Presentaci

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SPECTRAL FLOW IN THE SL(2,R) WZW MODEL
Carmen A. Núñez I.A.F.E. UBA
WORKSHOP New Trends in Quantum
Gravity  
Instituto de Fisica, Sao Paulo      
Septembre 2005  

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MOTIVATIONS
CFT based on affine SL(2)k, not only for k ? Z
and unitary integrable representations (j ? Z
or Z½).

• SL(2) symmetry is rather general
• String Theory on AdS3 ? SL(2,R) WZW model
• Black holes in string theory
• Liouville theory of 2D quantum gravity
• 3D gravity
• Certain problems in condensed matter

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RATIONAL vs NON-RATIONAL CFTs

• RCFT finite number of representations of
  modular group
• (e.g. c1 on circle of rational R2
  extended algebra).
• Non-RCFT are qualitatively different

• CFTs with SL(2) symmetry
• simplest models beyond the well studied
  RCFT

• Verma module is reducible there are null
  vectors free field rep.

• Continuous families of primary fields
• No highest or lowest weight representations
• No singular vectors ? fusion rules cannot be
• determined algebraically
• OPE of primary fields involves integrals
• over continuous sets of
  operators.

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STRING THEORY ON AdS3
This string theory is special in many respects

• Simplest string theory in time dependent
  backgrounds
• Concept of time in string theory
• String theories in more complicated geometries
• In the context of AdS/CFT it is special because
  
• Worldsheet theory can be studied beyond
  sugra
• It does not require turning on RR
  backgrounds
• BCFT is 2D ? infinite dimensional algebra

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Important lessons from
stringy analyses Observables in spacetime
theory Fundamental string
excitations
Worldsheet correlation functions
Greens functions of operators (in flat
spacetime interpreted as
in spacetime CFT S-matrix elements in
target space)
Spacetime CFT has
Constraints in worldsheet theory
non-local features
These restrictions are not understood from the
string theory point of view.
Is string theory on AdS3 consistent (unitary)?
Is the OPA closed over unitary states?
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STATUS OF STRING THEORY ON AdS3

• Unitary spectrum of physical states
• (spectral flow symmetry) J. Maldacena, H.
  Ooguri hep-th/0001053
• 
  
• Modular invariant partition function
• J. Maldacena,
  H.Ooguri, J. Son hep-th/0005183
• Product of characters of SL(2,R)
  representations?
• D. Israel, C. Kounnas,
  P.Petropoulos hep-th/0306053
• Correlation functions
• J. Maldacena, H.
  Ooguri hep-th/0111180
• 
  
• Analytic continuation of J.
  Teschner, hep-th/0108121
• 
  
• Generalization of bootstrap to
• 
  

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CORRELATION FUNCTIONS

• SL(2,R) WZW model ? WZW
  model
• (actions related by analytic
  continuation of fields)
• States in H of SL(2,R) ?
  non-normalizable states in H3
• Not all states in the SL(2,R) WZW model
  can be obtained
• by analytic continuation from
  spectral flowed states
• AdS/CFT Consistency of BCFT implies awkward
  constraints
• on worldsheet correlators. Factorization
  of 4-point functions is
• not unitary unless external states satisfy
  certain restrictions with
• no clear interpretation in worldsheet
  theory.

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WZW MODEL for SL(2, R)k
k level of the representation
Infinitely many symmetries generated by currents
Ja(z), Ja(z), a?,3
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Symmetry Algebra Virasoro ? Kac-Moody
Sugawara relation
And similarly for
Lie algebra of SL(2,R) can be represented
by differential operators
x isospin coordinate
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PRIMARY FIELDS
Form representations of the Lie algebra generated
by J0a(z)
keep track of SL(2) weights
AdS/CFT interpretation location
of operator in dual BCFT
Dj m j, j1, Dj- m j, j 1, Cj?
, m ? , ? 1,
?jm ? Unitary representations of SL(2,R)
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SPECTRAL FLOW
The transformation
with w ? Z, preserves the SL(2,R) commutation
relations
Sugawara ?
obey Virasoro algebra with same c
The spectral flow automorphism generates new
representations
and
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Hilbert space of SL(2,R) WZW model



w ? Z is the spectral flow parameter or winding
number
is an irreducible infinite dimensional
representation of the SL(2,R) algebra generated
from highest weight state jwgt defined by
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is generated from j,?wgt , 0lt ? lt1) and
And the Casimir is
and are conventional
discrete and continuous represent.
and are obtained by spetral flow
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• CFTs based on affine SL(2)k are well known in the
  case of
• Unitary integrable representations of
  SU(2)
• k ? ? and integer and
  half integer spins
• 
  A.B.Zamolodchikov V.A.Fateev (1986)
• Highest weight representations
• k ? C\0 and
• Admissible representations
• Rational level k2 p/q,
  p,q coprime integers
• V.G.Kac D.A.Kazhdan
• 
  F.G.Malikov, B.L.Feigin
  D.B.Fuchs
• H.Awata Y.Yamada
• All these are RCFT Null vector
  method applies.

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CORRELATION FUNCTIONS
The correlation functions in WZW theory obey
linear differential equations which follow from
the Sugawara construction of T(z). Knizhnik-Zamolo
dchikov equation
In SU(2) there are null vectors which impose
extra constraints and allow to determine the
fusion rules. But the space of vectors of
the unitary representations of SL(2,R)
with and
with
contains no null vectors. However the
spectral flow plays their rol.
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THE SPECTRAL FLOW OPERATOR
This is an auxiliary field (not physical)
which allows to construct operators in sectors w
1 and w 1 from operators in w 0 as follows
It satisfies the primary state conditions with
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NULL VECTOR METHOD
One can apply the null vector method to
correlators containing
What information can be obtained from this null
vector?
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3-POINT FUNCTIONS
N2 SL(2,C) conformal invariance of
the worldsheet and
target space determines the x and z dependence


This coincides with analytic
continuation of Teschners result. However it
does not determine the fusion rules ? need
4-point functions
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4-POINT FUNCTIONS
SL(2,C) conformal invariance of the
worldsheet and target space ? non-trivial
dependence on cross ratios
KZ reduces to
Teschner applied
generalization of bootstrap for Maldacena
Ooguri analyzed analytic continuation. Null
vector method?
A closed form for F(z,x) is not known for generic
values of ji
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Null vector method for 4-point functions
If one operator is
? there is one extra equation
and KZ equation simplifies because ?
The spectral flow operator is not
physical. It changes the winding number of
another operator by one unit. This gives a
3-point function violating winding number
conservation by one unit.
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Comments

• N-point functions may violate winding number
  conservation
• up to N-2 units Determined by
  SL(2,R) algebra
• Result agrees with free field approximation
  (Coulomb gas
• formalism). G. Giribet and C.N.,
  JHEP06(2000)010 JHEP06(2001)033
  
  
• Supersymmetric extension D. Hofman and
  C.N., JHEP07(2004)019
• Need 5-point functions to get information
  for 4-point function

• Coulomb gas is more practical method than
  bootstrap of BPZ
• It works in minimal models and SU(2) CFT
  due to singular vectors.
• Extension to SL(2,R) requires analytic
  continuation in the number
• of screening operators. It worked for
  3-point functions, but this
• is an experimental fact. There is no
  theoretical proof.

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OPEN PROBLEMS

• Computation of 4-point functions in w ? 0 sectors
  and factorization properties. Closure of OPA on
  unitary states
• Interpretation of unitarity constraints on
  worldsheet correlators

• They do not correspond to well defined objects
  in BCFT
• if

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• Factorization of 4-point functions is not
  unitary unless
• and

j3
j1
J
j4
j2
Non-physical J not well defined
objects in BCFT
Each leg imposes additional constraints
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• Higher genus Riemann surfaces

• Modular properties?

• Factorization properties?

• Verlinde theorem?

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THE END