Constraining theories with higher spin symmetry Juan Maldacena

Institute for Advanced Study Based on http://arxiv.org/abs/1112.1016

by J. M. and A. Zhiboedov & to appear. & to appear

• Elementary particles can have spin. • Even massless particles can have spin. • Interactions of massless particles with spin are very highly constrained.

Spin 1 = Yang Mills Spin 2 = Gravity Spin s>2 (higher spin) = No interacting theory in asymptotically flat space • Coleman Mandula theorem : The flat space S-matrix cannot have any extra spacetime symmetries beyond the (super)poincare group. Needs an S-matrix. Higher spin gauge symmetries  become extra global symmetries of the S-matrix. • Yes go: Vasiliev: Constructed interacting theories with massless higher spin fields in AdS 4 .

• • • Witten Sundborg Sezgin Sundell AdS 4  dual to CFT 3 Polyakov – Klebanov Giombi Yin … Massless fields with spin s ≥ 1  conserved currents of spin s on the boundary. Conjectured CFT 3 singlet sector dual: N free fields in the • • This corresponds to the massless spins fields in the bulk. 1/N = ħ = coupling of the bulk gravity theory. The bulk theory is not free.

• • • • • What are the CFT’s with higher spin symmetry (with higher spin currents) ? We will answer this question here: They are simply free field theories This is the analog of the Coleman Mandula theorem for CFT’s, which do not have an S matrix. We will also constrain theories where the higher spin symmetry is “slightly broken”.

Why do we care ?

• • • • • This is an interesting phase of gravity, or spacetime. Any boundary CFT that has a weak coupling limit has a higher spin conserved currents at zero coupling. In examples, such as N=4 SYM, this is smoothly connected to the phase where the higher spin fields are massive. Presumably by some sort of Higgs mechanism. In weakly coupled string theory, at high energies, we expect to have higher spin ``almost massless’’ fields. So it is interesting to understand the implications of this spontaneously broken symmetry. We will not address these more interesting questions here. We will just address the more restricted question posed in the previous slide.

• Vasiliev theory + boundary conditions that break the higher spin symmetry  Dual to the large N Wilson Fischer fixed point… Polyakov – Klebanov Giombi Yin • Two approaches to CFT’s : - Write Lagrangian and solve it in perturbation theory - Bootstrap: Use the symmetries to constrain the answer. Works nicely when we have a lot of symmetry. • We can also view this as constraining the asymptotic form of the no boundary wavefunction of the universe in AdS.

Assumptions

• • • • • We have a CFT obeying all the usual assumptions: Locality, OPE, existence of the stress tensor with a finite two point function, etc. The theory is unitary We have a conserved current of spin, s>2. We are in d=3 (We have only one conserved current of spin 2.)

Conclusions

• • • • There is an infinite number of higher spin currents, with even spin, appearing in the OPE of two stress tensors. All correlators of these currents have two possible forms: 1) Those of N free bosons in the singlet sector 2) Those of N free fermions in the singlet sector

Outline

• • • Unitarity bounds, higher spin currents.

Simple argument for small dimension operators Outline of the full argument • Then: cases with slightly broken higher spin symmetry.

Unitarity bounds

• Scalar operator: Δ ≥ ½ (in d=3)

Bounds for operators with spin

• Operator with spin s . (Symmetric traceless indices) • • Bound: Twist = Δ -s ≥ 1 .

If Twist =1 , then the current is conserved • We consider minus components only: Spin s-1 , Twist =0

Removing operators in the twist gap

• • • Scalars with 1 > Δ ≥ ½ Assume we have a current of spin four. The charge acting on the operator can only give (same twist  only scalars ) • Charge conservation on the four point function implies (in Fourier space) Of course we also have:

• This implies that the momenta are equal in pairs  the four point function factorizes into a product of two point functions. • • • We can now look at the OPE as 1  2 , and we see that the stress tensor can appear only if Δ=½ .

So we have a free field !

Intuition: Transformation = momentum dependent translation Mandula theorem !

 momenta need to be equal in pairs. Same reason we get the Coleman

• • • • Observations: We need to constrain both the correlators and the action of the higher spin symmetry. Of course three point functions determine the action of the symmetry. We used twist conservation and unitarity to constrain the action of the generator. Then we used this to constrain the correlators.

• Now we have:

Twist one

• • Sum over S’’ has finite range Some c’s are non-zero , e.g.

Structure of three point functions

• Three point functions of three conserved currents are constrained to only three possible structures: - Bosons - Fermions Giombi, Prakash, Yin Costa, Penedones, Poland, Rychkov - Odd (involves the epsilon symbol). - We have more than one because we have spin - The theory is not necessarily a superposition of free bosons and free fermions (think of s=2 !)

Brute Force method

• Acting with the higher spin charge, and writing the most general action of this higher spin charge we get a linear combination of the rough form Coefficients in Transformation law • • The three point functions are constrained to three possible forms by conformal symmetry  the relative coefficients of various terms. lead to a large number of equations that typically fix many of The equations separate into three sets, one for the bosons part, one for the fermion part and one for the odd part.

• In this way one constrains the transformation laws. • Then one constrains the four point function. • Same as in a theory with N bosons or fermions. One can also show that N is an integer.

Quantization of Ñ, or the coupling in Vasiliev’s theory

• • • We can show that the single remaining parameter, call it Ñ, is an integer. It is simpler for the free fermion theory It has a twist two scalar operator • Consider the two point function of • • If Ñ is not an integer some of these are negative. So Ñ=N

Conclusions

• Thus, we have proven the conclusion of our statement. Proved the Klebanov-Polyakov conjecture (without ever saying what the Vasiliev theory is !). • Generalizations: - More than one conserved spin two current of two)  expect the product of free theories (we did the case - Higher dimension.

Almost conserved higher spin currents

• • • There are interesting theories where the conserved currents are conserved up to 1/N corrections. Vasiliev’s theory with bounday conditions that break the higher spin symmetry N fields coupled to an O(N) chern simons gauge field at level k. • ‘t Hooft-like coupling Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin Aharony, Gur-Ari, Yacoby

Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin Aharony, Gur-Ari, Yacoby

Fermions + Chern Simons

(6.20), (6.14). All t hat remains is t he int egral of t he right hand side of (7.1). In order for • Spectrum of ``single trace’’ operators as in the free case. So t his relat ive coefficient is Violation of current conservation: conformal generat or K − a 2 = − 2 5 a 1 .

(7.3) fixed in this simple way, for all λ, to leading order in 1/ N .

(2pt fns set to 1 ) annihilat es t he right hand side of (7.1). We have spelled it out in order t o illust rat e t he use of t he broken symmet ry.

Breaks parity As a less t rivial example, consider t he insert ion of t he same broken charge conservat ion ident ity in t he t hree point funct ion of t he st ress t ensor. We will do t his t o leading order in • Insert this into correlation functions i j n − − − j 2 (x 1 )j 2 (x 2 )j 2 (x 3 ) ∼ √ a 1 N d 3 x [ ∂˜j 0 j 2 − 2 5 ˜j 0 ∂j 2 ](x)j 2 (x 1 )j 2 (x 2 )j 2 (x 3 ) .

S i (7.4) Now let ’stakethelargeN limit in this equation. In theleft hand sidewe can substitute t he act ion of t he charges on each of t he operat ors. T his gives [Q 4 , j 2 j 2 j 2 ] ∼ √ 1 N ∂ 3 j 2 j 2 j 2 + ∂j 4 j 2 j 2 + act ion on t he ot her j 2 s Not ice t hat t his is of order 1 N .

In t he right hand side of (7.4) t he order 1 N t erms come from d 3 x( ∂ j 2 (x)j 2 (x 1 ) ) ˜j 0 (x)j 2 (x 2 )j 2 (x 3 ) ∼ ∂ 5 1 d 3 x |x 1 1 − x| ˜j 0 (x)j 2 (x 2 )j 2 (x 3 ) (7.5) (7.6) where we have int egrat ed by part s and used t hat all indices are minus ( Now t he ∂ 1 = ∂ x − 1 ).

final result of the integral in the right hand side of (7.6) is the same as t he t hree point funct ion j 0 j 2 j 2 . Namely, t he t hree point funct ion involving a twist one scalar, as opposed t o t he twist two scalar ˜j 0 t hat we st art ed wit h. T his can be seen by t he fact t hat t he int egral in (7.6) (before t aking t he ∂ 5 1 derivat ive) has t he same conformal propert ies as j 0 j 2 j 2 .

If t he current were exact ly conserved, we would set (7.5) t o zero. In t hat case, t he only solut ion is t he one corresponding t o t he free fermion st ruct ure in t he t hree point funct ions.

T he reason is t hat t he free boson, or odd solut ions of t his charge conservat ion ident ity require a twist one operat or, j 0 . Recall t hat we had said in sect ion six t hat t hese t hree point funct ion ward ident it ies have t hree independent solut ions involving only t he fermion, 37

• Conclusion: All three point functions are • Two parameter family of solutions • We do not know the relation to the microscopic parameters N, k.

• • As we can rescale the operator and we get the large N limit of the Wilson Fischer fixed point. The operator becomes the operator which has dimension two (as opposed to the free field value of one). It also becomes parity even.

•

Discussion

In principle, it could be extended to higher point functions…

Future

• • • • It is interesting to consider theories which have other ``single trace” operators (twist 3) that can appear in the right hand side of the divergence of the currents. (e.g. Chern Simons plus adjoint fields).

These are Vasiliev theories + matter. What are the constraints on “matter’’ theory added to a system with higher spin symmetry?. Conjecture : String theory-like. Of course, this will be an alternative way of doing usual perturbation theory. The advantage is that one deals only with gauge invariant quantities.

Conclusions

• • Proved the analog of Coleman Mandula for CFT’s. Higher spin symmetry  Free theories. Used it to constrain Vasiliev-like theories • A similar method constrains theories with a higher spin symmetry violated at order 1/N.

A final conjecture

• • • Assume that we have a theory in flat space with a weakly coupled S-matrix. The the theory contains massive higher spin fields , s > 2 . The tree level S-matrix does is well behaved at high energies. • Then it should be a kind of string theory.