Parabolic Whittaker Functions and Topological Field Theories I

First, we define a generalization of the standard quantum Toda chain inspired by a construction of quantum cohomology of partial flags spaces GL(\ell+1)/P, P a parabolic subgroup. Common eigenfunctions of the parabolic quantum Toda chains are generalized Whittaker functions given by matrix elements of infinite-dimensional representations of gl(\ell+1). For maximal parabolic subgroups (i.e. for P such that GL(\ell+1)/P=\mathbb{P}^{\ell}) we construct two different representations of the corresponding parabolic Whittaker functions as correlation functions in topological quantum field theories on a two-dimensional disk. In one case the parabolic Whittaker function is given by a correlation function in a type A equivariant topological sigma model with the target space \mathbb{P}^{\ell}. In the other case the same Whittaker function appears as a correlation function in a type B equivariant topological Landau-Ginzburg model related with the type A model by mirror symmetry. This note is a continuation of our project of establishing a relation between two-dimensional topological field theories (and more generally topological string theories) and Archimedean (\infty-adic) geometry. From this perspective the existence of two, mirror dual, topological field theory representations of the parabolic Whittaker functions provide a quantum field theory realization of the local Archimedean Langlands duality for Whittaker functions. The established relation between the Archimedean Langlands duality and mirror symmetry in two-dimensional topological quantum field theories should be considered as a main result of this note.Comment: Section 1 is extended and Appendices are added, 23 page

Behavior of Cosmic Rays and Propellant-Free Microwave Thruster Can Support the Hypothesis of Crystalline Vacuum

The paper suggests an explanation of the efficiency of the EmDrive device based on the hypothesis of a crystalline vacuum, previously successfully used to explain the cutoff of the cosmic-ray spectrum. The hypothesis of crystalline vacuum enables to transform part of momentum directly to vacuum crystalline lattice giving rise to reaction force which  allow to ensure the fulfillment of the momentum conservation law during EmDrive functioning. Therefore crystalline vacuum plays the role of a supporting medium for all wave vectors of electromagnetic oscillations in a conical resonator, which causes the appearance of forces and momentums of rotation, the optimization of which can open up a tempting prospect for creating of unique aircraft

Models and Laws of the Development of Scientific Knowledge

The problem of the dynamics of scientific knowledge is one of the central problems in modern methodology of science. This problem involves three main issues. The first concerns the essence of the process of science: whether it is a gradual evolutionary change( i.e. expansion of the scope and content of scientific truths), or describes a more complex model with jumps, revolutions, qualitative differences in views on the same subject [14]? This question may be formulated otherwise: is the dynamics of science the process of cumulative or, rather, anticumulative (including the waiver of some previous scientific views as unacceptable the position of new theories)? [7]. The second question concerns the explanation of the dynamics of scientific knowledge: whether it is possible to interpret it by appealing exclusively to action intrascientific (internal) factors or you must recognize a significant impact on scientific knowledge of a number of non-scientific (external), in particular, socio-cultural, factors? [16;19]. The third question involves the search for general laws of development of scientific knowledge and specific patterns of development of different fields of science [6]. The answers to the above-formulated problem cannot be obtained without the involvement of the factual material of the history of science. But the appeal to history of science assures us that the dynamics of scientific knowledge science is not a purely logical process of the unfolding of the content of scientific knowledge, and cognitive changes that take place in historical space and time. However, it is equally clear that historical material always needs some philosophical interpretation, as can be rationale reconstructs in different ways [15]

Baxter operator formalism for Macdonald polynomials

We develop basic constructions of the Baxter operator formalism for the Macdonald polynomials associated with root systems of type A. Precisely we construct a dual pair of mutually commuting Baxter operators such that the Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter operators is closely related to the dual pair of recursive operators for Macdonald polynomials leading to various families of their integral representations. We also construct the Baxter operator formalism for the q-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained by degenerations of the Macdonald polynomials associated with the type A_l root system. This note provides a generalization of our previous results on the Baxter operator formalism for the Whittaker functions. It was demonstrated previously that Baxter operator formalism for the Whittaker functions has deep connections with representation theory. In particular the Baxter operators should be considered as elements of appropriate spherical Hecke algebras and their eigenvalues are identified with local Archimedean L-factors associated with admissible representations of reductive groups over R. We expect that the Baxter operator formalism for the Macdonald polynomials has an interpretation in representation theory of higher-dimensional arithmetic fields.Comment: 22 pages, typos are fixe