Lagrangian formulation of Higher Spin Theories on AdS Space

Kamal L. Panigrahi

Based on hep-th/0607248In collaboration with A. Fotopoulos, and M. Tsulaia

PLAN of this talk

• Motivation

• Tensionless limit and bosonic string triplet

• Massless fields on flat space time (a simple example)

• Fields on AdS

• A triplet in ambient space

• A massless vector field in ambient space

• Conclusions and open questions

Motivation • HS gauge theories are classical Field Theories describing

interacting massless fields with arbitrary spin• They are understood very well on AdS(d) space [Vasiliev’90]• Any connection with string theory / M-theory ?• Conjecture: Massless HS theory is the most symmetric phase of

string theory• String theory as a spontaneously broken phase of HS gauge theory• String theory at low energy limit infinite tension limit described

by sugravity• How about the high energy limit ?• String theory on AdS with T0 (high energy) and with a small AdS

radius (L0)• String theory contains in its spectrum an infinite tower of massive

HS particles In the frame work of AdS/CFT correspondence:

• Large AdS radius Higher spin fields becomes extremely massive and decouple from supergravity In the CFT side this correspond to Large ‘t Hooft limit

• Hence one can make predications about the strongly coupled CFT using the AdS/CFT correspondence.

• What happens in the opposite limit ? • String spectrum in AdS5 X S5 background at small AdS radius [Beisert, Bianchi,......04] which precisely matches with the operator spectrum of free N=4 SYM in the planar limit.• The limit of zero YM coupling has been conjectured to be dual to the massless HS field theory in AdS space.• Turning on coupling in YM side corresponds to Higgs phenomena in AdS (massless higher spin fields develop mass, by eating the lower spin gauge fields (La Grande Bouffe)) [Beisert, Bianchi,......04]

•So it is interesting and instructuve to know about the bulk physics

Tensionless limit and bosonic string triplets [Francia, sagnotti, Tsulaia,..]

• Start with the usual commutators• Virasoro generator• We are interested in the tensionless limit, where the full

gauge symmetry of the massive string spectrum is recovered.

• Let us introduce the reduced generators

• They satisfy

• Notice that the central charge vanishes.

BRST charge and tensionless limit• Introduce ghost mode C of ghost number 1 and anti

ghost B of ghost number -1 with• BRST charge is given by

• And the open bosonic field satisfies • The coresponding ghost vacuum satisfies

BRST charges • Make the rescaling (for non zero k)

• for k = 0

• The anti-commutation relation is not affected by this rescaling, but allows a non-singular limit

that defines a nilpotent BRST charge

• It is convenient to write the Q concisely as

• With

• The string field and the gauge parameter can also be decomposed as

• Now the string field theory equations of motion and coresponding gauge transformations become

Totally symmetric tensor

• We work with oscillator pair and effectively the constraint reduced to triplet.

• The string field and gauge parameter now involve the ghost mode and antighost mode

• The limiting form of the BRST charge Q implies that the field equations describe independent triplet

of symmetric tensors of rank (s, s-1, s-2), defined by

• The gauge transformation parameter

• Now expanding the field equations lead to

• And the corresponding gauge transformations

• This type of structure was first noticed by A. Bengtsson (without the field C)

• These field equations follow from the Lagrangian

• In compact notations it reads

• One can obtain an equivalent description in terms of a pair of two symmetric tensors. The Fronsdal kinetic operators

• The field equations become

• Which follows from

Massless fields on flat space-time• Triplet construction the gauge invaiant description of massless

fields with spins (s, s-1, s-2, ...1/0), requires in addition to a tensor field of rank s, two more auxillary fields of rank (s-1) and rank (s-2).

• After complete gauge fixing one is left only with the physical polarizations of higher spin fields with spin (s, s-1, s-2, ..., 0/1)

• To illustrate this let us take the simplest non-trivial example: ranks 2, 1, 0 fields:

• The triplet equations take the form

Gauge transformations

• The system is invariant under the following gauge transformations

• Let us introduce a traceless field

• Light cone gauge fixing: eliminate

using the gauge transformation parameter • Field equations eliminate and

Generalization

• Finally one is left with the physical polarizations

• spin 2 field and a gauge invariant scalar

• For arbitrary spin the field equations look like

• And the gauge transformations

Action

• The field equations can be derived from the action

• For the higher spin case one has• Let us add one more condition • It makes the the gauge transformation parameter

constrained it obeys the vanishing trace condition in addition to field eqns gives the possibility to

gauge away the and one obtains the propagation of a single irreducible

higher spin mode.

Fields on AdS

• AdS space is a vacuum solution of Einstein equations with a negative cosmological constant

• Riemann tensor has a form

• correspond to the flat space limit.• Conveniently represent the D-dimensional AdS space

as a hyperboloid in D+1 dimensional flat space.• AdS isometry group is noncompact unitary rep. is

infinite dimensional

• Let us rewrite the SO(D-1, 2) algebra:

• In a different way by defining

• Which look like

• AdS isometry group has a maximal compact subgroup spanned by H and J.

Massless and massive fields on AdS

• Infinte dimensional unitary rep of AdS group are obtained from lowest weight states |E0, s >, with

• All the states are formed by • One has to check the norm of the state• Unitary bound: below which all the states have negative

norm and therefore excluded from the spectrum• States that saturate unitary bound are massless states• Fields whose energy is above the unitary bound are the

massive rep of AdS

• To obatain wave equations describing massless fields with an arbitrary integer value of spin on AdS background one has to solve

• where

• Auxillary space is spanned by the oscillators with

• Consider a state

• Generators

A Triplet in ambient space of AdS [Fronsdal’ 79, Metsaev’94] • We consider the reducible rep of AdS group and the embedding is

specified by the condition

• x-space coordinates are homogeneous solutions of degree zero in ambient space.

• The field transforms to the ambient space as

• Inverse tranformation

• Where

• Take a state in the Fock space

• The commutation relation • A state in ambient space satisfies

• Ordinary derivative is replaced by• Momentum operator acting on a state produces proper

covariant derivative

• D’ Alembertian operator

• Divergence operator

• Symmetrized exterior derivative operators

• Having these operators at hand, and their algebra one can construct the BRST charge one gets the Lagrangian density.

Construction in ambient space

• Having all the transformation rules between the x and y-space, one gets

• Construction of the BRST charges Q

• where

• Further

• The Lagrangian

• Gauge transformation

• Total vacuum is a direct product of ghost and alpha vacuum

• One can see that in order for the Lagrangian to have ghost number 0, the field should have ghost number 0

and gauge tranformation parameter should have the ghost number -1.

• Their expansion in terms of ghost variable

Now the Lagrangian looks like

• Gauge transformations

• Equations of motion

• Final form of the Lagrangian (adding )

A massless vector field on AdS

• Take a massless vector field in AdS background, so that the Lagrangian now contains one physical field and an auxillary field C. In ambient space:

• Equations of motion

• Gauge transformations

Puzzle and Solution

• when making embedding into the higher dimensional space, some extra degrees of freedom appear.

• For exp: A vector in ambient space (y-space) may correspond to a vector and a scalar in the x-space.

• However, in D+1 dimensional ambient space there are extra gauge freedom that allows to eliminate the extra degrees of freedom.

• In particular one can use vanishing of gauge freedom to eliminate the a scalar from the spectrum.

• Now one is left with a gauge parameter with the constraint

• Using this one can further gauge away C and check that this further gauge fixing procedure is consistent with equations of motion.

• Finally one is left with the following that describes a massless vector field on AdS:

• In the x-space which means the following Lagrangian that contains a physical field and an auxillary field C:

• Note that this is invariant under the gauge transformations

• Equations of motion derived from the above action

• Using the gauge transformations, one can gauge away the field c and finally left with the equations of motion of a single massless field

Conclusion and open problems

• Lagrangian formulation of higher spin fields in AdS background by using the triplet method.

• Use the ambient space formulation to embed the D-dimensional AdS space into (D+1) dimensions in ambient space.

• Demonstrated the equivalence by taking the simplest example of U(1) gauge field.

• Self interacting triplets on AdS [Buchbinder, Fotopoulos, Petkou, Tsulaia’ 06]

• Mixed symmetry fields on AdS

• Possible connection with the massless and massive HS theory with the superstring and M-theory