Rectangular W-algebras of types so and sp and dual coset CFTs Takahiro Uetoko (Ritsumeikan Univ.) Based on: [arXiv:1906.05872] w/ Thomas Creutzig (Alberta Univ.), Yasuaki Hikida (YITP, Kyoto Univ.) Aug. 22 (2019) @YITP “Strings and Fields 2019” 1
Introduction • Strings and Higher spins String theory Higher spin gravity First Regge trajectory Vasiliev theory Tensionless limit [Gross ’88] 2
Introduction • Strings and Higher spins String theory Higher spin gravity First Regge trajectory Vasiliev theory Tensionless limit [Gross ’88] - String spectrum First Regge trajectory ( mass ) 2 ( mass ) 2 Vasiliev theory ( spin ) ( spin ) 3
Introduction • Strings and Higher spins String theory Higher spin gravity How to explain First Regge trajectory Vasiliev theory the higher Regge trajectories? Tensionless limit [Gross ’88] - String spectrum First Regge trajectory ( mass ) 2 ( mass ) 2 Vasiliev theory ( spin ) ( spin ) 4
Introduction • Strings and Extended higher spins Higher spin gravity String theory Vasiliev theory All Regge trajectory with � matrix valued fields M × M 5
Introduction • Strings and Extended higher spins Higher spin gravity String theory Vasiliev theory All Regge trajectory with � matrix valued fields M × M • Matrix extension of 3d Prokushkin-Vasiliev theory may be analyzed with the infinite dimensional symmetry of 2d CFT [Creutzig-Hikida ’13] 6
Introduction • Strings and Extended higher spins Higher spin gravity String theory Vasiliev theory All Regge trajectory with � matrix valued fields M × M • Matrix extension of 3d Prokushkin-Vasiliev theory may be analyzed with the infinite dimensional symmetry of 2d CFT [Creutzig-Hikida ’13] - Dual model is 2d Grassmannian-like coset With � , this reduce to su ( N + M ) k M = 1 the Gaberdiel-Gopakumar duality su ( N ) k ⊕ u (1) kNM ( N + M ) [Gaberdiel-Gopakumar ’10] - Evidence: spectrum, asymptotic symmetry, … [Creutzig-Hikida-Rønne ’13, Creutzig-Hikida ’18] 7
Introduction • Strings and Extended higher spins Can we generalize Higher spin gravity String theory Vasiliev theory All Regge trajectory this analysis to other models? with � matrix valued fields M × M • Matrix extension of 3d Prokushkin-Vasiliev theory may be analyzed with the infinite dimensional symmetry of 2d CFT [Creutzig-Hikida ’13] - Dual model is 2d Grassmannian-like coset With � , this reduce to su ( N + M ) k M = 1 the Gaberdiel-Gopakumar duality su ( N ) k ⊕ u (1) kNM ( N + M ) [Gaberdiel-Gopakumar ’10] - Evidence: spectrum, asymptotic symmetry, … [Creutzig-Hikida-Rønne ’13, Creutzig-Hikida ’18] 8
Introduction • Our question and summary Can we generalize this analysis to other models? An answer (my talk) • We consider 2 ways to truncate the DOF - Restricted matrix extensions; � so ( M ), sp ( M ) - Even spin truncation of � hs [ λ ] • We propose the dual coset model and examine the asymptotic symmetry 9
Plan of talk 1. Introduction 2. HS gravity with � gauge sector sl ( M ) 3. Some generalization for extended HS gravity 4. Summary 10
Plan of talk 1. Introduction 2. HS gravity with � gauge sector sl ( M ) 3. Some generalization for extended HS gravity 4. Summary 11
HS gravity with � gauge sector sl ( M ) • Chern-Simons description of HS gravity - 3d gravity: � CS theory sl (2) [Witten ’88] 12
� HS gravity with � gauge sector sl ( M ) • Chern-Simons description of HS gravity - 3d gravity: � CS theory sl (2) [Witten ’88] hs [ λ ] = B [ λ ] ⊖ 1 Include higher spin U ( sl (2)) B [ λ ] = ⟨ C 2 − 1 4 ( λ 2 − 1) 1 ⟩ - 3d HS gravity: [Prokushkin-Vasiliev ’98] hs [ λ ] 13
� � HS gravity with � gauge sector sl ( M ) • Chern-Simons description of HS gravity - 3d gravity: � CS theory sl (2) [Witten ’88] hs [ λ ] = B [ λ ] ⊖ 1 Include higher spin U ( sl (2)) B [ λ ] = ⟨ C 2 − 1 4 ( λ 2 − 1) 1 ⟩ - 3d HS gravity: [Prokushkin-Vasiliev ’98] CS theory with gravitational � hs [ λ ] sl ( n ) sl (2) [ � � ] λ = n ( n = 2,3,…) Ex) principal embedding sl ( n ) = sl (2) ⊕ ( ) n ⨁ g ( s ) s =3 spin-(s-1) representation 14
� � HS gravity with � gauge sector sl ( M ) • Chern-Simons description of HS gravity - 3d gravity: � CS theory sl (2) [Witten ’88] hs [ λ ] = B [ λ ] ⊖ 1 Include higher spin U ( sl (2)) B [ λ ] = ⟨ C 2 − 1 4 ( λ 2 − 1) 1 ⟩ - 3d HS gravity: [Prokushkin-Vasiliev ’98] CS theory with gravitational � hs [ λ ] sl ( n ) sl (2) [ � � ] λ = n ( n = 2,3,…) Ex) principal embedding Matrix extension sl ( n ) = sl (2) ⊕ ( ) n ⨁ g ( s ) [Gaberdiel-Gopakumar ’13, Creutzig-Hikida-Rønne ’13] - 3d HS gravity with � fields: M × M s =3 spin-(s-1) representation hs M [ λ ] ≃ gl ( M ) ⊗ B [ λ ] ⊖ 1 M ⊗ 1 ≃ sl ( M ) ⊗ 1 ⊕ 1 M ⊗ hs [ λ ] ⊕ sl ( M ) ⊗ hs [ λ ] 15
� � HS gravity with � gauge sector sl ( M ) • Chern-Simons description of HS gravity - 3d gravity: � CS theory sl (2) [Witten ’88] hs [ λ ] = B [ λ ] ⊖ 1 Include higher spin U ( sl (2)) B [ λ ] = ⟨ C 2 − 1 4 ( λ 2 − 1) 1 ⟩ - 3d HS gravity: [Prokushkin-Vasiliev ’98] CS theory with gravitational � hs [ λ ] sl ( n ) sl (2) [ � � ] λ = n ( n = 2,3,…) Ex) principal embedding Matrix extension sl ( n ) = sl (2) ⊕ ( ) n ⨁ g ( s ) [Gaberdiel-Gopakumar ’13, Creutzig-Hikida-Rønne ’13] - 3d HS gravity with � fields: M × M s =3 spin-(s-1) representation hs M [ λ ] ≃ gl ( M ) ⊗ B [ λ ] ⊖ 1 M ⊗ 1 ≃ sl ( M ) ⊗ 1 ⊕ 1 M ⊗ hs [ λ ] ⊕ sl ( M ) ⊗ hs [ λ ] sl ( M ) ⊗ 1 n ⊕ 1 M ⊗ sl ( n ) ⊕ sl ( M ) ⊗ sl ( n ) ≃ sl ( Mn ) [ � ] λ = n Include gravitational � sl (2) 16
� � � HS gravity with � gauge sector sl ( M ) • Chern-Simons description of HS gravity - 3d gravity: � CS theory sl (2) [Witten ’88] hs [ λ ] = B [ λ ] ⊖ 1 We examine � CS theory Include higher spin sl ( Mn ) U ( sl (2)) B [ λ ] = ⟨ C 2 − 1 4 ( λ 2 − 1) 1 ⟩ - 3d HS gravity: [Prokushkin-Vasiliev ’98] decomposed as CS theory with gravitational � hs [ λ ] sl ( n ) sl (2) [ � � ] λ = n ( n = 2,3,…) Ex) principal embedding Matrix extension sl ( n ) = sl (2) ⊕ ( sl ( Mn ) ≃ sl ( M ) ⊗ 1 n ⊕ 1 M ⊗ sl ( n ) ⊕ sl ( M ) ⊗ sl ( n ) ) n ⨁ g ( s ) [Gaberdiel-Gopakumar ’13, Creutzig-Hikida-Rønne ’13] - 3d HS gravity with � fields: M × M s =3 spin-(s-1) representation hs M [ λ ] ≃ gl ( M ) ⊗ B [ λ ] ⊖ 1 M ⊗ 1 ≃ sl ( M ) ⊗ 1 ⊕ 1 M ⊗ hs [ λ ] ⊕ sl ( M ) ⊗ hs [ λ ] sl ( M ) ⊗ 1 n ⊕ 1 M ⊗ sl ( n ) ⊕ sl ( M ) ⊗ sl ( n ) ≃ sl ( Mn ) [ � ] λ = n Include gravitational � sl (2) 17
HS gravity with � gauge sector sl ( M ) • Gauge field (Solution of EOM) - 3d gravity: A = e − ρ L 0 a ( z ) e ρ L 0 dz + L 0 d ρ z , ¯ z ρ → ∞ 18
HS gravity with � gauge sector sl ( M ) • Gauge field (Solution of EOM) - 3d gravity: A = e − ρ L 0 a ( z ) e ρ L 0 dz + L 0 d ρ z , ¯ z Include higher spin - 3d HS gravity: A = e − ρ V 2 0 a ( z ) e ρ V 2 0 dz + V 2 0 d ρ ρ → ∞ Ex) � case of � V s s = 2 − s +1, ⋯ , s − 1 V 2 0,±1 ≡ L 0,±1 19
HS gravity with � gauge sector sl ( M ) • Gauge field (Solution of EOM) - 3d gravity: A = e − ρ L 0 a ( z ) e ρ L 0 dz + L 0 d ρ z , ¯ z Include higher spin - 3d HS gravity: A = e − ρ V 2 0 a ( z ) e ρ V 2 0 dz + V 2 0 d ρ ρ → ∞ Matrix extension Ex) � case of � V s s = 2 − s +1, ⋯ , s − 1 V 2 0,±1 ≡ L 0,±1 - 3d HS gravity with � fields: M × M A = e − ρ ( 1 M ⊗ V 2 0 ) a ( z ) e ρ ( 1 M ⊗ V 2 0 ) dz + ( 1 M ⊗ V 2 0 ) d ρ 20
HS gravity with � gauge sector sl ( M ) • Asymptotically AdS condition Asymptotically AdS ( A − A AdS ) | ρ →∞ = 𝒫 (( e ρ ) 0 ) - 3d gravity: A = e − ρ L 0 a ( z ) e ρ L 0 dz + L 0 d ρ - 3d HS gravity: A = e − ρ V 2 0 a ( z ) e ρ V 2 0 dz + V 2 0 d ρ - 3d HS gravity with � fields: M × M A = e − ρ ( 1 M ⊗ V 2 0 ) a ( z ) e ρ ( 1 M ⊗ V 2 0 ) dz + ( 1 M ⊗ V 2 0 ) d ρ 21
HS gravity with � gauge sector sl ( M ) • Asymptotically AdS condition Asymptotically AdS ( A − A AdS ) | ρ →∞ = 𝒫 (( e ρ ) 0 ) - 3d gravity: A = e − ρ L 0 a ( z ) e ρ L 0 dz + L 0 d ρ a ( z ) = L 1 + 1 T ( z ) L − 1 k CS Virasoro generator - 3d HS gravity: A = e − ρ V 2 0 a ( z ) e ρ V 2 0 dz + V 2 0 d ρ - 3d HS gravity with � fields: M × M A = e − ρ ( 1 M ⊗ V 2 0 ) a ( z ) e ρ ( 1 M ⊗ V 2 0 ) dz + ( 1 M ⊗ V 2 0 ) d ρ 22
� HS gravity with � gauge sector sl ( M ) • Asymptotically AdS condition Asymptotically AdS ( A − A AdS ) | ρ →∞ = 𝒫 (( e ρ ) 0 ) - 3d gravity: A = e − ρ L 0 a ( z ) e ρ L 0 dz + L 0 d ρ a ( z ) = L 1 + 1 T ( z ) L − 1 k CS Virasoro generator - 3d HS gravity: W N generators A = e − ρ V 2 0 a ( z ) e ρ V 2 0 dz + V 2 0 d ρ n ∑ a ( z ) = V 2 W ( s ) ( z ) V s 1 + − s +1 s =2 - 3d HS gravity with � fields: M × M A = e − ρ ( 1 M ⊗ V 2 0 ) a ( z ) e ρ ( 1 M ⊗ V 2 0 ) dz + ( 1 M ⊗ V 2 0 ) d ρ 23
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