World-sheet duality for supersphere -models Thomas Quella - PowerPoint PPT Presentation

World-sheet duality for supersphere σ -models Thomas Quella (University of Amsterdam) Miniworkshop: Integrability in String Theory Galileo Galilei Institute Workshop on “Low-dimensional Quantum Field Theories and Applications” Based on arXiv:0809.1046 (with V. Mitev and V. Schomerus)

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? Outline Outline and Introduction String theory/gauge theory dualities Generalized symmetric spaces A new world-sheet duality? Supersphere σ -models The large volume limit Dual description at strong coupling Interpolation Outlook Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? Outline and Introduction String theory/gauge theory dualities Generalized symmetric spaces A new world-sheet duality? Supersphere σ -models The large volume limit Dual description at strong coupling Interpolation Outlook Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? String theory/gauge theory dualities Strong curvature Weak curvature String theory in 10 D ( σ -model with constraints) 1 / R λ Gauge theory Strong coupling Weak coupling Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? String theory/gauge theory dualities Strong curvature Weak curvature String theory in 10 D ( σ -model with constraints) 1 / R g “Some dual 2D theory” Strong coupling Weak coupling λ Gauge theory Strong coupling Weak coupling Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? A prominent example: AdS 5 × S 5 AdS 5 × S 5 N = 4 super Yang-Mills theory PSU (2 , 2 | 4) PSU (2 , 2 | 4) Symmetry α ′ , g s Parameters Gauge group SU ( N ) Radius R Coupling g YM t’Hooft coupling λ = Ng 2 YM [Maldacena] [...] [Metsaev,Tseytlin] [...] [Minahan,Zarembo] [Beisert,Staudacher] [...] Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? Another prominent example: AdS 4 × CP 3 AdS 4 × CP 3 N = 6 Chern-Simons theory Symmetry OSP (6 | 2 , 2) OSP (6 | 2 , 2) α ′ , g s Parameters Gauge group U ( N ) × U ( N ) Radius R Level k t’Hooft coupling λ = 2 π 2 N / k N M 2-branes probing C 4 / Z k Interpretation [Arutyunov,Frolov] [Stefanski] [Fre,Grassi] [...] [Aharony,Bergman,Jafferis,Maldacena] [...] Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? And a common structure... Both space are actually supercosets of the form PSU (2 , 2 | 4) AdS 5 × S 5 = SO (1 , 4) × SO (5) OSP (6 | 2 , 2) AdS 4 × CP 3 = U (3) × SO (1 , 3) The definition of these cosets is as follows: � � � � gh ∼ g , h ∈ H G / H = g ∈ G Note that G / H still admits an action of G : g = hg Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? (Generalized) symmetric spaces Let G be a Lie (super)group, Ω : G → G an automorphism, H = Inv Ω ( G ) = { h ∈ G | Ω( h ) = h } the invariant subgroup. Ω being of finite order, Ω L = id. Then the coset G / H is called a generalized symmetric space . Theorem If G has vanishing Killing form then the coset G / H is classically integrable and quantum conformally invariant, at least to the lowest non-trivial order in perturbation theory. [Young] [Kagan,Young] Examples: Cosets of PSL ( N | N ), OSP (2 S + 2 | 2 S ), D (2 , 1; α ). Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? A simple example: Superspheres Superspheres S M | 2 N ⊂ R M +1 | 2 N can be introduced as follows:   � x X 2 = � x 2 + 2 � � � η 2 = R 2   → X = η 1 � η 1 � η 2 � From this one derives their realization as a symmetric space : S M | 2 N = OSP ( M + 1 | 2 N ) OSP ( M | 2 N ) Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? Superspheres: Conformal invariance ( M , N ) = (2 S + 1 , S ) ⇒ Family of conformal σ -models � � ◮ Relation to O M − 2 N = O (2) σ -models ◮ There is no topological Wess-Zumino term ◮ There is one free parameter, the radius R In this talk: Focus on S 3 | 2 = OSP (4 | 2) OSP (3 | 2) Question: How can this theory be quantized? [Read,Saleur] [Mann,Polchinski] [Candu,Saleur] [Mitev,TQ,Schomerus] Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? A new world-sheet duality? Supersphere σ -model Large volume Strong coupling 1/R g 2 Strong coupling Weak coupling OSP (2 S + 2 | 2 S ) Gross-Neveu model [Candu,Saleur] 2 [Mitev,TQ,Schomerus] Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? A new world-sheet duality? Supersphere σ -model Large volume Strong coupling 1/R Z σ ( q , z , R ) R 2 = 1 + g 2 g 2 Z GN ( q , z , g 2 ) Strong coupling Weak coupling OSP (2 S + 2 | 2 S ) Gross-Neveu model [Candu,Saleur] 2 [Mitev,TQ,Schomerus] Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? Summary of existing evidence for the duality Large volume Strong coupling 1/R � �� � Lattice formulation Free theory Free ghosts Affine symmetry Combinatorics [Candu,Saleur] 2 [Mitev,TQ,Schomerus] Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? Summary of existing evidence for the duality Large volume Strong coupling 1/R � �� � Lattice formulation Free theory Free ghosts Affine symmetry Combinatorics [Candu,Saleur] 2 [Mitev,TQ,Schomerus] Certain partition functions can be determined for all R � ψ σ Z σ ( q , z , R ) = Λ ( q , R ) χ Λ ( z ) Λ Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction String theory/gauge theory dualities Supersphere σ -models Generalized symmetric spaces Outlook A new world-sheet duality? Interpolation of the spectrum We have to show that the following two spectra are continuously connected by the deformation: L 0 L 0 Fundamental ⊗ Adjoint Adjoint ∞ many representations 1 1 Fundamental 1 / 2 1 / 2 Algebra of functions on S 3 | 2 Trivial WZNW model σ -model at R → ∞ Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction The large volume limit Supersphere σ -models Dual description at strong coupling Outlook Interpolation Outline Outline and Introduction String theory/gauge theory dualities Generalized symmetric spaces A new world-sheet duality? Supersphere σ -models The large volume limit Dual description at strong coupling Interpolation Outlook Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models

Outline and Introduction The large volume limit Supersphere σ -models Dual description at strong coupling Outlook Interpolation Definition The model is defined by the action � X 2 = R 2 ∂� X · ¯ ∂� � S σ = X with Properties of this σ -model: ◮ There is no topological term ◮ Conformal invariance for each value of R ◮ Central charge: c = 1 ◮ Non-unitarity [Read,Saleur] [Polchinski,Mann] [Candu,Saleur] 2 [Mitev,TQ,Schomerus] Thomas Quella (University of Amsterdam) World-sheet duality for supersphere σ -models