Lagrangian formulation of Higher Spin Theories on AdS Space

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Lagrangian formulation of Higher Spin Theories on
AdS Space

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

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• 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

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Motivation

• HS gauge theories are classical Field Theories
  describing interacting massless fields with
  arbitrary spin
• They are understood very well on AdS(d) space
  Vasiliev90
• 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 T?0 (high energy) and
  with a small AdS radius (L?0)
• String theory contains in its spectrum an
  infinite tower of massive HS particles
• In the frame work of AdS/CFT correspondence

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• 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 N4 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

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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.

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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

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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

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• 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

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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

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• 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)

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• 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

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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

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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
  

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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

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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.

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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 D1 dimensional flat
  space.
• AdS isometry group is noncompact ? unitary rep.
  is infinite dimensional

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• 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.

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Massless and massive fields on AdS

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

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• 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

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A Triplet in ambient space of AdS Fronsdal 79,
Metsaev94

• 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

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• 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

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• 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.

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Construction in ambient space

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

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• Construction of the BRST charges Q
• where

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• Further
• The Lagrangian
• Gauge transformation
• Total vacuum is a direct product of ghost and
  alpha vacuum

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• 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

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• Gauge transformations
• Equations of motion
• Final form of the Lagrangian (adding
  )

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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

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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 D1 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.

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• 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

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

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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 (D1) 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