From Random Matrices to Supermanifolds

1. From Random Matrices to Supermanifolds • Why random matrices? What random matrices? • Which supermanifolds? • How do random matrix problems lead to questions about supermanifolds and supersymmetric field theories? ZMP Opening Colloquium (Hamburg, Oct 22, 2005)

2. 232 Total cross section versus c.m. energy for scattering of neutrons on Th. The resonances all have the same spin 1/2 and positive parity. Wigner ´55 Poisson NDE 1726 spacings Nearest-neighbor spacing distribution for the ``Nuclear Data Ensemble´´ comprising 1726 spacings. For comparison, the RMT prediction labelled GOE and the result for a Poisson distribution are also shown. GOE

3. Universality of spectral fluctuations In the spectrum of the Schrödinger, wave, or Dirac operator for a large variety of physical systems, such as • atomic nuclei (neutron resonances), • disordered metallic grains, • chaotic billiards (Sinai, Bunimovich), • microwaves in a cavity, • acoustic modes of a vibrating solid, • quarks in a nonabelian gauge field, • zeroes of the Riemann zeta function, one observes fluctuations that obey the laws given by random matrix theory for the appropriate Wigner-Dyson class and in the ergodic limit.

4. Spacing distribution of the Riemann zeroes GUE from A. Odlyzko (1987)

5. Wigner-Dyson universality • Wigner-Dyson symmetry classes: • A : complex Hermitian matrices (‘unitary class’, GUE) • AI : real symmetric matrices (‘orthogonal class’, GOE) • AII : quaternion self-dual matrices (‘symplectic class’, GSE) Dyson (1962, The 3-fold way): ``The most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of three known types.’’ This classification has proved fundamental to various areas of theoretical physics, including the statistical theory of complex many-body systems, mesoscopic physics, disordered electron systems, and the field of quantum chaos.

6. Outline • Motivation: universality of disordered spectra • Symmetry classes of disordered fermions: from Dyson‘s threefold way to the 10-way classification • Riemannian symmetric superspaces as target spaces of susy nonlinear sigma models • Howe duality: ratios of random characteristic polynomials • Spontaneous symmetry breaking of a hyperbolic sigma model

7. Symmetry classes of disordered fermions • Consider one-particle Hamiltonians (fermions): • Canonical anti-commutation relations: • Applications/examples: • Hartree-Fock-Bogoliubov theory of superconductors • Dirac equation for relativistic spin ½ particles Classify such Hamiltonians according to their symmetries! What are the irreducible blocks that occur? Theorem [Heinzner, Huckleberry & MRZ, CMP 257 (2005) 725]: Every irreducible block that occurs in this setting corresponds to one of a large family of irreducible symmetric spaces.

8. form of complex Hermitian real symmetric quaternion self-adjoint Z complex symmetric, W=W* Z complex symmetric, W=0 Z complex skew, W=W* Z complex skew, W=0 Z complex pxq, W=0 Z real pxq, W=0 Z quaternion 2px2q, W=0 Ten large families of symmetric spaces family symmetric space

9. Physical realizations AI : electrons in a disordered metal with conserved spin and with time reversal invariance A : same as AI, but with time reversal broken by a magnetic field or magnetic impurities AII: same as AI, but with spin-orbit scatterers CI : quasi-particle excitations in a disordered spin-singlet superconductor in the Meissner phase C : same as CI but in the mixed phase with magnetic vortices DIII: disordered spin-triplet superconductor D : spin-triplet superconductor in the vortex phase, or with magnetic impurities AIII: massless Dirac fermions in SU(N) gauge field background (N > 2) BDI: same as AIII but with gauge group SU(2) or Sp(2N) CII : same as AIII but with adjoint fermions, or gauge group SO(N) Altland, Simons & MRZ: Phys. Rep. 359 (2002) 283

10. Conjecture 2: In dimension metallic behavior is stable, i.e. states remain extended under perturbation by weak disorder Open Mathematical Problems Conjecture 1: All states (at arbitrarily weak disorder) are localized in two space dimensions for symm. classes A, AI, C, CI for any symmetry class Conjecture 3: In the so-called ergodic regime the level-correlation functions are universal. The universal laws are given by the invariant Gaussian random-matrix ensemble of the appropriate symmetry class

11. Random matrix methods Methods based on the joint probability density for the eigenvalues of a random matrix: • Orthogonal polynomials + Riemann-Hilbert techniques • (Scaling limit) reduction to integrable PDE’s (Painleve-type) In contrast, superanalytic methods apply to band random matrices, granular models, random Schrödinger operators etc. • Hermitian (or Hamiltonian) disorder: Schäfer-Wegner method (1980) , Fyodorov’s method (2001) see MRZ, arXiv:math-ph/0404057 (EMP, Elsevier, 2006) • Unitary (scattering, time evolution) disorder: color-flavor transformation (1996), Howe duality (2004)

12. Hermitian random matrices for a lattice with orbitals per site Hilbert space Orthogonal projectors Fourier transform of probability measure : Local gauge invariance Wegner’s N-orbital model (class A)

13. Unitary vector space The space of all orthogonal decompositions is a Grassmann manifold Pseudo-unitary vector space of signature (p,q). The pseudo-orthogonal decompositions form a non-compact Grassmannian Globally symmetric Riemannian manifold Symmetric supermanifolds: an example (type AIII)

14. Minimal case: The algebra of sections carries a canonical action of the Lie superalgebra Riemannian symmetric superspace Example (cont’d) Vector bundle A point determines Fibre

15. Complex Lie superalgebra ( -grading) (with Cartan involution) Pick such real Lie groups that is Riemannian symmetric space in the geometry induced by the Cartan-Killing form. acts on by Ad. Form the associated vector bundle canonically acts on ‘superfunctions’ Universal construction of symmetric superspaces

16. Graded-commutative algebra of sections Susy sigma model is functional integral of maps Invariance w.r.t. to action on determines metric tensor Supersymmetric nonlinear sigma models Action functional is given by the metric tensor in the usual way. Riemannian structure is important for stability!

17. RME A AI AII C CI D DIII AIII BDI CII noncomp. AIII BDI CII DIII D CI C A AI AII compact AIII CII BDI CI C DIII D A AII AI The 10-Way Table Correspondence between random matrix models and supersymmetric nonlinear sigma models: susy NLsM MRZ, J. Math. Phys. 37 (1996) 4986

18. Conjecture 1: All states (at arbitrarily weak disorder) are localized in two space dimensions for symm. classes A, AI, C, CI. Nonlinear sigma model has mass gap. Conjecture 2: In dimension metallic behavior is stable, i.e. states remain extended under perturbation by weak disorder for any symmetry class. Noncompact symmetry is spontaneously broken. Conjecture 3: In the so-called ergodic regime the level-correlation functions are universal. The universal laws are given by the invariant Gaussian random-matrix ensemble of the appropriate symmetry class. RG flow takes nonlinear sigma model to Gaussian fixed point. Open problems –in sigma model language

19. Smooth counting function Conrey, Farmer & MRZ, math-ph/0511024 Supersymmetric Howe pairs: motivation Riemann zeta function on critical line: Farmer’s conjecture for the autocorrelation function of ratios:

20. graded vector space representation has character representation has character Super Fock space Autocorrelation of ratios as a character

21. Notice advantage: linearization! acts here The actions of and on commute. acts there is susy dual pair in the sense of R. Howe [Howe (1976/89): Remarks on classical invariant theory] Propn. is character of highest-weight irreducible representation of on the invariants in . Set

22. Main idea: the function uniquely extends to radial analytic section of symmetric superspace • Properties of : • lies in the kernel of the full ring of -invariant differential operators for • has convergent weight expansion where and Determining the character Details in: Conrey, Farmer & MRZ Heuristic picture: naive transcription of Weyl character formula to this situation gives the correct answer!

23. space(time) target symmetric space (noncompact) Energy (action) function: Regularization: lattice Targets Noncompact nonlinear sigma models d = 1: M. Niedermaier, E. Seiler, arXiv:hep/th-0312293 d = 2: Duncan, Niedermaier & Seiler, Nucl. Phys. B 720 (2005) 235 d = 3: Spencer & MRZ, Commun. Math. Phys. 252 (2004) 167

24. Theorem (Spencer & MRZ): if and if and is not too small. Consider the simplest case of Discrete field Define action via geodesic distance of : Gibbs measure: Proof: Use Iwasawa decomposition for . Integrate out nilpotent degrees of freedom, resulting in convex action for torus variables. Apply Brascamp-Lieb inequality. Spontaneous Symmetry Breaking 1d chain