1 ) q 1 / 2 + (5 y 2 + 16 + 5 y − 2 ) q 1 Z 1 ( q, y ) = + (6 y 3 + 58 y + 58 y − 1 + 6 y − 3 ) q 3 / 2 + (6 y 4 + 128 y 2 + 315 + 128 y − 2 + 6 y − 4 ) q 2 + (6 y 5 + 198 y 3 + 1030 y + 1030 y − 1 + 198 y − 3 + 6 y − 5 ) q 5 / 2 + (6 y 6 + 240 y 4 + 2290 y 2 + 4724 + 2290 y − 2 + 240 y − 4 + 6 y − 6 ) q 3 + O ( q 3 ) .