Conformal Superspace σ-Models Thomas Quella University of Amsterdam Edinburgh, 20.1.2010 Based on arXiv:0809.1046 (with V. Mitev and V. Schomerus) and arXiv:09080878 (with C. Candu, V. Mitev, H. Saleur and V. Schomerus) [This research received funding from an Intra-European Marie-Curie Fellowship] σ-models in a nutshell World-sheet Target space 2D surface (Pseudo-)Riemannian manifold (w/wo boundaries or handles) (extra structure: gauge fields, ...) embedding strip cylinder embedding σ-models = (quantum) field theories Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 2/40 A simple example: The circle The moduli space of circle theories 2 R0 R R0 =R Radius Quantum regime Classical regime S1 =∼ S3 Two lessons 2 There is an equivalence: R $ R0 =R (\T-duality") In the quantum regime geometry starts to loose its meaning Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 3/40 A simple example: The circle The moduli space of circle theories 2 R0 R R0 =R Radius Quantum regime Classical regime S1 =∼ S3 An open string partition function 2 X w P Energy(R) 1 w 2R2 Z(q; zjR) = tr z q = η(q) z q w2Z Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 3/40 Adding curvature and supersymmetry... Appearances of superspace σ-models String theory Quantization of strings in flux backgrounds [Berkovits] String theory / gauge theory correspondence [Maldacena] Moduli stabilization in string phenomenology [KKLT] [...] Disordered systems Quantum Hall systems Self avoiding random walks, polymer physics, ... Efetov's supersymmetry trick Conformal invariance String theory: Diffeomorphism + Weyl invariance Statistical physics: Critical points / 2nd order phase transitions Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 4/40 String quantization: The pure spinor formalism Ingredients Superspace σ-model encoding geometry and fluxes Pure spinors: Curved ghost system 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) Conformal Superspace σ-Models 5/40 5 3 Strings on AdS5 × S and AdS4 × CP Spectrum accessible because of integrability Factorizable S-matrix 3 Structure fixed (up to a phase) by SU(2j2) n R -symmetry Bethe ansatz, Y-systems, ... Open issues String scattering amplitudes? 2D Lorentz invariant formulation? Other backgrounds? ! Conifold, nil-manifolds, ... Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 6/40 The standard perspective on AdS/CFT Overview Gauge theory String theory 5 N = 4 Super Yang-Mills AdS5 × S 3 N = 6 Chern-Simons AdS4 × CP 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) Conformal Superspace σ-Models 7/40 An alternative perspective on AdS/CFT Proposal: Two step procedure... Weakly coupled 4D gauge theory O Feynman diagram expansion Weakly coupled 2D theory (Topological σ-model) O \Well-established machinery" Strongly curved 2D σ-model [Berkovits] [Berkovits,Vafa] [Berkovits] Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 8/40 Summary: String theory/gauge theory dualities ? Weak curvature Strong curvature String theory in 10D (σ-model with constraints) 1=R λ Gauge theory Strong coupling Weak coupling ? Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 9/40 Summary: String theory/gauge theory dualities Weak curvature Strong curvature String theory in 10D (σ-model with constraints) 1=R g \Some dual 2D theory" Strong coupling Weak coupling Gauge theory λ Strong coupling Weak coupling Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 9/40 The structure of this talk Outline 1 Supercoset σ-models Occurrence in string theory and condensed matter theory Ricci flatness and conformal invariance 2 Particular examples Superspheres Projective superspaces 3 Quasi-abelian perturbation theory Exact open string spectra World-sheet duality for supersphere σ-models Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 10/40 Appearance of supercosets String backgrounds as supercosets... 5 3 2 Minkowski AdS5 × S AdS4 × CP AdS2 × S super-Poincar´e PSU(2;2j4) OSP(6j2;2) PSU(1;1j2) Lorentz SO(1;4)×SO(5) U(3)×SO(1;3) U(1)×U(1) [Metsaev,Tseytlin] [Berkovits,Bershadsky,Hauer,Zhukov,Zwiebach] [Arutyunov,Frolov] Supercosets in statistical physics... IQHE Dense polymers Dense polymers 2S+1j2S S−1jS (non-conformal) S CP U(1;1j2) OSP(2S+2j2S) U(SjS) U(1j1)×U(1j1) OSP(2S+1j2S) U(1)×U(S−1jS) [Weidenm¨uller][Zirnbauer] [Read,Saleur] [Candu,Jacobsen,Read,Saleur] Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 11/40 A unifying construction Definition of the cosets G=H : gh ∼ g Some additional requirements for conformal invariance H ⊂ G is invariant subgroup under an automorphism Ricci flatness (\super Calabi-Yau") , vanishing Killing form f f = 0 Examples: Cosets of PSU(NjN), OSP(2S + 2j2S), D(2; 1; α). Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 12/40 Properties of conformal supercoset models The moduli space of generic supercoset theories Radius Quantum regime Classical regime Properties at a glance Supersymmetry G: g 7! kg (realized geometrically) Conformal invariance [Kagan,Young] [Babichenko] Integrability [Pohlmeyer] [L¨uscher]... [Bena,Polchinski,Roiban] [Young] Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 13/40 Properties of conformal supercoset models The moduli space of generic supercoset theories Radius Quantum regime Classical regime The general open string partition function Cartan Energy(R) X G Z(q; zjR) = tr z q = Λ(q; R) χΛ (z) Λ | {z } | {z } Dynamics Symmetry Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 13/40 Sketch of conformal invariance The β-function vanishes identically... X β = = 0 certain G-invariants Ingredients: Invariant form: κµν Structure constants: f µνλ Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 14/40 Sketch of conformal invariance The β-function vanishes identically... f f = 0 Ingredients: Invariant form: κµν Structure constants: f µνλ Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 14/40 Sketch of conformal invariance The β-function vanishes identically... f = 0 There is a unique invariant rank 3 tensor! [Bershadsky,Zhukov,Vaintrob'99] [Babichenko'06] Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 14/40 Sketch of conformal invariance The β-function vanishes identically... ∼ f f = 0 There is a unique invariant rank 3 tensor! [Bershadsky,Zhukov,Vaintrob'99] [Babichenko'06] Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 14/40 Supersphere σ-models Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 15/40 The supersphere S 3j2 3j2 4j2 Realization of S as a submanifold of flat superspace R 0 ~x 1 ~ ~ 2 2 2 X = @η1A with X = ~x + 2η1η2 = R η2 Symmetry super- O(4) × SP(2) −−−−−−−−−−−−! OSP(4j2) symmetrization Realization as a supercoset OSP(4j2) S3j2 = OSP(3j2) Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 16/40 The supersphere σ-model Action functional Z µ 2 2 Sσ = @µX~ · @ X~ with X~ = R The space of states for freely moving open strings Y ai Y bj Y 2 ck ~ 2 2 X @t X @t X ··· and X = R ) Products of coordinate fields and their derivatives Large volume partition function \Single particle energies" add up ! # derivatives Partition function is pure combinatorics [Candu,Saleur] [Mitev,TQ,Schomerus] Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 17/40 The large volume limit Keeping track of quantum numbers... Symmetry OSP(4j2) ! SP(2) × SO(4) =∼ SU(2) × SU(2) × SU(2) Classify states according to the bosonic symmetry: ~ 1 1 1 X = (~x; η1; η2): V = 0; 2 ; 2 ⊕ 2 ; 0; 0 | {z } | {z } bosons fermions Other quantum numbers: Energy qE Polynomial grade tn (broken by X~ 2 = R2) Use this to characterize all monomials Y ai Y bj Y 2 ck ~ 2 2 X @t X @t X ··· with X = R Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 18/40 Constituents of the partition function A useful dictionary Field theoretic quantity Contribution Representation ±1 1 2 Fermionic coordinates t z1 2 ±1 ±1 ±1 ∓1 1 1 4 Bosonic coordinates t z2 z3 , t z2 z3 2 ; 2 Derivative @ q Constraint X~ 2 = R2 1 − t2 Constraint @nX~ 2 = 0 1 − t2qn t $ polynomial grade z1; z2; z3 $ SU(2) quantum numbers Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 19/40 The full σ-model partition function Summing up all contributions... " 1 − 1 Y 2 n Zσ(R1) = lim q 24 (1 − t q )× t!1 n=0 1 n −1 n # Y (1 + z1tq )(1 + z tq ) × 1 n −1 n −1 n −1 −1 n n=0 (1 − z2z3tq )(1 − z2z3 tq )(1 − z2 z3tq )(1 − z2 z3 tq ) The problem... Organize this into representations of OSP(4j2)! Thomas Quella (University of Amsterdam) Conformal Superspace σ-Models 20/40 Decomposition into representations of OSP(4j2) Since the model is symmetric under OSP(4j2) the partition function may be decomposed into characters of OSP(4j2): X Z (R ) = σ (q) χ (z) σ 1 [j1;j2;j3] [j1;j2;j3] [j1;j2;j3] All the non-trivial information is encoded in −C[j ;j ;j ]=2 1 σ q 1 2 3 X m+n m (m+4j +2n+1)+ n +j − 1 (q)= (−1) q 2 1 2 1 8 [j1;j2;j3] η(q)4 n;m=0 (j − n )2 (j + n +1)2 (j − n )2 (j + n +1)2 × q 2 2 − q 2 2 q 3 2 − q 3 2 Thomas Quella (University of Amsterdam)