Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins

Transcription

1 Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson Chalmers University of Technology, Gothenburg Talk at "Miami 2013", Fort Lauderdale, Fl December 13, 2013 Talk based on: "Towards an exact frame formulation of conformal higher spins in three dimensions", arxiv:1312.xxxx Critical solutions in topologically gauged N = 8 CFTs in three dimensions, arxiv: "Topologically gauged superconformal Chern-Simons matter theories" with Ulf Gran, Jesper Greitz and Paul Howe, arxiv: in JHEP "Aspects of topologically gauged M2-branes with six supersymmetries: towards a "sequential AdS/CFT"?, arxiv: [hep-th]

2 Background for the discussion: classical Two examples of classical CFTs in three dimensions: with N = 8 supersymmetries: BLG: [Bagger, Lambert] [Gustavsson] superconformal Chern-Simons (CS)-matter theory [Schwarz] only with SO(4) = SU(2) SU(2) gauge group (i.e. N = 2) parity symmetric no U(1) factor as a superconformal theory in 3d, any level k is possible relation to stacks of M2-branes tricky reducing to 6 susy s the M2-brane connection is clear [ABJM] ABJ(M) is a quiver theory with gauge groups like U k (N) U k (N), for any k and any N SU k (N) SU k (N), for any k and any N (for N = 2 and k = 1, 2 classically equivalent to BLG) SU k (M) SU k (N) U(1), for any k and any M, N Sp(N) U(1), for any N

3 Background for the discussion: quantum The (non-perturbative) quantum picture for N = 8 is better: monopole operators [ABJM, BKKS] can lead to enhanced symmetries for ABJM theories and relations to BLG This can be checked by comparing moduli spaces [Lambert et al] or partition functions and superconformal indices [Kapustin et al]: supersymmetry enhancement: Ex: ABJM U k (N) U k (N) has 2 extra susy s for k = 1, 2 U(1) enhancement parity enhancement

4 Motivation: some questions Questions at the classical level: why is classical BLG restricted to only SO(4)? can CS(supergravity) help? (recall the role of CS(gauge)) what would such a CS-gravity construction (=topological gauging) mean in M/string theory? in AdS/CFT (spin 2)? in the context of AdS/CFT and HS (higher spin)? 3d bosonization? [Chang et al],[aharony et al] using a set of boundary non-parity symmetric CS/matter theories to connect the A and B models (Klebanov, Polyakov (2002) and Sezgin, Sundell (2005)) in Vasiliev s 4d HS-theory all-spin anomaly cancelations in AdS/CFT [Giombi et al (2013)],[Giombi-Klebanov(2013)],[Tseytlin(2013)] also in the 2d W N context [Pope(1991)],[Bergshoeff et al(1991)]

5 Outline and summary Here we will consider topologically gauged "BLG" theories: [Gran,BN(2008)] i.e. matter/chern-simons gauge theory with N = 8 ("BLG") superconformal symmetry coupled to conformal supergravity we find new features like: SO(N) gauge groups for any N (instead of just the BLG SO(4)) [Gran, Greitz, Howe, BN(2012)],[BN(2013)] higgsing to topologically massive supergravity (super-tmg) [Chu, BN(2009)] a number of possible "critical" backgrounds [BN(2013)] with indications of [BN(2012)] a "sequential AdS/CFT" using Neumann boundary conditions (also [Vasiliev(2000,2012)], [Compere, Marolf(2008)]) a connection to higher spin

6 3-dim N = 8 superconformal field theory : field content Review of 3d BLG theory: the field content: scalars X i a i: SO(8) R-symmetry vector index a: three-algebra index related to [T a, T b, T c ] = f abc dt d (structure constants f here antisymmetric in a, b, c) spinors ψ a (2-comp Majorana) with a hidden R-symmetry chiral spinor index (also real 8-dim), vector gauge potential à µ a b = A µcd f cda b conformal dimensions (deduced from their kinetic terms): 1/2 for X i a 1 for ψ a 1 for A µ ("kinetic term" = Chern-Simons term) [Schwarz]

7 3d N = 8 superconformal field theory: Lagrangian The BLG Lagrangian (with interactions and CS from f efg d) L = 1 2 (D µx i a)(d µ X i a) + i 2 Ψ a γ µ D µ Ψ a εµνλ ( f abcd A µab ν A λcd f cda g f efgb A µab A νcd A λef ), i 4 Ψ b Γ ij X i cx j dψ a f abcd V BLG where D µ = µ + Ã µ and the potential V (st) BLG = 1 12 (Xi ax j bx k c f abc d)(x i ex j f X k g f efg d) two triple products but a "single trace" (st) can introduce a (quantized) level k by rescaling f abc d, large k = weak coupling, reduction from 11d to 10d [BN, Pope],[ABJM] no other free parameters!

8 BLG transformation rules The BLG transformation rules for (global) N = 8 supersymmetry are δxi a = i ɛγ i Ψ a, δψ a = D µ Xaγ i µ Γ i ɛ Xi b Xj c Xd k Γijk ɛ f bcd a. Demanding cancelation on the (Cov.der.) 2 terms in δl implies δã µ a b = i ɛγ µ Γ i X i cψ d f cda b Full susy => the fundamental identity [Bagger, Lambert], [Gustavsson] f abc g f efg d = 3f ef [a g f bc]g d, only one finite dim. realization: A 4 with SO(4) gauge symmetry (i.e. with levels (k, k)) [Papadopoulos][Gauntlett,Gutowski]

9 3-dim N = 8 superconformal gravity To gauge the global symmetries of the BLG theory we need 3d N = 8 conformal supergravity: On-shell Lagrangian = three CS-like terms [Gran,BN(2008)] L = 1 2 ɛµνρ Tr α ( ω µ ν ω ρ ω µ ω ν ω ρ ) ie 1 ɛ αµν ( D µ χ ν γ β γ α D ρ χ σ )ɛ βρσ ɛ µνρ Tr i (B µ ν B ρ B µb ν B ρ ), supercovariant spin connection: ω µαβ (e µ α, χ i µ) CS terms are of 3rd, 2nd and 1st order in derivatives, respectively OK for any number N of supersymmetries [Lindström,Roček(1989)]

10 Topologically gauged BLG theory: the Lagrangian Searching for solutions the interesting bosonic terms in L are L = 1 g LSUGRA conf + L BLG cov e 16 X2 R V new the scalar potential has a single-trace (st) contribution from L BLG cov V (st) BLG = λ2 12 (Xi ax j bx k c ɛ abcd )(X i ex j f X k g ɛ efg d) and a new triple-trace (tt) term from the topological gauging [Gran,Greitz,Howe,BN(2012)] V (tt) new = eg ( (X 2 ) 3 8(X 2 )X j b Xj cx k cx k b + 16Xi cx i ax j ax j b Xk b Xk c )

11 Topologically gauged BLG theory: new properties New theories? [Gran,Greitz,Howe,BN(2012)] The gauge sector is deformed: L CS(A) = 1 a L CS(A L ) + 1 a L CS(A R ) where a := g 8 λ, a := g 8 + λ SO(3) theories for certain values of the parameters for λ = 0 the three-algebra indices can be extended arbitrarily: => gauge group SO(N) for any N

12 Topologically gauged N = 6 superconformal ABJ(M) The BLG type of new potential was found first for ABJ(M) [Chu, BN(2009)] Matter fields now complex with A, B,.. indices in 4 of SU(4) The topologically gauged ABJM Lagrangian has about 25 new terms including interaction terms for the complex scalars Z a A (explicit λ and g) recall the original ABJ(M) potential (single trace in 3-alg.) V (st) ABJ(M) = 2 3 ΥCD Bd 2, Υ CD Bd = λf ab cdz C a Z D b Z c B+λf ab cdδ [C B ZD] a Z E b Z c E. the new terms with one structure constant are (double trace) V (dt) new = 1 8 gλf ab cd Z 2 Z C a Z D b Z c CZ d D 1 2 gλf ab cdz B a Z C b (Z D e Z e B) Z c C Z d D. and without structure constant (triple trace) V (tt) new = g 2 ( ( Z 2 ) Z 2 Z Z 6 ). also new Yukawa-like terms without structure constants

13 Higgsing of topologically gauged ABJM: the chiral point The chiral point for ABJ(M): VEV < Z >= v[chu, BN] L ABJM higgsed = L CS(grav) e 8 v2 R e 256 v6 Compare to the TMG version of LSS: [Li, Song, Strominger] thus L LSS TMG = 1 κ 2 ( 1 µ L CS(grav) (R 2Λ)), Λ = 1 l 2 µl = 1 (1) the signs of the Einstein-Hilbert and cosmological terms => negative energy black holes (unitarity?) these features are dictated by the sign of the ABJM scalar kinetic terms (via conformal invariance)! introducing parameters (λ and g M ) does not help

14 Higgsing of topologically gauged theories with 8 supersymmetries: the chiral points of the SO(N) model Is there a chiral point also for the N = 8 SO(N) theory? [Gran,Greitz,Howe,BN(2012)],[BN(2013)] L SO(N) = 1 g L CS(grav) 1 16 X2 R g ((X2 ) 3 8X 2 X X 6 ) where X i a (a=1,2,...,n and i=1,2,..,8) is used as follows X ij = X i ax j a, X 2 = tr(x ij ), X 4 = X ij X ij, X 6 = X ij X jk X ki Compare to TMG/LSS: VEV < X >= v =>µl = 1/3??

15 Higgsing of topologically gauged theories with 8 supersymmetries: the chiral points of the SO(N) model Is there a chiral point also for the N = 8 SO(N) theory? [Gran,Greitz,Howe,BN(2012)],[BN(2013)] L SO(N) = 1 g L CS(grav) 1 16 X2 R g ((X2 ) 3 8X 2 X X 6 ) where X i a (a=1,2,...,n and i=1,2,..,8) is used as follows X ij = X i ax j a, X 2 = tr(x ij ), X 4 = X ij X ij, X 6 = X ij X jk X ki Compare to TMG/LSS: VEV < X >= v =>µl = 1/3?? There are two well-known critical points on the market: L LSS = 1 κ 2 ( 1 µ L CS(grav) (R 2Λ)), Λ = 1 l 2 chiral AdS with µl = 1 [Li, Song, Strominger] null-warped AdS with µl = 3 [Anninos, Compere, de Buyl, Detourney, Guica]

16 New critical points of the SO(N) model X i a is an 8 N rectangular matrix [BN(2013)]=> generalize the VEV to a matrix: for I, A = 1, 2,.., p 8 < X i a > vδ I A, => µl = 1/ 1 4 p p = 1, 2, 3, 4, 5, 6, 7, 8 give µl = 1 3, 1, 3,, 5, 3, 7 3, 2

17 New critical points of the SO(N) model X i a is an 8 N rectangular matrix [BN(2013)]=> generalize the VEV to a matrix: for I, A = 1, 2,.., p 8 < X i a > vδ I A, => µl = 1/ 1 4 p p = 1, 2, 3, 4, 5, 6, 7, 8 give µl = 1 3, 1, 3,, 5, 3, 7 3, 2 corresponding to critical AdS for p = 2 null-warped AdS (or Schödinger(z=2)) for p = 3, 6 Minkowski for p = 4 (a BMS limit??)

18 New critical points of the SO(N) model X i a is an 8 N rectangular matrix [BN(2013)]=> generalize the VEV to a matrix: for I, A = 1, 2,.., p 8 < X i a > vδ I A, => µl = 1/ 1 4 p p = 1, 2, 3, 4, 5, 6, 7, 8 give µl = 1 3, 1, 3,, 5, 3, 7 3, 2 corresponding to critical AdS for p = 2 null-warped AdS (or Schödinger(z=2)) for p = 3, 6 Minkowski for p = 4 (a BMS limit??) but in fact µl = 5 is also known! [Ertl, Grumiller, Johansson(2010)]

19 New critical points of the SO(N) model X i a is an 8 N rectangular matrix [BN(2013)]=> generalize the VEV to a matrix: for I, A = 1, 2,.., p 8 < X i a > vδ I A, => µl = 1/ 1 4 p p = 1, 2, 3, 4, 5, 6, 7, 8 give µl = 1 3, 1, 3,, 5, 3, 7 3, 2 corresponding to critical AdS for p = 2 null-warped AdS (or Schödinger(z=2)) for p = 3, 6 Minkowski for p = 4 (a BMS limit??) but in fact µl = 5 is also known! [Ertl, Grumiller, Johansson(2010)] in a similar context a new solution with µl = 2 found recently (vector fields crucial) [Deger, Kaya, Samtleben, Sezgin(Nov-2013)]

20 The Higgsed Chern-Simons sector Symmetry breaking: (compare to [Mukhi, Papageorgakis],[Mukhi]) conformal > AdS SO(N) SO R (8) SO(N p) SO R (8 p) SO diag (p) gives (with m = gv 2 ) 2ɛF(A) + m(a B) = gxd(a, B)X ɛg(b) + m(a B) = gxd(a, B)X Eliminating B (by first solving the first equation above) gives for zero coupling g = 0 the exact solution ɛf = 4 m ɛp(ɛf) + 8 m 2 ɛ(ɛf, ɛf) (2) where P = d + A and for non-zero coupling g m(b A) = Σ n 0 ( X v )n (2ɛF gxp(a)x)( X v )n (3)

21 Questions and speculations "Sequential AdS/CFT": AdS 4 /CFT 3 AdS 3 /CFT 2??: a) dynamical [BN(2012)] b) from AdS foliations ([Compere, Marolf(2008)] c) from higher spin algebra/unfolding [Vasiliev(2000, 2012)]) how is the breaking from CFT 3 to AdS 3 seen in the AdS 4 bulk? Change of foliation in AdS 4 = 3d higgsing? [Andrade, Uhlemann (2011)] [Ohl, Uhlemann (2012)] Cotton tensors on the boundary can be related to Neumann b.c. for the AdS 4 bulk metric [Witten],[Compare, Marolf],[de Haro] Neumann boundary conditions for all spins used in recent anomaly computations (all-spin cancellations) [Giombi et al], [Giombi, Klebanov], [Tseytlin] the AdS 4 bulk should be an N = 8 higher spin theory (see work by Vasiliev and Sezgin-Sundell) >

22 AdS 4 Vasiliev higher spin theories Structure of AdS 4 Vasiliev systems very schematically! Action formulation not known in general! Interaction ambiguity and θ parameters (all products ): master fields: W (gauge 1-form), B (scalar), J (current) dw + W 2 = J + cc, db + WB Bπ(W) = 0 J = f (B)dz 2 with f (B) = B e θ(b) θ(b) = θ 0 + θ 2 B The parity preserving cases are (with θ 2n = 0 for n 1) θ 0 = 0: dual to free scalar O(N) model on the boundary (φ 2 a = 1 operator): bulk scalar with N bc > CFT UV [Klebanov-Polyakov (2002)] θ 0 = π 2 : dual to free fermion O(N) model on the boundary (ψ 2 a = 2 operator): bulk scalar with D bc > CFT IR [Sezgin-Sundell (2005)]

23 Possible connection to AdS 4 Vasiliev higher spin theories: non-trivial θ The CFT 3 to BULK dictionary: [Chang, Minwalla, Sharma, Yin (2012)] CS with finite level k < > θ 0 non-trivial: parity broken! adding double and triple trace deformations ((φ 2 ) 3, φ 2 ψ 2 ) < > changing the bulk scalar field b.c. Bosonization-like features arise when comparing the different free and interacting boundary theories! [Aharony, Gur-Ari, Yacoby (2012)] Susy works only for N 6 CFTs [Chang, Minwalla, Sharma, Yin (2012)] top gauged N = 8 vector model < > bulk?? (work in progress)

24 Conformal higher spins in 3d: covariant action Vasiliev s theory with Neumann b.c. on gauge fields gives conformal 3d higher spin theory on the boundary! Poisson bracket construction [Blencowe(1989), Pope-Townsend(1989)] Conformal algebra in 3d: SO(3, 2) Realized with Poisson brackets and classical variables q α, p α Even polynomials in q α, p α => higher spin (HS) algebra gauge all generators using a field A F = da + A A = 0 gives for any spin s = n + 1: generalized Cotton equations for e a1...an µ (the HS frame fields) constraints

25 Conformal higher spins in 3d: covariant action As an example consider spin 2: the generators P a, M a, D, K a are bilinear in q, p: Example : M a (1, 1) = 1 2 (σa ) α β q α p β (4) Decompose F = da + A 2 = 0: F a (2, 0) = De a (2, 0) = 0 (zero torsion constraint) F L a(1, 1) = R a (1, 1) + {e(2, 0), f (0, 2)} M a = 0 F D (1, 1) = db(1, 1) + {e(2, 0), f (0, 2)} D = 0 (gauge b = 0 => constraint on Schouten tensor) F a (0, 2) = Df a (0, 2) = 0 (Cotton equation)

26 Conformal higher spins in 3d: covariant action generalized Cotton equations for e a 1...a n µ (the HS frame fields) the fields in F = 0 are f a 1...a n µ (0, 2n), f a 1...a n µ (1, n 1),..., ω a 1...a n µ (n, n),..., e a 1...a n µ (2n, 0) the first one is the HS "Schouten tensor" the last one is the HS frame field in terms of which all other fields are expressed!! => spin s conformal field equations with 2s 1 derivatives Use HS "spin connections", i.e. the "second order" fields ω(n, n)(e), to write a Chern-Simons type action [BN(2013)] Only (spin 2) covariant tensors appear This action is not consistent since the Poisson bracket field equations are not integrable!

27 Conformal higher spins in 3d: covariant action Solution: Quantization [BN(2013)] Quantize q α, p α Expand the F = 0 field equations in terms of multi-commutators Should give the full star product action now in covariant form S = ( Ωd Ω Ω Ω Ω) where Ω = a s ω(s 1, s 1) s=2 if the interaction terms require other fields than ω (as when coupling to scalars) this action is a convenient starting point

28 This construction can be used also for coupling to scalar fields The conformal coupling Rφ in the scalar field equation is due to quantization and unfolding: DΦ(x; p) 0 >= 0 HS analogues of this coupling easy to derive (in principle) Star product formulation of the back reaction in HS field equations obtained by just writing down the action which makes use of f fields in the level just above ω However: the star product formulation of the field equations is not known (as it is in AdS for non-conformal HS)

29 Summary Main points: Topologically gauged CFT3 with N = 8 susy are SO(N) vector models (BLG only SO(4)) They have a potential giving rise to a set of special/critical background solutions "Sequential" AdS/CFT based on Neumann boundary conditions 3d conformal HS theory seems to play a role: a covariant star product Lagrangian can be constructed starting from a Poisson bracket formulation coupling HS to scalars still tricky!

30 Summary Main points: Topologically gauged CFT3 with N = 8 susy are SO(N) vector models (BLG only SO(4)) They have a potential giving rise to a set of special/critical background solutions "Sequential" AdS/CFT based on Neumann boundary conditions 3d conformal HS theory seems to play a role: a covariant star product Lagrangian can be constructed starting from a Poisson bracket formulation coupling HS to scalars still tricky! Thanks for your attention!