Chern-Simons vector models and higher spins - Ginzburg ...

Chern-Simons vector models and higher spinsSimone GiombiPerimeter InstituteGinzburg Conference, Moscow, June 1 2012Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 1 / 24

OutlineThe Klebanov-Polyakov-Sezgin-Sundell conjectures:HS gravity in AdS 4 ↔ 3d vector modelsVasiliev’s higher spin gauge theory in 4dthe “Type A” and “Type B” modelsParity violating modelsChern-Simons theory with vector fermion matterExact planar thermal free energy on R 2Higher spin symmetry at large N and conjectural AdS dualSummary and conclusionsBased on work with S. Minwalla, S. Prakash, S. Trivedi, S. Wadia, X. YinSimone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 2 / 24

Klebanov-Polyakov-Sezgin-Sundell (’02) conjecture:Vasiliev’s minimal bosonic HS gravity in AdS 4 is dual tofree or critical 3d O(N) vector model, in the O(N) singlet sector.S = 1 2∫d 3 x∂ µ φ i ∂ µ φ i ↔ “type A” HS gravity∫S =(∆, S) = (1, 0) + + ∑ s even(s + 1, s)d 3 xψ i γ µ ∂ µ ψ i ↔ “type B” HS gravity(∆, S) = (2, 0) − + ∑ s even(s + 1, s)Critical theories: interacting fixed points reached after perturbing these freetheories by quartic interaction. Correspond to change of boundary conditionon the bulk scalar field in the HS gravity side.Non-minimal versions (all integer spins): vector models with complex fieldsin U(N) singlet sector.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 3 / 24

Klebanov-Polyakov-Sezgin-Sundell (’02) conjecture:Vasiliev’s minimal bosonic HS gravity in AdS 4 is dual tofree or critical 3d O(N) vector model, in the O(N) singlet sector.Why vector models? A free gauge theory of SYM type also has HScunserved currents J s ∼ TrΦ∂ s Φ. But in addition there are manymore single trace operators Tr Φ∂ k 1Φ∂ k 2Φ · · · ∂ kn Φ, which should bedual to massive fields in the bulk.In a vector theory, operators of the form (φ i ∂ · · · ∂φ i )(φ j ∂ · · · ∂φ j ) areanalogous to multi-trace operators and should be thought asmulti-particle states from bulk point of view.A vector model has precisely the right spectrum to be dual to a pureHS gauge theory!Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 4 / 24

Klebanov-Polyakov-Sezgin-Sundell (’02) conjecture:Vasiliev’s minimal bosonic HS gravity in AdS 4 is dual tofree/critical 3d O(N) vector model, in the O(N) singlet sector.The restriction to singlet sector is important to match boundary andbulk spectrum. It may be implemented by gauging the O(N)symmetry and taking a limit of zero gauge coupling. In practice, wemay couple the vector field to a Chern-Simons gauge field at level k,and take the limit k → ∞.This suggests it may be interesting to study more generally vectormodels coupled to Chern-Simons at finite coupling (i.e. finiteλ = N/k in the large N limit).Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 5 / 24

The Vasiliev’s equationsMaster fields:1. W (x|y, ȳ, z, ¯z) = W µ dx µ 1-form in space-time2. S(x|y, ȳ, z, ¯z) = S α dz α + S ˙α d¯z ˙α 1-form in (z, ¯z)-space3. B(x|y, ȳ, z, ¯z) scalarx µ : spacetime, y α , ȳ ˙α , z α , ¯z ˙α : twistor variables.Collecting W and S into the 1-form A = W µ dx µ + S α dz α + S ˙α d¯z ˙α ,Vasiliev’s equation can be written asdA + A ∗ A = V(B ∗ κ)dz 2 + ¯V(B ∗ ¯κ)d¯z 2dB + A ∗ B − B ∗ π(A) = 0Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 6 / 24

The Vasiliev’s equationsdA + A ∗ A = V(B ∗ κ)dz 2 + ¯V(B ∗ ¯κ)d¯z 2dB + A ∗ B − B ∗ π(A) = 0Up to field redefinitions, V(X ) can be put in the formV(X ) = X exp ∗ (iΘ(X )) ,Θ(X ) = θ 0 + θ 2 X ∗ X + θ 4 X ∗ X ∗ X ∗ X + . . .An infinite family of HS gravity theories in 4d. Same spectrum, but achoice of Θ(X ) characterizes the interactions in the theory.(e.g. θ 0 affects 3-point interactions. θ 2 , θ 4 , . . . enter in higher-pointfunctions)Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 7 / 24

ParityIf we impose that the theory has a parity symmetry only twoinequivalent choices are left• Θ(X ) = 0, i.e. V(X ) = X if B is parity even• Θ(X ) = π , i.e. V(X ) = iX if B is parity odd2which correspond respectively to the “type A” and “type B” models,conjecturally dual to scalar/fermion vector models (free or critical).If we do not require parity symmetry, we have a large class of possibleparity breaking HS gravity theories parameterized by a choice of thefunction Θ(X ), or parameters θ 0 , θ 2 , . . ..At least classically, these are all consistent HS theories in AdS 4 . Onemay ask what are the dual CFTs.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 8 / 24

Chern-Simons vector modelSG, S. Minwalla, S. Prakash, S. Trivedi, S. Wadia, X. Yin 2011Consider the 3d theory of a fundamental massless fermion coupled toa U(N) Chern-Simons gauge field at level kS = k ∫4π S CS(A) + d 3 x ¯ψ i γ µ D µ ψ i i = 1, . . . , NIn 3d, ψ has dimension 1, and the only marginal coupling is theChern-Simons coupling k. This cannot run because it is quantized tobe integer.Fine-tuning the mass of the fermion to zero, we obtain a family ofinteracting CFT’s labelled by two integers k, N.Taking k → ∞, this reduces to the singlet sector of the free fermionicvector model dual to Vasiliev’s type B theory.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 9 / 24

Chern-Simons vector modelS = k ∫4π S CS(A) + d 3 x ¯ψ i γ µ D µ ψ i i = 1, . . . , NWe will be interested in the large N ’t Hooft limitN → ∞, k → ∞ with λ = N k fixedIn this limit, we effectively have a continuous line of non-susy CFT’sparameterized by λ. At λ = 0 we reduce to the free fermionic vectormodel.All I said so far applies for fermion being in any representation, e.g.the adjoint. However, working with a vector fermion entails severalsimplifications so that exact results become possible.The analogous Chern-Simons bosonic vector model has been studiedin parallel to our work in Aharony et. al., 2011. Also, interesting workin progress on susy extensions of these Chern-Simons vector models(see X. Yin talk).Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 10 / 24

Chern-Simons vector modelI will discuss in particular two interesting results about the large Nlimit of this Chern-Simons vector model1. The exact free energy of the theory on R 2 at finite temperatureF = −T log Z R 2 ×S 1 β = − h(λ) NV 2T 3h(λ) is a non-trivial function which we can compute exactly in λ.2. At N → ∞, for all λ, the theory admits an ∞-dimensional higher spinsymmetry, i.e. there is an infinite tower of HS currentsJ s , s = 1, 2, 3, . . . which are conserved at large N, so that∆(J s ) = s + 1 + O( 1 N )∀ λSimone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 11 / 24

Exact thermal free energyThe Chern-Simons gauge field does not carry propagating degrees offreedom, so the theory is still essentially a vector model, and weexpect it to be simpler than a typical large N gauge theory.However, the cubic self-interaction of the CS gauge field still makesperturbation theory complicated in general.Drastic simplifications can be achieved in a convenient gauge. Weemploy the “light-cone gauge”A − = 0 x ± = x 1 ± ix 2Here x 1 , x 2 are the Euclidean coordinates on R 2 . The Euclidean timedirection is x 3 , which will be compactified on a circle of radiusβ = 1/T .In this gauge, the cubic self-interaction vanishes, and the large N freeenergy can be solved exactly.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 12 / 24

Exact fermion propagatorThe basic ingredient we need to get the free energy is the exactfermion propagator〈ψ(p) i ¯ψ(−p) j〉 = δ j i1ip µ γ µ + Σ(p)Σ(p) is the exact fermion self-energy. In the light-cone gauge and inthe planar limit, it receives contributions only from 1PI rainbowdiagrams+ + . . .Note that diagrams with matter loops do not contribute at leadingorder at large N, because the fermion is in the fundamental.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 13 / 24

Exact fermion propagatorIt is not difficult to see that the sum of rainbow diagramscontributing to Σ(p) satisfies the Schwinger-Dyson equationΣ(p) = N 2∫d 3 ()q(2π) 3 γ µ 1iγ α q α + Σ(q) γν G µν (p − q)Here G µν (p) is the light-cone A µ propagator: G +3 = −G 3+ = 4πikp. +At finite temperature, we impose antiperiodic b.c. on the fermion, soq 3 = 2π ∫ ∫(n + 1/2), d 3 q → d 2 q ∑βZ+1/2Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 14 / 24

Exact fermion propagatorEmploying the “dimensional reduction” scheme to regulate the loopintegrals (shown to be consistent in CS-matter theories by Chen,Semenoff, Wu ’92 up to 2-loops), we solved the Schwinger-Dysonequation explicitly.The solution takes the formwithΣ(p) = f (βp s)p s + i g(βp s)p − γ + p 2 s ≡ p 2 1 + p 2 2f (y) = 2λ ( [ 1y log √])2 cosh c2 + y2 2 , g(y) = c2y − f 2 (y)2(c = 2λ log 2 cosh c )2The equation determining c = c(λ) has no solutions for |λ| ≥ 1. Weconclude that the CFT is defined only for 0 ≤ |λ| < 1.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 15 / 24

Exact thermal free energyOnce we have the exact fermion self-energy Σ, one may show by pathintegral or diagrammatically that the free energy is given in terms ofΣ byF = NV ∑ ∫ [ d 2 q2T(2π) Tr log [iγ µ q 2 µ + Σ(q)] − 1 ()]2 Σ(q) 1iγ µ q µ + Σ(q)nPerforming the integral and sum, the final result isF = − NV 2T 3 [c 3 1 − λ ∫ ∞+ 6 dy y log ( 1 + e −y )] ≡ −NV 2 T 3 h(λ)6π λwhere c = c(λ) is the constant introduced earlier.cSimone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 16 / 24

FNV 2 T 30.250.200.150.100.050.2 0.4 0.6 0.8 1.0 Λh(λ) = 3ζ(3) 2(log 2)3− λ 2 (log 2)4− λ 4 + . . . λ ≪ 14π 3π 2π(1 − λ)h(λ) ∼ log 3 (1 − λ) + . . . λ → 16πThe function h(λ) decreases monotonically from the free field valueto zero at λ = 1. Extreme thinning of d.o.f. at “strong coupling”.For comparison, in ABJM model we have h(λ) ∼ 1/ √ λ at λ → ∞.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 17 / 24

Higher spin symmetry at large NRecall that in the free theory (λ = 0), the spectrum of U(N) invariantsingle trace primaries isJ 0 = ¯ψ i ψ i , J s ∼ ¯ψ i γ (µ1 ∂ µ2 · · · ∂ µs)ψ i + . . .In the interacting theory, these can be made gauge invariant by∂ µ → D µ . The CS sector does not add any further single-traceprimaries, because (F µν ) i j ∼ 1 k ¯ψ j γ ρ ψ i ɛ µνρ by e.o.m.In the free theory ∂ · J s = 0, i.e. J s are in short representations of theconformal algebra with (∆, S) = (s + 1, s).Turning on the interaction, we expect the currents not to beconserved any more and to acquire anomalous dimension∆ s = s + 1 + ɛ s (λ, N).Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 18 / 24

Higher spin symmetry at large NBut for the currents to become non-conserved at λ ≠ 0, we musthave∂ · J s ∼ λO (s+2,s−1)In other words, there must be an operator in the (s + 2, s − 1)representation with which J s can combine to form a longrepresentation.At N = ∞, single trace operators can only combine with other singletrace operators. But there are no single-trace primaries in thespectrum with quantum numbers (s + 2, s − 1)!Therefore we conclude that at N = ∞, for all λ, the currents are stillconserved, which implies∆(J s ) = s + 1 + O( 1 N )∀ λThe vector nature of ψ is essential for this to work.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 19 / 24

Higher spin symmetry at large NWhat happens is that, at finite N, J s can (and does) combine with“multi-trace” operators. The non-conservation equation takes theschematic form∂ · J s ∼ f √ (λ) ∑∂ m J s1 ∂ n J s2 + g(λ) ∑∂ m J s1 ∂ n J s2 ∂ p J s3N NThe argument above implies that the HS currents do not haveanomalous dimensions in the planar limit. But one can in fact arguethat the scalar J 0 has protected dimension as well∆(J 0 ) = 2 + O( 1 N )which we have checked perturbatively to two-loop order.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 20 / 24

Comments on the holographic dualAt λ = 0, we know that the theory should be dual to the Vasiliev’s“type B” theory. So the holographic dual should be some deformationof it.Turning on λ, we have seen that the spectrum of “single trace”primaries is(∆, S) = (2 + O( 1 ∞ N ), 0) + ∑(s + 1 + O( 1 ), s)Nwhich implies that the dual bulk spectrum should contain classicallymassless higher spin fields and a m 2 = −2 scalar.The HS fields (and the scalar) can acquire mass via loop-corrections,but the bulk classical equations of motion should have exact higherspin gauge symmetry (to decouple longitudinal polarizations).Hence, the holographic dual should still be a higher spin gauge theory(with HS symmetry broken at quantum level), and it should breakparity due to the boundary Chern-Simons term.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 21 / 24s=1

Comments on the holographic dualThe only parity breaking higher spin gravity theory currently known isVasiliev’s theory specified by the general “interaction phase”Θ(X ) = θ 0 + θ 2 X ∗ X + . . .A natural conjecture is then that our Chern-Simons vector model isdual to the parity breaking Vasiliev’s theory with some specific choiceθ 0 (λ) , θ 2 (λ) , . . .with the condition that θ 0 (λ → 0) = π 2 , θ 2,4,...(λ → 0) = 0.We do not know a priori how to determine the phase as a function ofλ. But we can in principle compute perturbatively correlators on bothsides and compare.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 22 / 24

Comments on the holographic dualFrom considerations based on the softly broken HS symmetry purelyon CFT side, Maldacena-Zhiboedov showed that 3pt functions shouldbe a sum of free boson, free fermion and a parity odd tensor structure〈J s1 J s2 J s3 〉 = cos 2 θ 0〈J s1 J s2 J s3 〉 B + sin 2 θ 0〈J s1 J s2 J s3 〉 F + sin θ 0 cos θ 0〈J s1 J s2 J s3 〉 oddConfirmed by a direct 2-loop calculation in the CS-fermion theory,which gives θ 0(λ) = π 2 (1 − λ) + O(λ3 ).From the bulk calculation in Vasiliev’s theory with general phase θ 0 ,we get such a decomposition, with 〈JJJ〉 B and 〈JJJ〉 F correctlycoming out. However currently the coefficient of sin θ 0 cos θ 0 appearsto vanish...The appearance of the odd structure should just follow fromsymmetries as shown by MZ, strongly suggesting that we are missingsomething in the bulk calculation.Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 23 / 24

Summary and conclusionChern-Simons vector models define lines of interacting CFT’s withlagrangian description. They have approximate higher spin symmetryat large N.We proposed a generalization of the KPSS conjecture which involvesa parity breaking version of Vasiliev’s HS gravity. Partial evidence,still work in progress.Some future directionsHigher-point functions from bulk and CFTStudy of exact solutions and their CFT interpretationFree energy from the bulk HS theory? (Bulk action?)Susy extensions and relation to string theoryExtensions to higher dimensions. . .Simone Giombi (PI) Chern-Simons and higher spins Lebedev Inst., Jun 1 2012 24 / 24