Unidirectional amplification with acoustic non-Hermitian space−time varying metamaterial

Space−time modulated metamaterials support extraordinary rich applications, such as parametric amplification, frequency conversion, and non-reciprocal transmission. The non-Hermitian space−time varying systems combining non-Hermiticity and space−time varying capability, have been proposed to realize wave control like unidirectional amplification, while its experimental realization still remains a challenge. Here, based on metamaterials with software-defined impulse responses, we experimentally demonstrate non-Hermitian space−time varying metamaterials in which the material gain and loss can be dynamically controlled and balanced in the time domain instead of spatial domain, allowing us to suppress scattering at the incident frequency and to increase the efficiency of frequency conversion at the same time. An additional modulation phase delay between different meta-atoms results in unidirectional amplification in frequency conversion. The realization of non-Hermitian space−time varying metamaterials will offer further opportunities in studying non-Hermitian topological physics in dynamic and nonreciprocal systems.

On the Oort conjecture for Shimura varieties of unitary and orthogonal types

In this paper we study the Oort conjecture concerning the non-existence of Shimura subvarieties contained generically in the Torelli locus in the Siegel modular variety${\mathcal{A}}_{g}$. Using the poly-stability of Higgs bundles on curves and the slope inequality of Xiao on fibered surfaces, we show that a Shimura curve$C$is not contained generically in the Torelli locus if its canonical Higgs bundle contains a unitary Higgs subbundle of rank at least$(4g+2)/5$. From this we prove that a Shimura subvariety of$\mathbf{SU}(n,1)$type is not contained generically in the Torelli locus when a numerical inequality holds, which involves the genus$g$, the dimension$n+1$, the degree$2d$of CM field of the Hermitian space, and the type of the symplectic representation defining the Shimura subdatum. A similar result holds for Shimura subvarieties of$\mathbf{SO}(n,2)$type, defined by spin groups associated to quadratic spaces over a totally real number field of degree at least$6$subject to some natural constraints of signatures.

ARGUMENTS FOR F-THEORY

After a brief review of string and M-theory we point out some deficiencies. Partly to rectify them, we present several arguments for "F-theory", enlarging spacetime to (2, 10) signature, following the original suggestion of C. Vafa. We introduce a suggestive supersymmetric 27-plet of particles, associated to the exceptional symmetric hermitian space E6/ Spin c(10). Several possible future directions, including using projective rather than metric geometry, are mentioned. We should emphasize that F-theory is yet just a very provisional attempt, lacking clear dynamical principles.

Representation of Algebras with Involution

Let K be a field with an involution J. A *-algebra over K is an associative algebra A with an involution * satisfying (α.a)* = αJ.a*. A large class of examples may be obtained as follows. Let (V, φ) be an hermitian space over K consisting of a vector space V and a left hermitian (w.r.t. J) form φ on V which is nondegenerate in the sense that φ(V,v) = 0 implies v = 0. An endomorphism f of V may have an adjoint f* w.r.t. φ, defined by φ(f(u),v) = φ(u,f*(v)); due to the nondegeneracy of φ, f* is unique if it exists. The set B(V, φ) of all endomorphisms of V which do have an adjoint is easily verified to be a *-algebra.

Conformally Kähler manifolds

Introduction. The present paper is concerned with the conformal geometry of Hermitian spaces. In the first part we find a necessary and sufficient condition for a Hermitian space to be conformally Kähler, that is, conformal to some Kähler space. The condition is that a certain conformal tensor, , vanishes identically. Then, defining a Hermitian manifold as in Hodge (3), we consider such a manifold where the restriction is made that at every point the tensor is zero. This will be called a conformally Kähler manifold, and conditions under which it may be given a Kähler metric are obtained. It is found that any conformally Kähler manifold may be given a Kähler metric provided it is simply-connected or that its fundamental group is of finite order.