The Unreasonable Effectiveness of Supersymmetry Volker Schomerus Symposium
The Unreasonable Effectiveness of Supersymmetry Volker Schomerus Symposium Budapest 2013
Prologue There is a story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that? " was his query. "And what is this symbol here? " "Oh, " said the statistician, "this is pi. " "What is that? " "The ratio of the circumference of the circle to its diameter. " "Well, now you are pushing your joke too far, " said the classmate, "surely the population has nothing to do with the circumference of the circle. " "The Unreasonable Effectiveness of Mathematics in the Natural Sciences, " in Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960).
Prologue Consider diffusion in a 1 -dim system: t=0 t>0 SO(2) symmetry Rotational symmetry of 2 -dim plane can appear in 1 -dim physics! Also SUSY can appear in non-SUSY world …
Supersymmetry and Disorder Consider a particle in d-dimensional space Hamiltonian Green’s fct constant random potential Gaussian d-dimensional interacting QFT ↔ spectral density; [(TE)-1] ↔ transport. . (Super)symmetry depends on quantity to be computed
Conclusion Part I Disorder average leads to interacting QFT with internal supersymmetry perturbed WZW non-unitary Often low energy (IR) physics at strong coupling Dirac fermion in random potential Quantum Hall plateau transitions RG flows difficult to control in non-unitary QFT but SUSY can help …
Perturbed WZWk model WZW(G, k) is 2 D conformal QFT based on field Q: ∑ → G GD 2 D surface Lie (super-)group GL x GR symmetry: Noether currents JL , JR Perturb by ϕ = -g ∫ d 2 z ‹JL, JR› preserves GD & conf. sym. for G = U(N|N), OSP(2 N+2|2 N)
Results on Spectrum For operators in atypical representation of GD e. g. G = OSP(4|2), k = 1: [Candu, Mitev, VS] atypical and finite at g 2=1 States in highest weight mod. of osp(4|2)k=1 obey stable spectrum
Duality & Emergent Geometry [Candu, Saleur] [Quella, Mitev, VS] [Canazzo, Tlapac, VS] ∞ no of zero modes HA of supersphere! 1) trivial, fundamental , … At g = 1 spectrum of sigma model on S 3|2 New perturbative description at strong coupling ~ sine. Gordon/Thirring duality in terms of (super-)geometric target space data. ~ Ad. S/CFT correspondence
Conclusion Interacting QFTs with internal supersymmetry arise from disorder averaged electron systems also in Ad. S/CFT correspondence Supersymmetry can help exact computations & gives insights into non-perturbative physics e. g. non-perturbative dualities
- Slides: 10