Topologically gauged CFTs in 3d: solutions, AdS/CFT and ... · Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson Chalmers University of Technology,

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

Bengt E.W. NilssonChalmers 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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Background for the discussion: classical

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)


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Background for the discussion: quantum

The (non-perturbative) quantum picture for N = 8 is better:

monopole operators [ABJM(2008), BKKS(2008), BKK(2009)]can lead to enhanced symmetries for ABJM theories

and to N = 8 susy for N-stacks of M2 branes

This can be checked by comparing moduli spaces [Lambert et al]or partition functions/superconformal indices [Kapustin et al]:

Examples:supersymmetry enhancement:ABJM Uk(N)× U−k(N) has 2 extra susy’s for k = 1, 2U(1) enhancementparity enhancement


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Motivation: some questions

Questions at the classical level:

why is classical BLG restricted to only SO(4)?

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)]


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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)?



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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)?



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Outline and Results

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)]


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3d N = 8 superconformal field theory: BLG

Review: 3d BLG theory in terms of structure constants f abcd

scalars Xia (real)

i: SO(8) R-symmetry vector indexa: enumerate the 3-algebra elements Ta with triple product[Ta,Tb,Tc] = f abc

dTd (antisymmetric in a, b, c)spinors ψa (2-comp Majorana)

with a hidden R-symmetry chiral spinor index (also real 8-dim),

vector gauge potential Aµab = Aµcd f cda

b

conformal dimensions (deduced from their kinetic terms):1/2 for Xi

a

1 for ψa

1 for Aµ


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3d N = 8 superconformal field theory: Lagrangian

The BLG Lagrangian (with interactions and CS in terms of f abcd)

L = −12(DµXi

a)(DµXia) + i

2ΨaγµDµΨa + LCS(A)

− i4ΨbΓijXi

cXjdΨa f abcd − V(st)

BLG

where Dµ = ∂µ + Aµ and the Chern-Simons term

LCS(A) = 12εµνλ(f abcdAµab∂νAλcd + 2

3 f cdag f efgbAµabAνcdAλef

)and the potential

V(st)BLG = 1

12(XiaXj

bXkc f abc

d)(XieXj

f Xkg f efgd)

two triple products but a "single trace" (st)

introduce a level k by rescaling f abcd → λf abc

d where λ = 2πk

no other free parameters!Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson, Chalmers

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BLG transformation rules

The BLG transformation rules for (global) N = 8 supersymmetry are

δXai = iεΓiΨ

a,δΨa = DµXi

aγµΓiε+ 1

6 Xib Xj

c Xkd Γijkε f bcd

a.

Demanding cancelation on the (Cov.der.)2 terms in δL implies

δAµab = iεγµΓiXi

cψd f cdab

Full susy => the fundamental identity

f abcg f efg

d = 3f ef [ag f bc]g

d

only one finite dim. realization: A4 with SO(4) gauge symmetry(i.e. parity symmetric with levels (k,−k))[Papadopoulos(2008)][Gauntlett,Gutowski(2008)]


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3-dim N = 8 superconformal gravity

To gauge the global super-conformal symmetries of the BLG theorywe couple it to 3d N = 8 conformal supergravity: [Gran,BN(2008)]

On-shell Lagrangian = three CS-like terms

Lconfsugra = 1

2εµνρTr(ωµ∂ν ωρ + 2

3 ωµων ωρ)

−ie−1εαµν(DµχνγβγαDρχσ)εβρσ−εµνρTr(Bµ∂νBρ+23

BµBνBρ),

supercovariant spin-connection: ωµαβ(eµα, χiµ)

CS terms are of 3rd, 2nd and 1st order in derivatives, respectivelyOK for any number N of supersymmetries if triality rotated to εi

[Lindström,Rocek(1989)] (N = 1 by [Deser,Kay(1983)])


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Topologically gauged BLG theory

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)]


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Topologically gauged BLG theory: some details

Supersymmetry to order (Dµ)2 gives the conformal coupling− e16 X2R:

(f µ is the dual field strength of the spin 3/2 field χµ) [Gran, BN (2008)]

Ltop gaugedBLG = Lconf

sugra + LcovBLG

+ 1√2ieχµΓiγνγµΨaDνXia (”the supercurrent term”)

− i4εµνρχµΓijχν(Xi

aDρXja) + i√

2f µΓiγµΨaXi

a

− e16 X2R + i

4 X2 f µχµ

Obtaining δL = 0 at linear and zeroth order in Dµ will requireadding many new terms to L and the transformation rules =>


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Topologically gauged BLG theory: the transformation rules

The complete transformation rules:with level parameter λ = 2π

k and the gravitational coupling gNB: new matter term in δAµa

b [Gran, Greitz, Howe, BN(2012)]

δeµα = iεgγαχµ, δχµ = Dµεg,

δBijµ = − i

2e εgΓijγνγµf ν − ig4 χµΓk[iεgXj]

a Xka −

ig32 χµΓijεgX2

− ig16ΨaΓijkγµεmXk

a −3ig8 ΨaγµΓ[iεmXj]

a ,

δXia = iεmΓiΨa,

δΨa = γµΓiεm(DµXia − iAχµΓiΨa) + λ

6 ΓijkεXibXj

cXkd ε

bcda

+ g8ΓiεmXi

bXjbXj

a −g

32ΓiεmXiaX2,

δAµab = iλεmγµΓiXi

cΨd εcda

b − iλ2 χµΓijεgXi

cXjd ε

cdab

+ ig4 εmγµΓiψ[aXi

b] + ig8 χµΓijεgXi

aXjb.


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Topologically gauged BLG theory: the Lagrangian

The bosonic terms in L relevant for this talk are

L = 1g Lsugra

conf + LBLGcov − 1

16 X2R− V(tt)new

the total scalar potential has

a single-trace (st) contribution from LBLGcov

V(st)BLG = λ2

12 (XiaXj

bXkc ε

abcd)(XieXj

f Xkg ε

efgd)

and a new triple-trace (tt) term from the topological gauging

V(tt)new = eg2

2·32·32

((X2)3 − 8(X2)Xj

bXjcXk

cXkb + 16Xi

cXiaXj

aXjbXk

bXkc

)without structure constants!


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Topologically gauged BLG theory: new properties

New theories with N = 8 superconformal (local) symmetry?[Gran,Greitz,Howe,BN(2012)]The gauge sector is deformed:

LCS(A) =1a

LCS(AL) +1a′

LCS(AR)

wherea :=

g8− λ, a′ :=

g8

+ λ

no longer parity symmetric

by taking λ = 2πk → 0, or setting f abc

d = 0 =>the "three-algebra" indices can be extended arbitrarilyi.e. the gauge group is now SO(N) for any N


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Topologically gauged N = 6 superconformal ABJ(M)

The new kind of potential without structure constants was first foundfor ABJ(M): [Chu, BN(2009)]

scalars ZAa complex (A in 4 of SUR(4), a 3-algebra index)

potential terms (explicit λ and g)the original ABJ(M) potential (single trace in 3-alg.):

V(st)ABJ(M) = 2

3λ2|ΥCD

Bd|2, ΥCDBd = f ab

cdZCa ZD

b ZcB+f ab

cdδ[CB ZD]

a ZEb Zc

E .

new terms with one structure constant (double trace)

V(dt)new = − 1

8 gλf abcd|Z|2ZC

a ZDb Zc

CZdD − 1

2 gλf abcdZB

a ZCb (ZD

e ZeB)Zc

CZdD .

and without structure constant (triple trace)

V(tt)new = −g2( 5

12·64 (|Z|2)3 − 132 |Z|

2|Z|4 + 148 |Z|

6) .


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Higgsing of topologically gauged ABJM: the chiral point

The chiral point for ABJ(M): VEV < Z >= v [Chu, BN(2009)]

LABJMhiggsed = LCS(grav) − 1

8 v2R− 1256 v6

Compare to the TMG version of LSS (opposite over-all sign!):[Li, Song, Strominger(2008)]

LLSSTMG = 1

κ2 ( 1µLCS(grav) − (R− 2Λ)), Λ = − 1

l2

thusµl = 1 (the LSS chiral point)

the signs of the Einstein-Hilbert and cosmological terms=> negative energy black holes (unitarity ?)

these features are dictated by the sign of the ABJ(M) scalarkinetic terms (via conformal invariance)!


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Chiral points of the SO(N) model?

Is there a chiral point solution also for the N = 8 SO(N) theory?[Gran,Greitz,Howe,BN(2012)],[BN(2013)]

LSO(N)N=8 = 1

g LCS(grav) − 116 X2R− g2

2·32·32((X2)3 − 8X2X4 + 16X6)

where in terms of Xia (a = 1, 2, ...,N and i = 1, 2, .., 8)

Xij = XiaXj

a, X2 = Xii, X4 = XijXij, X6 = XijXjkXki

Set VEV < X8N >= v and compare to TMG/LSS =>

µl = 1/3 ??

There are two well-known critical points on the market:chiral AdS with µl = 1 [Li, Song, Strominger(2008)]

null-warped AdS with µl = 3[Anninos, Compere, de Buyl, Detourney, Guica(2010)]


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Chiral points of the SO(N) model?

Is there a chiral point solution also for the N = 8 SO(N) theory?[Gran,Greitz,Howe,BN(2012)],[BN(2013)]

LSO(N)N=8 = 1

g LCS(grav) − 116 X2R− g2

2·32·32((X2)3 − 8X2X4 + 16X6)

where in terms of Xia (a = 1, 2, ...,N and i = 1, 2, .., 8)

Xij = XiaXj

a, X2 = Xii, X4 = XijXij, X6 = XijXjkXki

Set VEV < X8N >= v and compare to TMG/LSS =>

µl = 1/3 ??

There are two well-known critical points on the market:chiral AdS with µl = 1 [Li, Song, Strominger(2008)]

null-warped AdS with µl = 3[Anninos, Compere, de Buyl, Detourney, Guica(2010)]


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New critical points of the SO(N) model

Xia is an 8× N rectangular matrix =>

generalize the VEV to a matrix: for I,A = 1, 2, .., p ≤ 8 [BN(2013)]

< Xia >→ vδI

A, => µl = 1/|1− 4p |

p = 1, 2, 3, 4, 5, 6, 7, 8 =>µl = 1

3 , 1, 3,∞, 5, 3,73 , 2

corresponding tocritical AdS for p = 2null-warped AdS (or Schödinger(z=2)) for p = 3, 6Minkowski for p = 4 (a BMS limit ??)

but in fact a special µl = 5 (p = 5) solution is also known![Ertl, Grumiller, Johansson(2010)]

in a similar context a new solution with µl = 2 (p = 8) foundrecently (vector fields crucial)[Deger, Kaya, Samtleben, Sezgin(Nov-2013)]


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< Xia >→ vδI

A, => µl = 1/|1− 4p |

p = 1, 2, 3, 4, 5, 6, 7, 8 =>µl = 1

3 , 1, 3,∞, 5, 3,73 , 2





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< Xia >→ vδI

A, => µl = 1/|1− 4p |

p = 1, 2, 3, 4, 5, 6, 7, 8 =>µl = 1

3 , 1, 3,∞, 5, 3,73 , 2





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< Xia >→ vδI

A, => µl = 1/|1− 4p |

p = 1, 2, 3, 4, 5, 6, 7, 8 =>µl = 1

3 , 1, 3,∞, 5, 3,73 , 2





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< Xia >→ vδI

A, => µl = 1/|1− 4p |

p = 1, 2, 3, 4, 5, 6, 7, 8 =>µl = 1

3 , 1, 3,∞, 5, 3,73 , 2





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The Higgsed Chern-Simons sector

Symmetry breaking:(compare to [Mukhi, Papageorgakis(2008)],[Mukhi(2011)])

conformal –> AdSSO(N)× SOR(8)→ SO(N − p)× SOR(8− p)× SOdiag(p)

field equations in the SOdiag(p) sector (with m = gv2)

2εF(A) + m(A− B) = gXD(A,B)X (1)

εG(B) + m(A− B) = −gXD(A,B)X (2)

For zero coupling g = 0, eliminating B gives the exact form of thefield equation (P = d + A):

εF = 4mεP(εF) + 8

m2 ε(εF, εF) (3)

while for g non-zero an iteration is needed:

m(B− A) = Σn≥0(Xv )n(2εF − gXP(A)X)(X

v )n (4)


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The Higgsed Chern-Simons sector: the spectrum

Split the indices: i→ (i, I), a→ (a,A)indices I and A identified under the diagonal SOdiag(p)

Scalars:

xia = (xi

a, xiA, xI

a, xIA) where

xIA = (z,w(IA), y[IA])

the physical ones after higgsing are

xia, z, wAB (some with masses equal to the BF bounds !)

Vector fields:the massive ones (YM+CS, see above)

AAaµ , BiJ

µ , AIJµ

while the rest are massless!

Leads to supermultiplets with an a index and without an a!


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Questions and speculations

"Sequential AdS/CFT": AdS4/CFT3 → AdS3/CFT2 ?:dynamical (relies on spin 2 Neumann b.c.) [BN(2012)]from AdS foliations [Compere, Marolf(2008)]from higher spin algebra/unfolding [Vasiliev(2000, 2012)]

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) —>


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AdS4 Vasiliev higher spin theories

AdS4 Vasiliev systems: (very schematically)

action formulation not known

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)]


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Possible connection to AdS4 Vasiliev higher spin theories:non-trivial θ

The CFT3 to BULK dictionary:

CS with finite level k <—> θ0 non-trivial: parity non-symmetric!

Bosonization-like features arise when comparing the different freeand interacting boundary theories! [Aharony, Gur-Ari, Yacoby (2012)]

by choosing b.c. one gets susy CFTs but only for N ≤ 6 CFTs[Chang, Minwalla, Sharma, Yin (2012)]

top gauged N = 8 vector model <—> bulk?? (work in progress)


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Conformal higher spins in 3d: covariant action

Vasiliev’s AdS4 theory with Neumann b.c. on HS gauge fields =>conformal 3d HS theory on the boundary!

What is this conformal theory? How can it be constructed?Is there a Lagrangian? Coupling to scalars?

Poisson bracket construction[Blencowe(1989), Pope-Townsend(1989)]

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


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Vasiliev’s AdS4 theory with Neumann b.c. on HS gauge fields =>conformal 3d HS theory on the boundary!

What is this conformal theory? How can it be constructed?Is there a Lagrangian? Coupling to scalars?

Poisson bracket construction[Blencowe(1989), Pope-Townsend(1989)]

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


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Consider spin 2:the generators Pa,Ma,D,Ka are bilinear in q, p:

Ma(1, 1) = − 12(σa)α

βqαpβ, etc (5)

Decompose F = dA + A2 = 0 using gauge fields e, ω, b, fFa(2, 0) = Dea(2, 0) = 0 (zero torsion constraint)FL

a (1, 1) = Ra(1, 1) + e(2, 0), f (0, 2)|Ma = 0FD(1, 1) = db(1, 1) + e(2, 0), f (0, 2)|D = 0(gauge b = 0 => constraint on Schouten tensor)Fa(0, 2) = Dfa(0, 2) = 0 (Cotton equation: Cµν = 0)


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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! →


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Solution: Quantization [BN(2013)]

quantize qα, pα: => multi-commutators

expand the F = 0 equations in terms of these multi-commutators

may (?) give the full star product action now in covariant form

S =

∫Tr(ΩdΩ +

23

Ω ∧ Ω ∧ Ω)

where

Ω =

∞∑s=2

asω(s− 1, s− 1)

if the interaction terms require other fields than ω (as whencoupling to scalars) this action is a convenient starting point


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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))


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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))


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Summary

Main points:

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!


Documents

Topologically gauged CFTs in 3d: solutions, AdS/CFT and ... · Topologically gauged CFTs in 3d: solutions, AdS/CFT and higher spins Bengt E.W. Nilsson Chalmers University of Technology,