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Constraining theories with higher spin symmetry Constraining theories with higher spin symmetry

Constraining theories with higher spin symmetry - PowerPoint Presentation

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Constraining theories with higher spin symmetry - PPT Presentation

Juan Maldacena Institute for Advanced Study Based on httparxivorgabs 11121016 amp to appear by J M and A Zhiboedov amp to appear Elementary particles can have spin ID: 192344

higher spin symmetry theory spin higher theory symmetry point free theories conserved currents twist fields functions operator current matrix

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Slide1

Constraining theories with higher spin symmetry

Juan MaldacenaInstitute for Advanced Study

Based on

http://arxiv.org/abs/

1112.1016

& to appear

by J. M. and A.

Zhiboedov

& to appear. Slide2

Elementary particles can have spin. Even massless particles can have spin.

Interactions of massless particles with spin are very highly constrained.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 AdS4 .

Spin 1 = Yang Mills

Spin 2 = Gravity

Spin s>2 (higher spin) = No interacting theory in asymptotically flat spaceSlide3

AdS4

 dual to CFT3 Massless fields with spin s ≥ 1  conserved currents of spin s on the boundary. Conjectured CFT3 dual: N free fields in the singlet sector

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.

Witten

Sundborg

Sezgin

Sundell

Polyakov

Klebanov

Giombi

Yin …Slide4

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”.Slide5

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. Slide6

Vasiliev theory + boundary conditions that break the higher spin symmetry

 Dual to the large N Wilson Fischer fixed point… 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.

Polyakov

Klebanov

Giombi

YinSlide7

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 unitaryWe have a conserved current of spin, s>2. We are in d=3(We have only one conserved current of spin 2.)Slide8

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 sectorSlide9

Outline

Unitarity bounds, higher spin currents.Simple argument for small dimension operatorsOutline of the full argumentThen: cases with slightly broken higher spin symmetry. Slide10

Unitarity bounds

Scalar operator: Δ ≥ ½ (in d=3) Slide11

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 =0Slide12

Removing operators in the twist gap

Scalars with 1 > Δ ≥ ½Assume 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

a

lso have:Slide13

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  momenta need to be equal in pairs. Same reason we get the Coleman Mandula theorem !Slide14

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

. Slide15

Twist one

Now we have:

Sum over S’’ has finite rangeSome c’s are non-zero , e.g.Slide16

Structure of three point functions

Three point functions of three conserved currents are constrained to only three possible structures: - Bosons - Fermions - Odd (involves the epsilon symbol).

We have more than one because we have spinThe theory is not necessarily a superposition of free bosons and free fermions (think of s=2 !)

Giombi

,

Prakash

, Yin

Costa,

Penedones

, Poland,

RychkovSlide17

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 The three point functions are constrained to three possible forms by conformal symmetry

 lead to a large number of equations that typically fix many of the relative coefficients of various terms. The equations separate into three sets, one for the bosons part, one for the fermion part and one for the odd part.

Coefficients in

Transformation lawSlide18

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. Slide19

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

=NSlide20

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  expect the product of free theories (we did the case of two)

- Higher dimension. ConclusionsSlide21

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 symmetryN 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,

YacobySlide22

Fermions + Chern

SimonsSpectrum of ``single trace’’ operators as in the free case. Violation of current conservation: (2pt fns

set to 1 ) Insert this into correlation functions

Breaks parity

Giombi

,

Minwalla

,

Prakash

,

Trivedi

,

Wadia

, Yin

Aharony

,

Gur

-Ari,

YacobySlide23

Conclusion: All three point functions are

Two parameter family of solutions We do not know the relation to the microscopic parameters N, k. Slide24

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. Slide25

Discussion

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

FutureSlide26

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. Slide27

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.