1. Principal Chiral Model on Superspheres
GGI Florence, Sep 2008
Volker Schomerus
based on work w. V. Mitev, T Quella; arXiv:  [hep-th]
& discussions with H. 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
D-dim SQFT
gauge theory
new
old
2D critical
system
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 → ψ η → 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
fund rep: Cf =1
f(R)
→ (ψ,β,γ)

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 φ θ 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