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

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

Thermal energy conversion by coupled shape memory and piezoelectric effects

International audienceThis work gives experimental evidence of a promising method of thermal-to-electric energy conversion by coupling shape memory effect (SME) and direct piezoelectric effect (DPE) for harvesting quasi-static ambient temperature variations. Two original prototypes of thermal energy harvesters have been fabricated and tested experimentally. The first is a hybrid laminated composite consisting of TiNiCu shape memory alloy (SMA) and macro fiber composite piezoelectric. This composite comprises 0.1 cm3 of active materials and harvests 75 µJ of energy for each temperature variation of 60 °C. The second prototype is a SME/DPE 'machine' which uses the thermally induced linear strains of the SMA to bend a bulk PZT ceramic plate through a specially designed mechanical structure. The SME/DPE 'machine' with 0.2 cm3 of active material harvests 90 µJ over a temperature increase of 35 °C (60 µJ when cooling). In contrast to pyroelectric materials, such harvesters are also compatible with both small and slow temperature variations

On q-deformed gl(l+1)-Whittaker function II

A representation of a specialization of a q-deformed class one lattice gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed. For \ell=1, this provides an interpretation of non-specialized q-deformed gl(2)-Whittaker function in terms of QM_d(\IP^1). In particular the (q-version of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Gamma-function as a substitute of topological genus in semi-infinite geometry. A relation with Givental-Lee universal solution (J-function) of q-deformed gl(2)-Toda chain is also discussed.Comment: Extended version submitted in Comm. Math. Phys., 24 page

Consensus on the reporting and experimental design of clinical and cognitive-behavioural neurofeedback studies (CRED-nf checklist)

Neurofeedback has begun to attract the attention and scrutiny of the scientific and medical mainstream. Here, neurofeedback researchers present a consensus-derived checklist that aims to improve the reporting and experimental design standards in the field.</p