STRING THEORY on ADS3 and ADS3 CFT2

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STRING THEORY on ADS3 and ADS3 /CFT2

• SOME OPEN PROBLEMS
• Carmen A. Núñez
• I.A.F.E CONICET-UBA
• II Workshop Quantum Gravity
• Sao Paulo, September 2009

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MOTIVATIONS

• String propagation on curved backgrounds has
  received much attention own interest and
  relation to gauge theories
• Despite important progress in various directions,
  several quite elementary questions remain
  unanswered AdS3
• Briefly review these problems and comment on
  status of the AdS3/CFT2 correspondence

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String theory on AdS3

• AdS3 geometry NS-NS two-form
• exact background
• AdS3 SL(2,R) group manifold
• 2d conformally invariant s-model WZNW model
• AdS3 simplest setting beyond flat space.
• General non-compact group manifolds define
  natural framework to study strings on non-trivial
  geometries.
• Restricting to simple groups, only SL(2,R)
  possesses a single time direction.

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AdS/CFT correspondence

• Sugra in bulk of AdSn large N SYM on
  boundary
• Low energy limit of more fundamental superstring
  th.
• Exact structure of string theory on AdSn?
• Connection to SYM theory on boundary?
• AdS3 plays special role
• Asymptotic isometry group is 8 dimensional
• Theory on boundary is 2d CFT (? 2d sigma model
  whose target space is bulk AdS3 on which the
  string propagates.)
• So far, only case where duality can be checked
  beyond sugra level
• Understanding relationship b/ these two CFT has
  led
• to set more precisely AdS/CFT correspondence
• to get feedback on structure of string theory on
  AdS3

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Bosonic string theory on AdS3

• Very little is known about WZNW models on
• non-compact groups
• Most works based on
• formal extension of the compact case
• in the framework of current-algebra techniques.
• Most works deal with Euclidean AdS3
  ,
• but string spectrum is very different.

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Brief review of AdS3 WZNW model

• Symmetry generated by SL(2,R)L ? SL(2,R)R current
  algebra
• Sugawara relation
• Virasoro algebra

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The spectrum

• Physical string states must be in unitary
  representations of SL(2,R)
• Hilbert space H decomposes into unitary rep of
  current algebra labeled by eigenvalues
  J. Maldacena and H. Ooguri (2000)
• Representations parametrized by j, related to
  second Casimir as
• Principal discrete representations (lowest and
  highest weight)
• 
  m j n
• 
  m j n n0,1,
• Principal continuous representations

Cj? , m ? , ? 1,
??(0,1
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Representations of the current algebra

• Primary states annihilated by Jn3, ?, n?1
• Acting with Jn3, ?, n?1 on primaries
  SL(2,R)
• 
  
• Eigenvalues of L0
• bounded below

Weight diagram of

• No-ghost theorem
• Evans, Gaberdiel, Perry (1997)

Bound on mass of string states Partition function
not modular invariant
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Spectral flow symmetry

• The transformation

with w ? Z preserves the commutation relations
Sugawara ?
and obey Virasoro algebra with same c
The spectral flow automorphism generates new
representations
and
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Spectral flowed representations

• Compact groups (SU(2)) the spectral flow maps
  positive energy
• 
  representation of current algebra into another.
• SL(2,R), the spectral flow with w1 L0
  is not bounded below

contain ghosts unless
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Unitary spectrum

• Only case one gets a rep. with L0 bounded below
  by spectral flow is with w?1

Spectral flow symmetry implies are
restricted to AdS/CFT Operators outside this
bound cannot be identified with
local operators in BCFT
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Physical spectrum of string theory on AdS3
w ? Z winding number
Short strings
Long strings
Virasoro constraints decouple negative norm
states unitary spectrum

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Correlation functions

• Spectrum verified by partition function
• 
  J.Maldacena,H. Ooguri, J.Son (2001).
• To determine consistency and unitarity of full
  theory,
• show that OPE closes on Hilbert space of the
  theory ? correlation functions
• Some two- and three-point amplitudes computed
• 
  J.Maldacena and H. Ooguri (2001)
• by analytic continuation from
  Teschner (1999, 2000)
• Spectrum of SL(2,R) non-normalizable states
  of
• No spectral flow representations in

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Some open questions

• Despite many efforts and apparent
  simplicity of the model, several important and
  elementary issues are still beyond our
  understanding
• Is the OPE of states closed in the spectrum of
  the theory?
• Scattering amplitudes beyond 3-point functions?
• construct four-point functions in different
  sectors and
• verify that intermediate states in the
  factorization belong to the spectrum and it
  agrees with spectral flow selection rules
• Can we apply techniques developed for RCFT?
• Fundamental problem
• OPE primary states sufficient in RCFT,
  descendants not strictly necessary.
• Spectral flow operation maps primaries
  into non-primaries

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Eucidean vs. Lorentzian theory

• Principal continuous rep.

• SL(2,R)

m ? , ?
1, ??(0,1
m j n, n 0,1,

No spectral flow or discrete rep.
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To achieve this goal

• Analytic and algebraic structure of SL(2,R)
  explored further.
• Conformal bootstrap approach requires knowing
  OPE and structure constants. Then one can
  construct any ngt3-point function in terms of two-
  and three-point functions.
• Coulomb gas approach Works well for RCFT
  (minimal models, SU(2)) but requires analytic
  continuation in models with continuous sets of
  primary fields.
• Studied the OPE in AdS3 by analytic continuation
  from
• adding w
  W. Baron, C.N. Phys. Rev. D79 (2009) 086004
• Reproduced exact three-point functions S.
  Iguri, C.N. Phys.Rev.D77 (2008)
• 
  
  066015
• Constructed w-conserving four-point functions for
  generic states in AdS3 using Coulomb gas
  formalism S. Iguri,
  C.N arXiv0908.3460

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Operator Product Expansion
W. Baron, C.N. Phys. Rev. D79 (2009) 086004
Generalized OPE of including spectral
flow
Both spectral flow preserving and
non-preserving 3-point functions
Maximal region in which parameters may vary such
that no poles hit contour of integration is
Admits analytic continuation to generic j1, j2
defined by deforming the contour. Deformed
countour original finite circles
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Fusion rules

• OPE closed in H
• Verifies several consistency checks
• k? 0 limit classical tensor
  products of SL(2,R) rep.
• w selection rules are reproduced

Full consistency of fusion rules should follow
from analysis of factorization and crossing
symmetry of four-point functions
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Four-point functions

• Bootstrap program based on above OPE gives
  four-point functions with w-preserving and
  violating channels.
• If correlators in AdS3 are obtained from those in
  by analytic continuation, both channels
  must give equivalent contributions.
• Explicit computation of w-conserving four-point
  functions using Coulomb gas approach (free
  fields) confirms this.
• 
  S. Iguri, C.N. arXiv0908.3460

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Coulomb gas realization

• Successful for minimal models and SU(2)-CFT, j ?
  Z/2, but analytic continuation in theories with
  continous sets of primary fields?
• Wakimoto realization in terms of free fields

Interaction term becomes negligible near the
boundary ???. Theory can be treated
perturbatively in this region
Vertex operators Spectral flow operators
screening charges
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Results
Norlund-Rice theorem
Meromorphic continuation of K(l) with simple
poles at x0,1,2,
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Analytic continuation

• Reproduces expression obtained by J. Teschner for
• Extends previous work by V. Dotsenko, NPB358
  (1991) 547
• Some particular four-point functions can also be
  written as

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Conformal blocks

• Multiple integral realization of
  conformal blocks.

• Relation found between conformal blocks in
  Liouville theory and Nekrasovs partition
  function of N2 theories revives longstanding
  idea that all CFT can be described with free
  fields.
• Alday,
  Gaiotto, Tachikawa, arXiv0906.3219
• Dotsenko-Fateev integrals and Nekrasovs
  functions provide a basis for hypergeometric
  integrals

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AdS3/CFT2 correspondence

• Type IIB superstring theory on AdS3 x S3 x T4
  dual
• 2d CFT describing the D1-D5 system on T4
• 
  J. Maldacena (1997)
• Low energy description of D1-D5 system is smodel
  with target space (T4)N/SN, SN permutation group
  of NN1N5 elements
• 
  J. Maldacena, A. Strominger (1998)

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Naive evidence from symmetries
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The symmetric orbifold

• Chiral spectrum of smodel built on that of
  single copy of T4 plus operators in twisted
  sectors
• Chiral operators are constructed as

?n 1, a
twist field of a single element of SN
Charged under R-symmetry group of SU(2)L ? SU(2)R
Two and three-point functions computed by O.
Lunin and S. Mathur (2001) and A. Jevicki, M.
Mihailescu, S.Ramgoolam (2000)
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The dual string theory

• Near horizon geometry of the D1-D5 system
  AdS3xS3xT4 plus fluxes of RR fields through
  the S3
• Progress in formulation of string propagation on
  RR backgrounds has been made, but explicit
  calculations in AdS3 geometry cannot be done yet
• Convenient to study string theory on the S-dual
  background
• Near horizon limit of NS1-NS5 system is AdS3 x S3
  x T4 with fluxes of NS field B??
• Worldsheet theory is WZNW model with
• SL(2,R) ? SU(2) ? U(1)4
  affine symmetry

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Chiral primaries

• H J
• Vertex operators of chiral primaries (in -1
  picture)
• C.
  Cardona, C.N., JHEP0906, 009 (2009)

SU(2) charge
Spacetime conformal dimension
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Three-point functions

• Factorize into products of SU(2), fermions,
  SL(2,R) fields.
• SU(2) correlators A.
  Zamolodchikov, V. Fateev (1986)
• Unflowed sector M.
  Gaberdiel, I. Kirsch (2007)
• 
  A. Dabolkhar, A. Pakman (2007)
• Spectral flowed sectors
• Fermionic correlators G. Giribet, A.
  Pakman, Rastelli (2007)
• SL(2,R) correlators
  C. Cardona, C.N. (2009)

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AdS3/CFT2 dictionary

• k N5
  Maldacena, Strominger, 1998
• gs2 N5 Vol(T4)/ N1
  Giveon, Kutasov, Seiberg, 1999
• n 2j1kw G. Giribet, A.
  Pakman, Rastelli, 2007
• The matching obtained so far reflects the
  cancellation of structure constants of AdS3 with
  those of S3. Fermionic contributions reduce to
  unity in all cases considered.
• Extend the dictionary to four-point functions and
  descendant states. Consider more general internal
  spaces.

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Conclusions

• Despite significant recent progress, string
  theory on AdS3 (one of
• the simplest examples beyond flat space) is
  not well understood.
• Several consistency checks have been performed
  to determine
• consistency but spectral flow sectors have to
  be studied further.
• We determined the fusion rules and computed
  w-conserving four-
• point functions using the Coulomb gas
  approach.

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Conclusions

• We showed that spectral flow preserving and
  violating channels give equivalent contributions,
  thus corroborating that w-conserving amplitudes
  in AdS3 can be obtained from by analytic
  continuation.
• We verified AdS3/CFT2 correspondence in arbitrary
  spectral flow sectors up to three-point
  functions, providing additional verification of
  AdS3/CFT2 duality conjecture.

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Open problems and future directions

• It is necessary to understand mechanism
  determining the decoupling of non-unitary states
• Verlinde theorem relating fusion
• coefficients with modular transformations.
• 
  Constructing and studying
• 
  w-violating four-point functions.
• Compute four-point functions in AdS3 x S3 x T4
  and compare with symmetric product Pakman,
  Rastelli, Razamat (2009)
• Construct more examples of AdS3/CFT2 duality
  (other internal geometries)

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THE END
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Vertex operators spectral flow selection rules
Vertex operators for primary states
Spectral flow operator
Vertex operators for w ? 0 states
at least one state in


all states in
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Fusion rules and interactions

•