A construction of the Hermitian symmetric superspaces which are the supermanifold analogs of the Cartan domains of type IV is presented herein. Natural generalizations of the Jordan triple product and Bergman operator for the superdomains are defined, and their properties are studied.
The construction of Euclidean Majorana fermions on a two‐dimensional cylindrical space–time is discussed, based on a given real‐time theory. After a review of the basics of defining Euclidean fermions in the free theory, a functional integral formulation for the Euclidean theory is introduced. This functional integral connects the expectation values of the Euclidean fields with the theory of relative Pfaffians. A proof, which draws on these ideas, is given for the Feynman–Kac formula of a general interacting theory.
We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spinc quantization. We prove the analog of Kodaira vanishing for the Spinc Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semiclassically. We also introduce a new quantization scheme, based on a rescaled Laplacian, for which we are able to prove strong semiclassical properties. The two quantizations are shown to be close semiclassically.
We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary conditions.
Consider in L 2(ℝ l ) the operator family H(ε):=P 0(ℏ,ω)+ε Q 0. P 0 is the quantum harmonic oscillator with diophantine frequency vector ω, Q 0 a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and ε ∈ ℝ. Then there exists ε*>0 with the property that if |ε|<ε* there is a diophantine frequency ω(ε) such that all eigenvalues E n (ℏ,ε) of H(ε) near 0 are given by the quantization formula where α is an l-multi-index.
About a year ago, a team of physicists reported in Science that they had observed "evidence for E8 symmetry'' in the laboratory. This expository article is aimed at mathematicians and explains the chain of reasoning connecting measurements on a quasi-1-dimensional magnet with a 248-dimensional Lie algebra.
We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.
We develop the scattering theory of a general conformally compact metric by treating the Laplacian as a degenerate elliptic operator (with non-constant indicial roots) on a compact manifold with boundary. Variability of the roots implies that the resolvent admits only a partial meromorphic continuation, and the bulk of the paper is devoted to studying the structure of the resolvent, Poisson, and scattering kernels for frequencies outside the region of meromorphy. For low frequencies the scattering matrix is shown to be a pseudodifferential operator with frequency dependent domain. In particular, generalized eigenfunctions exhibit L2 decay in directions where the asymptotic curvature is sufficiently negative. We explicitly construct the resolvent kernel for generic frequency in this part of the continuous spectrum.
We present a general theory of non-perturbative quantization of a class of hermitian symmetric supermanifolds. The quantization scheme is based on the notion of a super Toeplitz operator on a suitable 2-graded Hilbert spaces of super-holomorphic functions. The quantized supermanifold arises as the *-algebra generated by all such operators. We prove that our quantization framework reproduces the invariant super Poisson structure on the classical supermanifold as Planck′s constant tends to zero.