Reviews in Mathematical Physics

We study D-branes wrapping an exceptional four-cycle P(1,a,b) in a blown-up C³/Z_m non-compact Calabi–Yau threefold with (m;a,b)=(3;1,1), (4;1,2) and (6; 2, 3). In applying the method of local mirror symmetry we find that the Picard–Fuchs equations for the local mirror periods in the Z_3,4,6 orbifolds take the same form as the ones in the local E_6,7,8 del Pezzo models, respectively. It is observed, however, that the orbifold models and the del Pezzo models possess different physical properties because the background NS B-field is turned on in the case of Z_3,4,6 orbifolds. This is shown by analyzing the periods and their monodromies in full detail with the help of Meijer G-functions. We use the results to discuss D-brane configurations on P(1,a,b) as well as on del Pezzo surfaces. We also discuss the number theoretic aspect of local mirror symmetry and observe that the exponent which governs the exponential growth of the Gromov–Witten invariants is determined by the special value of the Dirichlet L-function.

Results are obtained on resolvent expansions around zero energy for Schrödinger operators H=-Δ+V(x) on L²(R^m), where V(x) is a sufficiently rapidly decaying real potential. The emphasis is on a unified approach, valid in all dimensions, which does not require one to distinguish between ∫V(x)dx=0 and ∫V(x)dx≠0 in dimensions m=1,2. It is based on a factorization technique and repeated decomposition of the Lippmann–Schwinger operator. Complete results are given in dimensions m=1 and m=2.

In this note we prove Strong Dynamical Localization for the almost Mathieu operator H_θ,λ,ω=-Δ+ λcos(2π(θ + xω)) for all λ>2 and Diophantine frequencies ω. This improves the previous known result [22, 13] which established Dynamical Localization for a.e. θ and for λ≥15.

We introduce a class of noncompact Riemannian manifolds on which we can argue quantum field theory for scalar particles in external fields. More precisely, we consider quantized linear Klein–Gordon fields subject to (non quantized) electromagnetic forces in a certain class of static space-time. This class is broad enough to include physically important examples of the Euclidean space, the hyperbolic space, and by passing to the natural Lorentzian structure, the Schwarzschild metric up to conformal equivalence. The S-matrix of the massive Klein–Gordon equation on these manifolds is unitarily implemented on the Fock space constructed via the spectrum of the Laplace–Beltrami operator with scalar curvature. We also give the same result for the massless case in the asymptotically flat and hyperbolic spaces.