Principal chiral model on superspheres
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Principal Chiral Model on Superspheres. GGI Florence, Sep 2008 Volker Schomerus. based on work w. V. Mitev, T Quella; arXiv:0809.1046 [hep-th] & discussions with H. Saleur. Statement of Mission. Aim: Solve the OSP(2S+2|2S) Non-linear Sigma Model.

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Principal Chiral Model on Superspheres

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Principal chiral model on superspheres

Principal Chiral Model on Superspheres

GGI Florence, Sep 2008

Volker Schomerus

based on work w. V. Mitev, T Quella; arXiv:0809.1046 [hep-th]

& discussions withH. Saleur


Statement of mission

Statement of Mission

Aim: Solve the OSP(2S+2|2S)

Non-linear Sigma Model

c=1 CFTs with continuously varying

exponents; interacting, non-unitary

SUSY

helps?

~ non-abelian version of O(2) model

2=2S+2-2S

Motivation - Model - Solution - Conclusion


Motivation gauge string dualities

Motivation: Gauge/String dualities

D+1-dim super

string theory

2D critical

system

D-dim SQFT

gauge theory

new

old

high/low T

duality

Remark: strong-weak coupling duality

anomalous dim ↔ string masses ↔ critical exp

space-time sym ↔ target sp. sym ↔ internal sym

Applications to strongly coupled gauge physics

QCD, quark gluon plasma, 3D ? cold atoms etc.


2d critical systems for ads st

2D Critical systems for AdS ST

N=4 Super-Yang-Mills in 4D ↔ Strings on:

AdS5 x S5 = (PSU(2,2|4)/SO(1,4)xSO(5))0

non-compact

target

Involved critical systems are non-rational

and they possess internal supersymmetry

e.g.

PSU(2,2|4)

perturbative gauge

theory regime

perturbative α’

string regime :

gravity + ...

Is there a weakly

coupled 2D dual?


2d critical systems for ads st1

2D Critical systems for AdS ST

N=4 Super-Yang-Mills in 4D ↔ Strings on:

AdS5 x S5 = (PSU(2,2|4)/SO(1,4)xSO(5))0

non-compact

target

Involved critical systems are non-rational

and they possess internal supersymmetry

e.g.

PSU(2,2|4)

perturbative gauge

theory regime

perturbative α’

string regime :

gravity + ...

Is there a weakly

coupled 2D dual?


Pcm on superspheres s 2s 1 2s

PCM on Superspheres S2S+1|2S

parameter R

Family of CFTs with continuously varying exp.

cp PCM on S3

massive flow

β(S2S+1|2S) = β(S1) = 0

Remark: S2S+1|2S→ family of interacting CFTs

non-abelian extension of free boson on S1=S1

R


Duality with gross neveu model

Duality with Gross-Neveu model

CompactifiedR free boson dual to massless Thirring:

R2 = 1+g2

Claim: [Candu,Saleur] Supersphere PCM dual to GN

2S+2 real fermions

S βγ-systems c=-1

hψ= hβ= hγ=1/2

c=1 CFT with affine

osp(2S+2|2S) ; k=1

~ JμJμ

rule: x → ψη → β,γ


R dependence of conf weights

R-dependence of conf. weights

Free Boson:

In boundary theory

bulk more involved

at R=R0 universal U(1) charge

Prop.: For boundary spectra of superspheres:

quadratic

Casimir

Deformation of conf. weights is `quasi-abelian’

[Bershadsky et al] [Quella,VS,Creutzig] [Candu, Saleur]

Example: mult. (x,η) ΔR = ΔR=∞ + (1/2R2) Cf = 0 + (1/2R2) 1 = 1/2R2

→ (ψ,β,γ)

fund rep: Cf =1

f(R)


Application to pcm gn duality

Application to PCM – GN duality

Cartan elements

fermionic contr.

zero modes

Euler fct ~ C

bosonic contr.

character

branching fct

Fact:

[Mitev,Quella,VS]

just as it is predicted by the PCM – GN duality !


Duality for non compact backgrounds

Duality for non-compact backgrounds?

Cigar (coset AdS3/R) Sine-Liouville theory

~

θ

θ

φ

φ

~

conjectured by [Fateev,Zamolodchikov2]proven in[Yasuaki Hikida,VS]

● Strong - weak coupling duality

● Sigma model ↔ FFT + exp-int

free σ-model: k →∞

↔ b2 = 1/(k-2) → 0

cp. Buscher rules


Summary open problems

Summary & Open Problems

  • Solved boundary sector of Supersphere PCM!

  • how about bulk ? deformation is quasi-abelian

  • but involves three Casimirs:

  • CL CR CD spin chain?

  • New dualities sigma ↔ non-sigma models

  • extension of Sine-Gordon – Thirring duality ?

massive version of PCM – GN


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