Higher spins in 3D: going from AdS to flat Andrea Campoleoni Université Libre de Bruxelles and International Solvay Institutes based on work with H.A. González, B. Oblak and M. Riegler arXiv:1512.03353 & arXiv:1603.03812 Workshop on Topics in Three Dimensional Gravity, ICTP Trieste, 24/3/2016
Higher spins in 3D: (Higher-spin) BMS modules in 3D going from AdS to flat Andrea Campoleoni Université Libre de Bruxelles and International Solvay Institutes based on work with H.A. González, B. Oblak and M. Riegler arXiv:1512.03353 & arXiv:1603.03812 Workshop on Topics in Three Dimensional Gravity, ICTP Trieste, 24/3/2016
BMS symmetry B ondi- M etzner- S achs group = asymptotic symmetries at null ∞ of asymptotically flat gravity Bondi, van der Burg, Metzner; Sachs (1962) Barnich, Oblak (2014) Garbarz, Leston (2015) Bagchi, Gopakumar, Mandal, Miwa (2010) Grumiller, Riegler, Rosseel (2014)
BMS symmetry B ondi- M etzner- S achs group = asymptotic symmetries at null ∞ S e e C o m p è r e ’ s t a l k of asymptotically flat gravity ! Bondi, van der Burg, Metzner; Sachs (1962) Barnich, Oblak (2014) Garbarz, Leston (2015) Bagchi, Gopakumar, Mandal, Miwa (2010) Grumiller, Riegler, Rosseel (2014)
BMS symmetry B ondi- M etzner- S achs group = asymptotic symmetries at null ∞ S e e C o m p è r e ’ s t a l k of asymptotically flat gravity ! Bondi, van der Burg, Metzner; Sachs (1962) Nice symmetry, but what about the quantum regime? (Unitary) representations of local BMS? Barnich, Troessaert (2009) Barnich, Oblak (2014) Induced representations Garbarz, Leston (2015) Bagchi, Gopakumar, Mandal, Miwa (2010) Limit of CFT representations Grumiller, Riegler, Rosseel (2014)
BMS symmetry B ondi- M etzner- S achs group = asymptotic symmetries at null ∞ S e e C o m p è r e ’ s t a l k of asymptotically flat gravity ! Bondi, van der Burg, Metzner; Sachs (1962) Nice symmetry, but what about the quantum regime? (Unitary) representations of local BMS? Barnich, Troessaert (2009) 3 Barnich, Oblak (2014) Induced representations Garbarz, Leston (2015) Bagchi, Gopakumar, Mandal, Miwa (2010) Limit of CFT representations Grumiller, Riegler, Rosseel (2014)
BMS symmetry B ondi- M etzner- S achs group = asymptotic symmetries at null ∞ S e e C o m p è r e ’ s t a l k of asymptotically flat gravity ! Bondi, van der Burg, Metzner; Sachs (1962) Nice symmetry, but what about the quantum regime? (Unitary) representations of local BMS? Barnich, Troessaert (2009) 3 Barnich, Oblak (2014) Induced representations Garbarz, Leston (2015) Bagchi, Gopakumar, Mandal, Miwa (2010) Limit of CFT representations Grumiller, Riegler, Rosseel (2014) g i e o N T . b y r t e o s o p l s a e e S
Why D = 3? And why higher spins? Motivation I: beauty In D = 3 the local BMS group is an Inonu-Wigner contraction of the AdS 3 local conformal symmetry at spatial infinity Brown, Henneaux (1986)
Why D = 3? And why higher spins? Motivation I: beauty In D = 3 the local BMS group is an Inonu-Wigner contraction of the AdS 3 local conformal symmetry at spatial infinity Brown, Henneaux (1986) Motivation II: …and the beast Several ways to obtain BMS as a limit of conformal symmetry: are they all equivalent? Henneaux, Rey; A.C., Pfenninger, Higher-spin fields → non-linear W algebras Fredenhagen, Theisen (2010) Extension of the symmetry → more control over the flat limit!
Why D = 3? And why higher spins? Motivation I: beauty In D = 3 the local BMS group is an Inonu-Wigner contraction of the AdS 3 local conformal symmetry at spatial infinity Brown, Henneaux (1986) Motivation II: …and the beast Several ways to obtain BMS as a limit of conformal symmetry: are they all equivalent? Henneaux, Rey; A.C., Pfenninger, Higher-spin fields → non-linear W algebras Fredenhagen, Theisen (2010) Extension of the symmetry → more control over the flat limit! techniques that may be useful in D = 4?
Asymptotic symmetries in flat space Asymptotic symmetries at spatial infinity in AdS 3 Brown, Henneaux (1986) ( m � n ) L m + n + c 12 m ( m 2 � 1) � m + n, 0 [ L m , L n ] = L m + n + ¯ c 12 m ( m 2 � 1) � m + n, 0 [ ¯ L m , ¯ ( m � n ) ¯ L n ] =
Asymptotic symmetries in flat space Asymptotic symmetries at spatial infinity in AdS 3 Brown, Henneaux (1986) ( m � n ) L m + n + c 12 m ( m 2 � 1) � m + n, 0 [ L m , L n ] = L m + n + ¯ c 12 m ( m 2 � 1) � m + n, 0 [ ¯ L m , ¯ ( m � n ) ¯ L n ] = Define new generators and central charges c 1 = c � ¯ c , P m ⌘ 1 L m + ¯ J m ⌘ L m � ¯ � � L − m L − m , c 2 = c + ¯ c ` `
Asymptotic symmetries in flat space Asymptotic symmetries at spatial infinity in AdS 3 Brown, Henneaux ∈ (1986) ( m − n ) J m + n + c 1 12 m ( m 2 − 1) δ m + n, 0 , [ J m , J n ] = ( m − n ) P m + n + c 2 12 m ( m 2 − 1) δ m + n, 0 , [ J m , P n ] = ` − 2 ( · · · ) [ P m , P n ] = 0 , Define new generators and central charges c 1 = c � ¯ c , P m ⌘ 1 L m + ¯ J m ⌘ L m � ¯ � � L − m L − m , c 2 = c + ¯ c ` `
Asymptotic symmetries in flat space Asymptotic symmetries at spatial infinity in AdS 3 null infinity in Minkowski 3 ∈ ( m − n ) J m + n + c 1 12 m ( m 2 − 1) δ m + n, 0 , [ J m , J n ] = it ` ! 1 ( m − n ) P m + n + c 2 12 m ( m 2 − 1) δ m + n, 0 , [ J m , P n ] = the bms 3 m [ P m , P n ] = 0 , Define new generators and central charges c 1 = c � ¯ c , P m ⌘ 1 L m + ¯ J m ⌘ L m � ¯ � � L − m L − m , c 2 = c + ¯ c ` `
Asymptotic symmetries in flat space Asymptotic symmetries at spatial infinity in AdS 3 null infinity in Minkowski 3 ∈ ( m − n ) J m + n + c 1 12 m ( m 2 − 1) δ m + n, 0 , [ J m , J n ] = it ` ! 1 ( m − n ) P m + n + c 2 12 m ( m 2 − 1) δ m + n, 0 , [ J m , P n ] = the bms 3 m [ P m , P n ] = 0 , Define new generators and central charges c 1 = c � ¯ c , P m ⌘ 1 L m + ¯ J m ⌘ L m � ¯ � � L − m L − m , c 2 = c + ¯ c ` ` Same result directly from flat gravity Barnich, Compere (2007) Everything extends to higher spins Afshar, Bagchi, Fareghbal, Grumiller, Rosseel; Gonzalez, Matulich, Pino, Troncoso (2013)
Outline The bms 3 algebra and its unitary irreps Ultrarelativistic vs Galilean limits of CFT Higher spins Characters & partition functions
Outline The bms 3 algebra and its unitary irreps Ultrarelativistic vs Galilean limits of CFT Ultrarelativistic vs Galilean limits of CFT Higher spins Higher spins Characters & partition functions Characters & partition functions
The bms 3 algebra The centrally extended bms 3 algebra ( m ∈ Z ) w ∈ ( m − n ) J m + n + c 1 12 m ( m 2 − 1) δ m + n, 0 , [ J m , J n ] = ( m − n ) P m + n + c 2 12 m ( m 2 − 1) δ m + n, 0 , [ J m , P n ] = [ P m , P n ] = 0 , c 2 plays an important role in representation theory e c 2 = 3 and doesn’t vanish in gravity: G bms . Simi
The bms 3 algebra The Poincaré subalgebra ( m = − 1 , 0 , 1) ∈ ( m − n ) J m + n + c 1 12 m ( m 2 − 1) δ m + n, 0 , ← Lorentz [ J m , J n ] = ( m − n ) P m + n + c 2 12 m ( m 2 − 1) δ m + n, 0 , [ J m , P n ] = [ P m , P n ] = 0 , P m → translations; J 1 and J -1 → boosts; J 0 → rotations
How to build representations of bms 3 ?
How to build representations of bms 3 ? Poincaré is a subalgebra…
How to build representations of bms 3 ? Poincaré is a subalgebra…
How to build representations of bms 3 ? It is a contraction Poincaré is a of the 2D local subalgebra… conformal algebra
How to build representations of bms 3 ? It is a contraction Poincaré is a of the 2D local subalgebra… conformal algebra
Poincaré unitary irreps in a nutshell Irreps of Poincaré group classified by orbits of momenta fy p 2 = � M 2 ta p µ all that satisfy for some mass ass M P 0 gives the energy and P 1 ,P -1 commute with it build a basis of eigenstates of momentum: y | p µ , s i . All plane waves can be obtained from a given one via U ( Λ ) | p µ , s i = e is θ | Λ µ ν p ν , s i , U ( ω ) = exp [ i ( ω J 1 + ω ∗ J − 1 )] is a unitary operator
Rest-frame state & Poincaré modules Massive representations 3 m k µ = ( M, 0 , 0) Representative for the momentum orbit choose as The corresponding plane wave satisfies y | M, s i P 0 | M, s i = M | M, s i , P − 1 | M, s i = P 1 | M, s i = 0 , J 0 | M, s i = s | M, s i choose as is annihilated by all P n aside P 0 ! y | M, s i
Rest-frame state & Poincaré modules Massive representations 3 m k µ = ( M, 0 , 0) Representative for the momentum orbit choose as The corresponding plane wave satisfies y | M, s i P 0 | M, s i = M | M, s i , P − 1 | M, s i = P 1 | M, s i = 0 , J 0 | M, s i = s | M, s i choose as ! o f n i e h t e v a S is annihilated by all P n aside P 0 ! y | M, s i
Rest-frame state & Poincaré modules Rest-frame state : P 0 | M, s i = M | M, s i , P − 1 | M, s i = P 1 | M, s i = 0 , J 0 | M, s i = s | M, s i choose as Irreps of the Poincaré algebra built upon y | M, s i Basis of the representation space: | k, l i = ( J − 1 ) k ( J 1 ) l | M, s i P n and J n act linearly on these states Irreducible? Yes, Casimirs commute with all J n Unitary? Change basis! | p µ , s i = U ( Λ ) | M, s i h p µ , s | q µ , s i = � µ ( p, q )
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