1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 S Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson Chalmers University of Technology, Gothenburg Talk at "Modern Developments in M-theory", Banff, Canada, January 14, 2014 Talk based on: "Towards an exact frame formulation of conformal higher spins in three dimensions", arXiv:1312.5883 “Critical solutions in topologically gauged N = 8 CFTs in three dimensions” , arXiv:1304.2270 "Topologically gauged superconformal Chern-Simons matter theories" with Ulf Gran, Jesper Greitz and Paul Howe, arXiv:1204.2521 in JHEP "Aspects of topologically gauged M2-branes with six supersymmetries: towards a "sequential AdS/CFT"?, arXiv:1203.5090 [hep-th] Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
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Main subject: Classical CFTs in 3 dimensionsRecall: N = 8 supersymmetries:
BLG: [Bagger, Lambert(2007), Gustavsson(2007)]superconformal Chern-Simons (CS)-matter theory[Schwarz(2004)]only with gauge group SO(4) = SU(2)× SU(2)parity symmetricno U(1) factorany level k possiblerelation to stacks of M2-branes tricky
consider instead N = 6: then the M2-brane connection is clear[ABJM(2008), ABJ(2008)]
ABJ(M) are quiver theories with gauge groups likeUk(N)× U−k(N), for any k and any NSUk(N)× SU−k(N), for any k and any N(N = 2 and k = 1, 2 case classically equivalent to BLG)
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
can CS(supergravity) help? (recall the role of CS(gauge))what would such a CS-gravity constructionmean in string/M theory?in AdS4/CFT3? (for spin 2)in AdS4/CFT3 for higher spins (HS)?
the A and B models in Vasiliev’s AdS4 HS-theory haveparity symmetric boundary theories[Klebanov, Polyakov (2002)], [Sezgin, Sundell (2005)]BUT parity non-symmetric CS theories interpolate between them[Chang et al(2012)], [Aharony et al(2012)]recent checks of the correspondence use Neumann b.c. for all spinsand all-spin CFTs [Giombi et al (2013)],[Giombi-Klebanov(2013)], [Tseytlin(2013)]
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
can this restriction be lifted?can CS(supergravity) help? (recall the role of CS(gauge))
what would such a CS-gravity constructionmean in string/M theory?in AdS4/CFT3? (for spin 2)in AdS4/CFT3 for higher spins (HS)?
the A and B models in Vasiliev’s AdS4 HS-theory haveparity symmetric boundary theories[Klebanov, Polyakov (2002)], [Sezgin, Sundell (2005)]BUT parity non-symmetric CS theories interpolate between them[Chang et al(2012)], [Aharony et al(2012)]recent checks of the correspondence use Neumann b.c. for all spinsand all-spin CFTs [Giombi et al (2013)],[Giombi-Klebanov(2013)], [Tseytlin(2013)]
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
can this restriction be lifted?can CS(supergravity) help? (recall the role of CS(gauge))
what would such a CS-gravity constructionmean in string/M theory?in AdS4/CFT3? (for spin 2)in AdS4/CFT3 for higher spins (HS)?
the A and B models in Vasiliev’s AdS4 HS-theory haveparity symmetric boundary theories[Klebanov, Polyakov (2002)], [Sezgin, Sundell (2005)]BUT parity non-symmetric CS theories interpolate between them[Chang et al(2012)], [Aharony et al(2012)]recent checks of the correspondence use Neumann b.c. for all spinsand all-spin CFTs [Giombi et al (2013)],[Giombi-Klebanov(2013)], [Tseytlin(2013)]
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
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’ll find new features like:SO(N) gauge groups for any N (instead of the SO(4) in BLG )[Gran, Greitz, Howe, BN(2012)]higgsing to topologically massive supergravity (super-TMG)and a number of possible "critical" backgrounds [BN(2013)]
such results first found in top gauged ABJM/ABJ[Chu, BN(2009)],[Chu, Nastase, Papageorgakis, BN(2010)]
with indications ofa "sequential AdS/CFT" using Neumann b.c. [BN(2012)](see also [Vasiliev(2000,2012)], [Compere, Marolf(2008)])a connection to higher spin [BN(2013)]
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
This supergravity theory has no propagating degrees of freedom!
clear in the light-cone gauge: all non-zero field components (plus∂+ on them) can be solved for [BN(2008)]=> "topologically gauged CFT3"
Conformal supergravity can be coupled to BLG byNoether methods
[Gran,BN(2008)] [Gran, Greitz, Howe, BN(2012)]or by other methods [Gran, Greitz, Howe, BN(2012)]
demanding on-shell susy (as originally done by BLG)superspace : "the Dragon window" in 3d (Cotton eq)[Cederwall, Gran, BN(2011)][Howe, Izquierdo, Papadopoulos, Townsend(1995)]
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
Neumann b.c. for the AdS4 bulk metric for spin 1 and 2 =>Chern-Simons or Cotton terms on the boundary[Witten(2003)],[Leigh, Petkou(2003,2007)],[Compare, Marolf(2008)],[de Haro(2008)]
Neumann b.c. for all spins used in recent anomaly computations(all-spin cancellations)[Giombi et al(2013)], [Giombi, Klebanov(2013)],[Tseytlin(2013)]
the AdS4 bulk should be an N = 8 higher spin theory(see work by Vasiliev and Sezgin-Sundell) —>
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
Interaction ambiguity given by θ parameters (all products ∧ and ?):3 master fields: W (gauge 1-form), B (scalar), J (current)
dW + W2 = J + cc,dB + WB− Bπ(W) = 0J = f (B)dz2 with f (B) = B eθ(B)
θ(B) = θ0 + θ2B2 + ..
The parity preserving A and B models are (with θ2n = 0 for n ≥ 1)θ0 = 0: dual to free scalar O(N) model on the boundary(φ2 a ∆ = 1 operator): bulk scalar with N bc –> CFTUV
[Klebanov-Polyakov (2002)]
θ0 = π2 : dual to free fermion O(N) model on the boundary
(ψ2 a ∆ = 2 operator): bulk scalar with D bc –> CFTIR
[Sezgin-Sundell (2005)]
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
conformal algebra in 3d: SO(3, 2)realized with Poisson brackets for qα, pα (bosonic spinors)even polynomials in qα, pα => higher spin (HS) algebragauge all generators using a HS gauge field A
then: F = dA + A ∧ A = 0 gives for any spin s = n + 1:generalized Cotton equations for ea1...an
µ (the HS frame fields)constraints
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
conformal algebra in 3d: SO(3, 2)realized with Poisson brackets for qα, pα (bosonic spinors)even polynomials in qα, pα => higher spin (HS) algebragauge all generators using a HS gauge field A
then: F = dA + A ∧ A = 0 gives for any spin s = n + 1:generalized Cotton equations for ea1...an
µ (the HS frame fields)constraints
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
Even polynomials in qα, pα => higher spin (HS) algebrageneralized exact Cotton equations for ea1...an
µ (= HS frame fields)after "solving" F = 0 the remaining fields in A are [BN(2013)]
f a1...anµ (0, 2n), f a1...an
µ (1, n− 1), ..., ωa1...anµ (n, n), ..., ea1...an
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(non-zero) fields are expressed!!spin 3 sector done in detail
=> spin s conformal field equations with 2s− 1 derivatives
this suggests using HS "spin connections" ω(n, n)(e) to write aChern-Simons type action
only (spin 2) covariant tensors appear
but such an action is not consistent with the Poisson bracket fieldequations! →
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
This strategy can be used also for coupling to scalar fields:
The conformal coupling Rφ in the scalar field equation is due toquantization and unfolding of the eq. (D = d + A):
DΦ(x; p)|0 >= 0
HS analogues of Rφ easy to derive (in principle)
What about the back reaction in the HS gauge field equation?(Star product version of F = dA + A2 = J(Φ))
Write down the action giving the above scalar field equation!(containing the fields f (n− 1, n + 1) in the level above ω(n, n))Is this consistent?Not clear since the star product formulation of the equation F = Jwith currents J constructed from Φ is not known!(which it is in AdS for non-conformal HS (Vasiliev))
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
This strategy can be used also for coupling to scalar fields:
The conformal coupling Rφ in the scalar field equation is due toquantization and unfolding of the eq. (D = d + A):
DΦ(x; p)|0 >= 0
HS analogues of Rφ easy to derive (in principle)
What about the back reaction in the HS gauge field equation?(Star product version of F = dA + A2 = J(Φ))
Write down the action giving the above scalar field equation!(containing the fields f (n− 1, n + 1) in the level above ω(n, n))Is this consistent?Not clear since the star product formulation of the equation F = Jwith currents J constructed from Φ is not known!(which it is in AdS for non-conformal HS (Vasiliev))
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers
Topologically gauged 3d free CFTs with N = 8 susy areSO(N) vector models (BLG only SO(4))
These theories have new scalar potentials =>special/critical background solutions forµl = 1
3 , 1, 3,∞, 5, 3,73 , 2
"Sequential" AdS/CFT based on N b.c. ?AdS4/CFT3: 3d conformal HS theory plays a role for N b.c.
a covariant star product Lagrangian can be constructed startingfrom a Poisson bracket formulation (CS again??)coupling 3d conformal HS to scalars still tricky!
Thanks for your attention!
Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers