SMOOTH SOLUTIONS IN VASILIEV THEORY Andrea Campoleoni Universit - PowerPoint PPT Presentation

SMOOTH SOLUTIONS IN “VASILIEV THEORY” Andrea Campoleoni Université Libre de Bruxelles & International Solvay Institutes A.C., T. Procházka, J. Raeymaekers, 1303.0880 Workshop on “Higher Spins, Strings and Duality”, Galileo Galilei Institute, Firenze, 7/5/2013

GRAVITY IN D = 2+1 Einstein-Hilbert action 1 e a ∧ R bc + 1 3 4 ⁄ 3 l 2 e a ∧ e b ∧ e c I = � abc 16 ⇥ G Field equations b + 1 d ⇤ ab + ⇤ ac ∧ ⇤ c l 2 e a ∧ e b = 0 ← constant curvature! R ab ≡ l de a + ⇤ a b ∧ e b = 0 T a ≡ Rewriting in terms of the metric 1 d 3 x √− g R + 2 3 4 ⁄ g µ ν = ⇥ ab e a µ e b I = ⇒ ν l 2 16 ⇤ G 2

GRAVITY IN D = 2+1 Einstein-Hilbert action 1 e a ∧ R bc + 1 3 4 ⁄ 3 l 2 e a ∧ e b ∧ e c I = � abc 16 ⇥ G A couple of useful tricks... a = 1 2 ✏ a b,c . ! µ bc ! µ so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R) Rewriting in terms of the metric 1 d 3 x √− g R + 2 3 4 ⁄ g µ ν = ⇥ ab e a µ e b I = ⇒ ν l 2 16 ⇤ G 2

GRAVITY IN D = 2+1 � � Einstein-Hilbert action Z ✓ ◆ 1 e a ∧ R a + 1 6 l 2 ✏ abc e a ∧ e b ∧ e c I = 8 ⇡ G A couple of useful tricks... a = 1 2 ✏ a b,c . ! µ bc ! µ so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R) Rewriting in terms of the metric 1 d 3 x √− g R + 2 3 4 ⁄ g µ ν = ⇥ ab e a µ e b I = ⇒ ν l 2 16 ⇤ G 2

GRAVITY IN D = 2+1 Einstein-Hilbert action Achúcarro, Townsend (1986); Witten (1988) ( e = e a J a ✓ ◆ Z 1 e ∧ R + 1 with I = 3 l 2 e ∧ e ∧ e tr ! = ! a J a 16 ⇡ G A couple of useful tricks... a = 1 2 ✏ a b,c . ! µ bc ! µ so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R) Rewriting in terms of the metric 1 d 3 x √− g R + 2 3 4 ⁄ g µ ν = ⇥ ab e a µ e b I = ⇒ ν l 2 16 ⇤ G 2

GRAVITY IN D = 2+1 Einstein-Hilbert action Achúcarro, Townsend (1986); Witten (1988) ( e = e a J a ✓ ◆ Z 1 e ∧ R + 1 with I = 3 l 2 e ∧ e ∧ e tr ! = ! a J a 16 ⇡ G A couple of useful tricks... a = 1 2 ✏ a b,c . ! µ bc ! µ so(2,2) ≃ so(1,2) ⊕ so(1,2) ≃ sl(2,R) ⊕ sl(2,R) e e Chern-Simons formulation on AdS 3 ⇣ ⌘ ⇣ ⌘ ⇤ = 1 e = l A − e A + e A A 2 2 2

HIGHER SPINS IN D = 2+1 Natural generalization of the gravity frame action Blencowe (1989) ✓ ◆ Z 1 e ∧ R + 1 with R = d ! + ! ∧ ! I = 3 l 2 e ∧ e ∧ e tr 16 ⇡ G For g = sl(N,R) describes fields of “spin” 2,3,...,N A J A dx µ = ab T ab + · · · 1 2 a J a + e µ dx µ e = e µ e µ A J A dx µ = ab T ab + · · · 1 2 a J a + ⌅ µ dx µ ⌅ = ⌅ µ ⌅ µ Example: the sl(3,R) algebra [ J a , J b ] = � abc J c [ J a , T bc ] = � m a ( b T c ) m J m � � [ T ab , T cd ] = ⇤ ⇥ a ( c � d ) bm + ⇥ b ( c � d ) am 3

HIGHER SPINS IN D = 2+1 Natural generalization of the gravity frame action Blencowe (1989) ✓ ◆ Z 1 e ∧ R + 1 with R = d ! + ! ∧ ! I = 3 l 2 e ∧ e ∧ e tr 16 ⇡ G For g = sl(N,R) describes fields of “spin” 2,3,...,N A J A dx µ = ab T ab + · · · 1 2 a J a + e µ dx µ e = e µ e µ A J A dx µ = ab T ab + · · · 1 2 a J a + ⌅ µ dx µ ⌅ = ⌅ µ ⌅ µ More in general: take any Lie algebra g with a non-degenerate Killing form and branch it under the adjoint action of sl(2,R) ↪ g ⇤ ⌅ ⇧ ⌥ g ( ⌅ ,a ) ⌃ g = sl (2 , R ) ⊕ dim = 2 ` + 1 ⌅ , a 3

⇒ HIGHER SPINS IN D = 2+1 Simple characterization in terms of Chern-Simons theories (for gauge fields) � � � � e = l , ω = 1 A − � A + � S = S CS [ A ] − S CS [ ⌃ A A A ] 2 2 Field equations → flatness conditions F = d e e A + e A ∧ e F = dA + A ∧ A = 0 . A Simple field equations, but rich space of solutions on AdS Non-trivial topology → black holes Gutperle, Kraus (2011) Gaberdiel, Gopakumar Boundary conditions → boundary dynamics, AdS/CFT... (2010) How to select “non-singular” solutions? Gutperle, Kraus (2011) Castro, Gopakumar, Gutperle, Raeymaekers (2011) 4

OUTLINE Coupling ∞ spins: hs[ λ ] Chern-Simons theories Smoothness criteria for asymptotically AdS solutions “Analytic continuation” of the sl(N) conical surpluses Conclusion

THE GAUGE SECTOR OF THE PROKUSHKIN-VASILIEV MODEL

⇒ ⇒ FRAME-LIKE DESCRIPTION FOR HS HS “vielbeins” and “spin connections” e µa 1 ... a s − 1 ω µb,a 1 ... a s − 1 Everything is traceless, then in D=2+1... a = 1 ≈ (example: ) 2 ✏ a b,c ! µ bc ! µ “Vielbeins” and “spin connections” have the same structure Structure of the higher-spin generators: e ab... traceless ⇒ T ab... traceless in ab... e ab... irreducible ⇒ [ J a , T b 1 ... b s − 1 ] = � m a ( b 1 T b 2 ... b s − 1 ) m 7

SL(N) HIGHER-SPIN THEORIES For sl(3,R) the Jacobi identity fixes the algebra but... J ( a J b ) − 2 ⇣ 3 η ab J c J c ⌘ √ ⇒ [ J a , T bc ] = � m a ( b T c ) m , T ab = − σ � 3-dim repr. for ⇒ J m � � [ T ab , T cd ] = ⇤ ⇥ a ( c � d ) bm + ⇥ b ( c � d ) am J a Consider traceless and symmetric polynomials in J a (+ traceless projection in the a n indices) T a 1 ... a s ∼ J ( a 1 . . . J a s ) N-dim repr. for ⇒ N 2 - 1ind. traceless matrices out of T’s with s<N J a Hoppe (1982) General lesson to build higher-spin algebras: choose a representation of so(1,2) ≃ sl(2,R) and compute products of the representatives 8

HIGHEST WEIGHT IRREPS OF SL(2,R) sl(2,R) algebra: [ J + , J − ] = 2 J 0 , [ J ± , J 0 ] = ± J ± 0 − 1 2( J + J − + J − J + ) = 1 Casimir: 4( λ 2 − 1) C 2 = J 2 λ≠ N ⇒ two pairs of conjugate irreps Realize the generators as e.g. ✓ ◆ J 0 = 1 J + = y @ x @ @ x − y @ J − = − x @ @ x , , @ y . 2 @ y and act on v i = x i y λ − i − 1 , v i = x λ − i − 1 y i , ¯ w i = x i y − ( λ + i +1) , w i = x − ( λ + i +1) y i , ¯ 9

HIGHEST WEIGHT IRREPS OF SL(2,R) sl(2,R) algebra: [ J + , J − ] = 2 J 0 , [ J ± , J 0 ] = ± J ± 0 − 1 2( J + J − + J − J + ) = 1 Casimir: 4( λ 2 − 1) C 2 = J 2 λ≠ N ⇒ two pairs of conjugate irreps Realize the generators as e.g. ✓ ◆ J 0 = 1 J + = y @ x @ @ x − y @ J − = − x @ @ x , , @ y . 2 @ y and act on v i = x i y λ − i − 1 , v i = x λ − i − 1 y i , ¯ w i = x i y − ( λ + i +1) , w i = x − ( λ + i +1) y i , ¯ 9

A FAITHFUL MATRIX REPR. OF HS[ ] λ ( J + ) jk = δ j, k +1 , Irrep of sl(2,R) with highest weight : ( J − ) jk = j ( j − λ ) δ j +1 , k , t 1 2 ( λ − 1). ( J 0 ) jk = 1 2( λ + 1 − 2 j ) δ j, k , Building the hs[ λ ] generators: Pope, Romans, Shen (1990) m = ( − 1) ℓ − m ( ℓ + m )! � � , ( J + ) ℓ ]] T ℓ J − , . . . [ J − , [ J − (2 ℓ )! � �� � ℓ − m terms Explicit realization: A.C., Procházka, Raeymaekers (2013) � [ ℓ ] n ℓ − m � ℓ − m � ( T ℓ m ) jk = ( − 1) ℓ − m [ ℓ − λ ] n [ j − m − 1 ] ℓ − m − n δ j, k + m , [ 2 ℓ ] n n n = 0 10

A HS[ ] CHERN-SIMONS THEORY λ The satisfy T ℓ [ J i , T ` m ] = ( i ` − m ) T ` m m + i N 6 Trace: � tr v = λ ( λ 2 − 1) lim v jj , N → λ j = 1 Chern-Simons theory with hs[ λ ] ⊕ hs[ λ ] gauge algebra as a model for the interactions of spins 2,..., ∞ Bergshoeff, Blencowe, Stelle (1990); Vasiliev (1991) Field equations: F = d ¯ ¯ A + ¯ A ∧ ¯ F = dA + A ∧ A = 0 . A = 0 11

A HS[ ] CHERN-SIMONS THEORY λ The satisfy T ℓ [ J i , T ` m ] = ( i ` − m ) T ` m m + i N 6 Trace: � tr v = λ ( λ 2 − 1) lim v jj , N → λ j = 1 Chern-Simons theory with hs[ λ ] ⊕ hs[ λ ] gauge algebra as a model for the interactions of spins 2,..., ∞ Bergshoeff, Blencowe, Stelle (1990); Vasiliev (1991) Field equations: F = d ¯ ¯ A + ¯ A ∧ ¯ F = dA + A ∧ A = 0 . A = 0 What are the “admissible” connections? 11

PROPERTIES OF THE HS[ ] MATRICES λ Properties of the hs[ λ ] matrices: m The non-zero elements belong to the diagonal ts ( T ℓ m ) j, j − m m ts ( T ℓ The are polynomials in j m ) j, j − m Is that enough? What kinds of linear combinations do we have to consider? What are their properties? 12

PROPERTIES OF THE HS[ ] MATRICES λ Properties of the hs[ λ ] matrices: Khesin, Malikov (1996) ∃ N such that if j > k+N at v j, k = 0 The matrix elements along a diagonal, for some fixed n, l, v j, j + n become polynomial in j for sufficiently large j 12

PROPERTIES OF THE HS[ ] MATRICES λ Properties of the hs[ λ ] matrices: Khesin, Malikov (1996) ∃ N such that if j > k+N at v j, k = 0 The matrix elements along a diagonal, for some fixed n, l, v j, j + n become polynomial in j for sufficiently large j N The trace is still well defined X tr v ∼ lim v jj N → λ j = 1 12

PROPERTIES OF THE HS[ ] MATRICES λ Properties of the hs[ λ ] matrices: Khesin, Malikov (1996) ∃ N such that if j > k+N at v j, k = 0 The matrix elements along a diagonal, for some fixed n, l, v j, j + n become polynomial in j for sufficiently large j N The trace is still well defined X tr v ∼ lim v jj N → λ j = 1 1) lim One can perform the substitution provided N → λ that is large enough + N . 12

SMOOTH ASYMPTOTICALLY ADS SOLUTIONS