The FZZ-duality conjecture- A Proof
Y. Hikida

Refs.
[1] YH-Schomerus, JHEP10(2007)064 [0706.1030].
[2] YH-Schomerus, JHEP12(2007)100 [0711.0338].
[3] YH-Schomerus, arXiv:0805.3931 [hep-th].
September 3rd (2008)@IPMU

1. Introduction
Closed string tachyon
condensation and the FZZ duality

Tachyon condensation in string theory• Tachyon condensation
Tachyoncondensation

Unstable vacuum

Stable vacuum• Open string tachyon condensation
Lowest mode istachyonic[ Sen ]

No D-brane

Unstable D-brane

What happens if closed string tachyon is condensed?
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FZZ-duality conjecture - A Proof2

Closed string tachyon condensation• Closed string tachyon condensation– Generically it is difficult to deal with since thebackground itself would change.
Localized tachyon condensation• Winding string tachyon[ Adams-Polchinski-Silverstein ]Tachyoncondensation

Tachyonic for– Singularities may be removed by tachyon condensation.
[ McGreevy-Silverstein, Horowitz-Silverstein ]– Difficult to say something concrete, need to understandstringy effects.
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The FZZ-duality conjecture• The Fateev-Zamolodchikov2 (FZZ) duality
Sine-Liouville theory

Cigar model– Condensation of winding string tachyon changes thegeometry from cylinder to cigar.
– Exact results in ’ corrections– Weak-strong duality w.r.t. k = R2 (like T-duality)
The aim of this talk is to proof this duality
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FZZ-duality conjecture - A Proof

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A solvable model for 2d black hole
• 2d black hole
– A solvable model is proposed by Witten, which can beconstructed by gauging the H +3 model Lorentzian black hole

 Euclidean black hole (cigar)

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Strategy• Strategy to proof the FZZ duality– The cigar model can be defined by the sum of H 3+modeland free boson (and ghosts) [ Dijkgraaf-Verlinde2 ]The action of H+3 model (describing strings on AdS3)– H 3+ - Liouville theory
N-pt. functionof H3+ model[ Stoyanovski, Ribaut-Teschner ](2N-2)-pt. function ofLiouville theory

We show that the combination of the twoabove facts leads to the FZZ duality.
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Plan of this talk
1. Introduction
2. H +3 - Liouville relation–––Relation between H 3+ model and Liouville theory
Path integral derivation
Few comments

3. The FZZ duality–––The cigar model as a gauged WZNW model
The cigar – Liouville relation
The duality between the cigar and Sine-Liouville

4. Conclusion
5. Appendix
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FZZ-duality conjecture - A Proof7+
2. H 3 - Liouville relation

The relation and a proof in path integralformulation

+H 3 -Liouville relation• H+3 -Liouville relation(Stoyanovsky-Ribault-Teschner relation)+N-pt. function of H 3 model(string theory on Euclidian AdS3)(2N-2)-pt. function of Liouville theorywith (N-2) degenerate fields• The results of our work
– Path integral derivation of the relation– Generalizations and applications• On Riemann surface of a higher genus• WZNW model on supergroup• The FZZ duality from the coset construction
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Liouville field theory
Liouville field theory is simplest model among non-compact CFTs and it
may serve as a building block for general CFT.

• Action• N-pt. scattering amplitudes…

=September 3, 2008

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+H 3 model
H3+ model (=SL(2,C)/SU(2) WZNW model) describes strings on
Euclidean AdS3

• The action of WZNW model

Introduce

Free action with an interaction term

• Vertex operator
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+H 3 - Liouville relation• H +3 - Liouville relation• N-2 degenerate fields V-1/2b inserted at yi
Sklyanin’s separationof variables
Transverse momentum  of AdS3 is mapped to the inserted point y.

• The shifts of parameters
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Path integral derivation (I)• Path integral form of N-point function• Integration over , Terms including :
Integration over  leads to  -function for 

Integration over world-sheet coordinates

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Path integral derivation (II)After integrating out , , the theory includes only the radial direction.
However, the theory is not the same as the Liouville field theory yet.

• Field redefinition of 1) Interaction term:2) Kinetic term:
The shift of j-momentum

Insertion of degenerated fields
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FZZ-duality conjecture - A Proof

Twist factor N
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Path integral derivation (III)• Shift of background charge
1) Introduce Weyl factor
2) Change  such that

3) Compute the kinetic term4) Read the shift of background charge

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Few comments• H +3 - Liouville relation– Relation between differential equations+• KZ eq. for H 3 model  BPZ eq. for Liouville theory– Extension to generic Riemann surface of genus g• The number of inserted fields is N-2+2g– Supersymmetric generalization• OSP(p|2) WZNW model  N=p super Liouville theory– Inclusion of spectrally flowed operators[ Ribaut ]• The number of inserted fields is N-2-S (total # of flow)
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3. The FZZ duality
Duality between the cigar model andSine-Liouville theory

Outline of proof• The FZZ duality
Cigar model

Sine-Liouville theory• Strategy+H 3 model + free boson

1) The cigar model2) H+3 - Liouville relation
Relation between the cigar
model and a new theory [Liouville + free boson]3) The new theory
Sine-Liouville theory
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The cigar model• The gauged WZNW model– H+3 WZNW model + Free boson– Primary fields• Basis change: -basism-basis• Gauge invariant operators– Correlation functions

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Correlation functions of cigar model• Map to Liouville + Free boson theories– Correlators of cigar model+= Correlators of Free boson (X) x H 3 model+

H3 - Liouville relation= Correlators of Free boson (X) x Liouville theory– Redefinition from X to 

– Jacobian for change of variables from  to y

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The cigar – Liouville correspondence• The correspondence– N-pt. functions of the cigar model– (2N-2)-pt. function of Liouville + free boson

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Three shortfalls• Three more steps to derive FZZ duality
1) The relation is not weak-strong duality for k
We utilize the self-duality of Liouville theory2) It relates N-pt. functions to (2N-2)-pt. functions
Insertions can be interpreted as an interaction3) The dual theory is not Sine-Liouville theory
Field redefinitions

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+FZZ-duality conjecture - A Proof

reflection relation

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Reflection relation• Field identification in Liouville theory– Reflection relation• A generic model– An interaction term

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can be reflected byFZZ-duality conjecture - A Proof

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The FZZ duality• The duality between the cigar and Sine-Liouville– Relation between correlators of primary fields– Duality between the theories• The both theories have the samesymmetry– Violation of winding number conservation[ Fateev-Lukyanov ]• We need to include spectral flow action to H +3 model.
( #(inserted degenerated fields) = N-2-|S| )September 3, 2008

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4. Conclusion
Summary and future problems

Conclusion• Summary– H +3 - Liouville relation• Path integral derivation• Generalizations and applications(Higher genus, supersymmetric model, …)– FZZ duality• Duality between the cigar model and Sine-Liouville theory• Condensation of winding string tachyon• Future problems– Generalizations• SL(N) WZNW model from SL(N) Toda theory• Other SUSY models and their applications• Generalizations of FZZ duality[ Fateev ]– Applications• Matrix model, AdS/CFT correspondence• Implication to geometric Langlands duality [ Giribet-Nakayama ]September 3, 2008

FZZ-duality conjecture - A Proof26

5. Appendix
Technical details

Relation between differential equations (I)• H +3 model– Knizhnik-Zamolodchikov equations

Sklyanin’s separation of variables +• Liouville field theory– Belavin-Polyakov-Zamolodchikov equations

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Relation between differential equations (II)• A sketch of proof– Sklyanin’s separation of variables– Sugawara singular vector at z=yj

– Change of variables + twist factor leads to BPZ equation
Separation of variables
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Extension to higher genus g (I)• H+3 -Liouville relation for

The number of extra insertion is N-2+2g• Meromorphic 1-form on surface with g#(pole) - #(zero) = 2-2g

– For g=0#(zero)=N-2+2g#(pole)=N
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Extension to higher genus g (II)• The map of parameters– For
 (momentum): N
-conservation: -1
N-1– For (momentum): N (holonomy): g
0 (zero mode): g-1
N-1+2gSeptember 3, 2008

Sklyanin’s
separationof variables

Sklyanin’s
separationof variables

u:1yj (pos. of deg. field): N-2
N-1u:
1yj (pos. of deg. field): N-2+2g
N-1+2g

FZZ-duality conjecture - A Proof31

OSP(1|2) – N=1 Liouville relation (I)• A suprsymmetric generalization
SL(2,C)/SU(2) WZNW model (H +3 model)
Target-space supersymmetry

OSP(1|2) WZNW model:

Fermionic generators

SL(2) generators• OSP(1|2) model from N=1 Liouville theory
OSP(1|2) model

N=1 Liouville theory+ free fermion

1. Integrate over
2. Redefine:September 3, 2008

FZZ-duality conjecture - A Proof32OSP(1|2) – N=1 Liouville relation (II)• OSP(1|2) – N=1 Liouville relation
N-pt. function of OSP(1|2) WZNW model(2N-2)-pt. function of N=1 Liouville+ free fermion with (N-2) degenerate fields• Comments– 2 and 3-point functions of OSP(1|2) model arecomputed and consistent with OSP(1|2) symmetry.
• The relevant results of N=1 Liouville are given in–[ Rashkov-Stanishkov, Poghosian, Fukuda-Hosomichi ]This is the first practical use of H +3 - Liouville relation.

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Winding number violation (I)• SL(2) current algebra

Vacuum state:• Spectral flow symmetry
Vacuum state:• Free field realization

(w) must have a zero of order S at w=0
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Winding number violation (II)• Ribault relation• Winding number violation of the cigar model– Insert a representation of the identity of the cigar model
The FZZ duality with non-zero S
Winding # violation
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Ribault relation
FZZ-duality conjecture - A Proof35

The gauged WZNW model• The cigar model as gauged WZNW model– The action of H +3 model

Gauging U(1) isometry:– Field redefinition(+ FP ghosts)
Shift the fields

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Relation to Sine-Liouville theory• Field redefinition• Reflection by V- is performed for VL
Interaction termfor Sine-Liouville• Reflection by V- is performed also for N vertex ops.

Field redefinition + reflection
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