Principal Chiral Model on Superspheres

1. 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

2. 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

3. 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.

4. 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?

5. 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?

6. 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

7. 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

8. 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)

9. 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 !

10. 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

11. 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