1 / 17

Superspace Sigma Models

Superspace Sigma Models. IST String Fest Volker Schomerus. based on work w. C. Candu, C.Creutzig, V. Mitev, T Quella, H. Saleur; 2 papers in preparation. Superspace Sigma Models. Aim: Study non-linear sigma models with target space supersymmetry not world-sheet.

ayala
Download Presentation

Superspace Sigma Models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Superspace Sigma Models IST String Fest Volker Schomerus based on work w. C. Candu, C.Creutzig, V. Mitev, T Quella,H. Saleur; 2 papers in preparation

  2. Superspace Sigma Models Aim: Study non-linear sigma models with target space supersymmetry not world-sheet Strings in AdS backgrounds [pure spinor] c = 0 CFTs; symmetry e.g. PSU(2,2|4) ... Focus on scale invariant QFT, i.e. 2D CFT Properties:Weird: logarithmic Conformal Field Theories! Remarkable: Many families with cont. varying exponents

  3. Plan of 2nd Lecture M O D E L S • SM on CP0|1 - Symplectic Fermions • Sigma Model on Superspace CP1|2 • Properties of the Chiral Field on CP1|2 • Conclusion & Some Open Problems Bulk and twisted Neumann boundary conditions M E T H O D S Exact boundary partition functions Quasi-abelian evolution; Lattice models; Symplectic fermions as cohomology

  4. 1.1 Symplectic Fermions - Bulk Only global u(1) sym. is manifest Has affine psu(1|1) sym: Action rewritten in terms of currents: U(1) x U(1) Possible modification: θ-angle trivial in bulk

  5. 1.1 Symplectic Fermions - Bulk Has affine psu(1|1) sym: Action rewritten in terms of currents: U(1) x U(1) Possible modification: θ-angle trivial in bulk

  6. ΘΘ x 1.2 SF – Boundary Conditions boundary term Implies twisted Neumann boundary conditions: A with currents: Results: 4 ground states In P0 of u(1|1) Θ1Θ2 -1 Ground states: λ(Θ1,Θ2) = λ(tr(A1A2 )) x twist fields [Creutzig,Quella,VS] [Creutzig,Roenne]

  7. 1.3 SF – Spectrum/Partition fct. c/24 U(1) gauging constraint branching functions characters

  8. 2.1 The Sigma Model on CPN-1|N zα, ηα→ ωzα, ωηα = → 2 parameter family of 2D CFTs;c = -2 a - non-dynamical gauge field D = ∂ - ia θ term non-trivial; θ=θ+2π Non-abelian extension of symplectic fermion

  9. 2.2 CPN-1|N – Boundary Conditions Boundary condition in σ-Model on target X is hypersurface Y + bundle with connection A Dirichlet BC ┴ to Y; Neumann BC || to Y; U(N|N) symmetric boundary cond. for CPN-1|N line bundles on Y = CPN-1|N with monopole Aμ Spectrum of sections e.g. N=2; μ = 0 μ integer ~ (1+14+1) + 48 + 80 + ... + (2n+1) x 16 + superspherical harmonics atypicals typicals of U(2|2)

  10. U(1) gauging constraint 2.3 Spectra/Partition Function Count boundary condition changing operators at R = ∞ (free field theory) Built from Zα,Zα,∂xZα, ... fermionic contr. monopole numbers Euler fct bosonic contr. u(2|2) characters for N=2 and μ = ν ~ χ(1+14+1) + χ48 + χ80 + ... q χad + ...

  11. Result: Spectrum at finite R universal indep of Λ,N fμν|μ-ν| ~ λ Characters of u(2|2) reps Branching fcts from R = ∞ λ = λ(Θ1,Θ2) from symplectic Fermions Casimir of U(2|2)

  12. 3.1 Quasi-abelian Evolution Free Boson: In boundary theory bulk more involved Prop.: Boundary spectra of CP1|2 chiral field : quadratic Casimir Deformation of conf. weights is `quasi-abelian’ [Bershadsky et al] [Quella,VS,Creutzig] [Candu, Saleur] e.g. (1+14+1) remains at Δ=0; 48, 80, .... are lifted

  13. 3.1 Quasi-abelian Evolution Free Boson: In boundary theory bulk more involved at R=R0 universal U(1) charge Prop.: Boundary spectra of CP1|2 chiral field : quadratic Casimir Deformation of conf. weights is `quasi-abelian’ [Bershadsky et al] [Quella,VS,Creutzig] [Candu, Saleur] e.g. (1+14+1) remains at Δ=0; 48, 80, .... are lifted

  14. 3.2 Lattice Models and Numerics boundary term f00 acts on states space: level 1 level 2 level 3 Extract fνμ(R) from Δhk= fνμ(R)(Clk-Cl0) l = μ - ν no sign of instanton effects w=w(R)

  15. 3.3 SFermions as Cohomology U(2|2): U(1) x U(1|2) in 3 of U(1|2) Result: [Candu,Creutzig,Mitev,VS] Cohomology of Q arises from states in atypical modules with sdim ≠ 0 all multiplets of ground states contribute → λ(Θ1,Θ2) from SF

  16. Conclusions and Open Problems • Exact Boundary Partition Functions for CP1|2 • For S2S+1|S there is WZ-point at radius R = 1 • Is there WZ-point in moduli space of CPN-1|N? • Much generalizes to PCM on PSU(1,1|2)! Techniques: QA evolution; Lattice; Q-Cohomology [Candu,H.Saleur] [Mitev,Quella,VS] osp(2S+2|2s) at level k=1 e.g. psu(N|N) at level k=1 CP0|1 = PSU(1|1)k=1 AdS3 x S3 QA evolution, simple subsector

  17. Examples: Super-Cosets [Candu] [thesis] Familiesv w. compact form, w.o. H-flux: cpct symmetric superspaces volume G/GZ 2 OSP(2S+2|2S) OSP(2S+1|2S) → S2S+1|2S OSP(2S+2|2S) OSP(2S+2-n|2S) x SO(n) c = 1 U(N|N) U(N-1|N)xU(1) → CPN-1|N U(N|N) U(N-n|N) x U(n) c = -2 . . • note: cv (GL(N|N)) = 0 = cv (OSP(2S+2|2S))

More Related