World-sheet Duality for Superspace σ -Models Thomas Quella University of Amsterdam Opening Conference: “New Perspectives in String Theory” Galileo Galilei Institute, Florence Based on arXiv:0809.1046 (with V. Mitev and V. Schomerus) and work in progress (with Candu, Mitev, Saleur and Schomerus) [This research receives funding from an Intra-European Marie-Curie Fellowship]
Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook Flux backgrounds Crucial problem: Quantization of strings in flux backgrounds AdS/CFT correspondence [Maldacena] [...] String theory on AdS-space ⇔ Gauge theory on boundary String phenomenology [Kachru,Kallosh,Linde,Trivedi] [...] Moduli stabilization through fluxes Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 2/34
Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook The pure spinor formalism Ingredients Superspace σ -model encoding geometry and fluxes Pure spinors: λ ∈ SO(10) / U(5) BRST procedure [Berkovits at al] [Grassi et al] [...] Features Manifest target space supersymmetry Manifest world sheet conformal symmetry Action quantizable, but quantization hard in practice Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 3/34
Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook Strings on AdS 5 × S 5 and AdS 4 × CP 3 Spectrum accessible because of integrability Factorizable S-matrix Structure fixed (up to a phase) by SU(2 | 2) ⋉ R 3 -symmetry Bethe ansatz, Y-systems, ... Open issues String scattering amplitudes? 2D Lorentz invariant formulation? Other backgrounds? → Conifold, nil-manifolds, ... Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 4/34
Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook The standard perspective on AdS/CFT Overview Gauge theory String theory AdS 5 × S 5 N = 4 Super Yang-Mills AdS 4 × CP 3 N = 6 Chern-Simons ⇐ ⇒ S-matrix, spectrum,... S-matrix, spectrum,... t’Hooft coupling λ , ... Radius R , ... Problem From this perspective, both sides need to be solved separately. Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 5/34
� � � � Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook An alternative perspective on AdS/CFT Proposal: Two step procedure... Weakly coupled gauge theory Feynman diagram expansion Weakly coupled 2D theory (Topological σ -model) “Well-established machinery” Strongly curved σ -model [Berkovits] [Berkovits,Vafa] [Berkovits] Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 6/34
Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook Summary: String theory/gauge theory dualities ? Strong curvature Weak curvature String theory in 10D ( σ -model with constraints) 1 / R λ Gauge theory Strong coupling Weak coupling ? Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 7/34
Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook Summary: String theory/gauge theory dualities Strong curvature Weak curvature String theory in 10D ( 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 Superspace σ -Models 7/34
Introduction Flux backgrounds Generalized symmetric spaces String theory/gauge theory dualities Supersphere σ -models The structure of this talk Conclusions and Outlook The structure of this talk Prospect... 1 Introduce a class of 2D σ -models which share many conceptual features with σ -models on AdS-spaces 2 Pick one example and show how exact spectra can be derived explicitly using a duality to a non-geometric CFT Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 8/34
Introduction String backgrounds as supercosets Generalized symmetric spaces Generalized symmetric spaces Supersphere σ -models Superspheres Conclusions and Outlook An interesting observation String backgrounds as supercosets... AdS 5 × S 5 AdS 4 × CP 3 AdS 2 × S 2 Minkowski PSU(2 , 2 | 4) OSP(6 | 2 , 2) PSU(1 , 1 | 2) super-Poincar´ e Lorentz SO(1 , 4) × SO(5) U(3) × SO(1 , 3) U(1) × U(1) [Metsaev,Tseytlin] [Berkovits,Bershadsky,Hauer,Zhukov,Zwiebach] [Arutyunov,Frolov] Definition of the cosets � � � � gh ∼ g , h ∈ H G / H = g ∈ G Geometric realization of supersymmetry: g �→ hg Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 9/34
Introduction String backgrounds as supercosets Generalized symmetric spaces Generalized symmetric spaces Supersphere σ -models Superspheres Conclusions and Outlook Generalized symmetric spaces Let G be a Lie (super)group, Ω : G → G an automorphism of finite order, Ω L = id. Let H = Inv Ω ( G ) = { h ∈ G | Ω( h ) = h } be the invariant subgroup. 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 PSU( N | N ), OSP(2 S + 2 | 2 S ), D(2 , 1; α ). Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 10/34
Introduction String backgrounds as supercosets Generalized symmetric spaces Generalized symmetric spaces Supersphere σ -models Superspheres Conclusions and Outlook Two compact examples Compact symmetric spaces with vanishing Killing form Superspheres S 2 S +1 | 2 S Projective superspaces CP S − 1 | S OSP(2 S +2 | 2 S ) U( S | S ) OSP(2 S +1 | 2 S ) U(1) × U( S − 1 | S ) f f = 0 Remark: One can write CP S − 1 | S = S 2 S − 1 | 2 S / U (1). Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 11/34
Introduction String backgrounds as supercosets Generalized symmetric spaces Generalized symmetric spaces Supersphere σ -models Superspheres Conclusions and Outlook Relation to AdS σ -models Similarities Family of CFTs with continuously varying exponents Completely new type of 2D conformal field theory Standard methods do not apply! Target space supersymmetry Symmetric superspaces Integrability Differences Compactness No string constraints imposed Thomas Quella (University of Amsterdam) World-sheet Duality for Superspace σ -Models 12/34
Introduction String backgrounds as supercosets Generalized symmetric spaces Generalized symmetric spaces Supersphere σ -models Superspheres Conclusions and Outlook Superspheres Realization of S M | 2 N as a submanifold of flat superspace R M +1 | 2 N � x X 2 = � x 2 + 2 � � � η 2 = R 2 X = η 1 � with η 1 � η 2 � 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 Superspace σ -Models 13/34
Introduction String backgrounds as supercosets Generalized symmetric spaces Generalized symmetric spaces Supersphere σ -models Superspheres Conclusions and Outlook Superspheres: Conformal invariance Analogy to O( K ) models S M | 2 N S M − 2 N ← → � � � � M + 1 − 2 N Similarity to O K = O σ -models There is no topological Wess-Zumino term Self-duality O( K ) ⇔ O(4 − K ) 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 Superspace σ -Models 14/34
Introduction A world-sheet duality for superspheres? Generalized symmetric spaces Dual description at strong coupling Supersphere σ -models Anomalous dimensions Conclusions and Outlook The large volume limit A world-sheet duality for superspheres? 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 Superspace σ -Models 15/34
Introduction A world-sheet duality for superspheres? Generalized symmetric spaces Dual description at strong coupling Supersphere σ -models Anomalous dimensions Conclusions and Outlook The large volume limit A world-sheet duality for superspheres? Supersphere σ -model Large volume Strong coupling 1/R Z σ ( q , z , R ) R 2 = 1 + g 2 Z GN ( q , z , g 2 ) 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 Superspace σ -Models 15/34
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