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## Über dieses Buch

​This book presents a selection of papers based on the XXXIII Białowieża Workshop on Geometric Methods in Physics, 2014. The Białowieża Workshops are among the most important meetings in the field and attract researchers from both mathematics and physics. The articles gathered here are mathematically rigorous and have important physical implications, addressing the application of geometry in classical and quantum physics. Despite their long tradition, the workshops remain at the cutting edge of ongoing research. For the last several years, each Białowieża Workshop has been followed by a School on Geometry and Physics, where advanced lectures for graduate students and young researchers are presented; some of the lectures are reproduced here. The unique atmosphere of the workshop and school is enhanced by its venue, framed by the natural beauty of the Białowieża forest in eastern Poland.

The volume will be of interest to researchers and graduate students in mathematical physics, theoretical physics and mathematmtics.

## Inhaltsverzeichnis

### Traces of Holomorphic Families of Operators on the Noncommutative Torus and on Hilbert Modules

Abstract
We revisit traces of holomorphic families of pseudodifferential operators on a closed manifold in view of geometric applications. We then transpose the corresponding analytic constructions to two different geometric frameworks: the noncommutative torus and Hilbert modules. These traces are meromorphic functions whose residues at the poles as well as the constant term of the Laurent expansion at zero (the latter when the family at zero is a differential operator) can be expressed in terms of Wodzicki residues and extended Wodzicki residues involving logarithmic operators. They are therefore local and contain geometric information. For holomorphic families leading to zeta regularised traces, they relate to the heat-kernel asymptotic coefficients via an inverse Mellin mapping theorem. We revisit Atiyah’s L 2- index theorem by means of the (extended) Wodzicki residue and interpret the scalar curvature on the noncommutative two torus as an (extended) Wodzicki residue.
Sara Azzali, Cyril Lévy, Carolina Neira-Jiménez, Sylvie Paycha

### r ∞-Matrices, Triangular L ∞-Bialgebras and Quantum∞ Groups

Abstract
A homotopy analogue of the notion of a triangular Lie bialgebra is proposed with a goal of extending basic notions of the theory of quantum groups to the context of homotopy algebras and, in particular, introducing a homotopical generalization of the notion of a quantum group, or quantum-group.
Denis Bashkirov, Alexander A. Voronov

### Normal Forms and Lie Groupoid Theory

Abstract
In these lectures I discuss the Linearization Theorem for Lie groupoids, and its relation to the various classical linearization theorems for submersions, foliations and group actions. In particular, I explain in some detail the recent metric approach to this problem proposed in [6].
Rui Loja Fernandes

### Higher Braces Via Formal (Non)Commutative Geometry

Abstract
We translate the main result of [11] to the language of formal geometry. In this new setting we prove directly that the Koszul resp. Börjeson braces are pullbacks of linear vector fields over the formal automorphism $$\varphi(a)=\mathrm{exp}(a)-1$$ in the Koszul, resp. $$\varphi(a)=a(1-a)^{-1}$$ in the Börjeson case. We then argue that both braces are versions of the same object, once materialized in the world of formal commutative geometry, once in the noncommutative one.
Martin Markl

### Conformally Rescaled Noncommutative Geometries

Abstract
Noncommutative geometry aims to provide a set of mathematical tools to describe spacetime, gravity and quantum field theory at small scales. The paper reviews the idea that noncommutative spaces are described in terms of algebras and their geometry, which is encoded as spectral triples. The latter are basic ingredients of the new notion of Riemannian spin geometry adapted to the language of operator algebras. Using this background we propose a new idea of conformally rescaled and curved spectral triples, which are obtained from a real spectral triple by a nontrivial scaling of the Dirac operator. The obtained family is shown to share many properties with the original spectral triple. We compute the Wodzicki residue and the Einstein–Hilbert functional for such family on the four-dimensional noncommutative torus.
Andrzej Sitarz

### On Some Quaternionic Coherent States and Wavelets

Abstract
As a natural extension of the affine groups of the reals and the complexes, leading to one and two-dimensional wavelets, we look at the quaternionic affine group and its representations on both complex and quaternionic Hilbert spaces. We then study the problem of coherent states and wavelets built from these representations.
S. Twareque Ali

### Supersymmetric Vector Coherent States for Systems with Zeeman Coupling and Spin-Orbit Interactions

Abstract
This work addresses a method for constructing supersymmetric vector coherent states (VCS) for a two-dimensional electron gas in a perpendicular magnetic field in the presence of both Rashba and Dresselhaus spin-orbit (SO) interactions, with an effective Zeeman coupling. The model Hamiltonian, decomposed into two conveniently defined operators, acts on a tensor product of two Hilbert spaces associated with corresponding chiral sectors. Supersymmetric (SUSY) VCS, related to a SUSY pair of Hamiltonians, are built. A generalized oscillator algebra is provided using quadrature operators. Mean-values of position and momentum operators and uncertainty relation in the SUSY VCS are obtained and discussed. Besides, the SUSY VCS time evolution is also given.
I. Aremua, E. Baloïtcha, M. N. Hounkonnou

### Energy and Stability of the Pais–Uhlenbeck Oscillator

Abstract
We study stability of higher-derivative dynamics from the viewpoint of more general correspondence between symmetries and conservation laws established by the Lagrange anchor. We show that classical and quantum stability may be provided if a higher-derivative model admits a bounded from below integral of motion and the Lagrange anchor that relates this integral to the time translation.
D. S. Kaparulin, S. L. Lyakhovich

### Deformation Quantization with Separation of Variables and Gauge Theories

Abstract
We construct a gauge theory on a noncommutative homogeneous Kähler manifold by using the deformation quantization with separation of variables for Kähler manifolds. A model of noncommutative gauge theory that is connected with an ordinary Yang–Mills theory in the commutative limit is given. As an examples, we review a noncommutative $$\mathbb{C}P^N$$ and construct a gauge theory on it. We also give details of the proof showing that the noncommutative $$\mathbb{C}P^N$$ constructed in this paper coincides with the one given by Bordemann, Brischle, Emmrich and Waldmann [1].
Yoshiaki Maeda, Akifumi Sako, Toshiya Suzuki, Hiroshi Umetsu

### Physics of Spectral Singularities

Abstract
Spectral singularities are certain points of the continuous spectrum of generic complex scattering potentials. We review the recent developments leading to the discovery of their physical meaning, consequences, and generalizations. In particular, we give a simple definition of spectral singularities, provide a general introduction to spectral consequences of $$\mathcal{PT}$$-symmetry (clarifying some of the controversies surrounding this subject), outline the main ideas and constructions used in the pseudo-Hermitian representation of quantum mechanics, and discuss how spectral singularities entered in the physics literature as obstructions to these constructions. We then review the transfer matrix formulation of scattering theory and the application of complex scattering potentials in optics. These allow us to elucidate the physical content of spectral singularities and describe their optical realizations. Finally, we survey some of the most important results obtained in the subject, drawing special attention to the remarkable fact that the condition of the existence of linear and nonlinear optical spectral singularities yield simple mathematical derivations of some of the basic results of laser physics, namely the laser threshold condition and the linear dependence of the laser output intensity on the gain coefficient.

### On the Moduli Space of Yang–Mills Fields on $$\mathbb{R}^4$$

Abstract
We consider the problem of description of the structure of the moduli space of Yang–Mills fields on $$\mathbb{R}^4$$ with gauge group G. According to harmonic spheres conjecture, this moduli space should be closely related to the space of harmonic spheres in the loop space ΩG. Since the structure of the latter space is much better understood, the proof of conjecture will help to clarify the structure of the moduli space of Yang–Mills fields. We propose an idea how to prove the harmonic spheres conjecture using the twistor methods.
Armen Sergeev

### On Covariant Poisson Brackets in Field Theory

Abstract
A general approach is proposed to constructing covariant Poisson brackets in the space of histories of a classical field-theoretical model. The approach is based on the concept of Lagrange anchor, which was originally developed as a tool for path-integral quantization of Lagrangian and non-Lagrangian dynamics. The proposed covariant Poisson brackets generalize the Peierls’ bracket construction known in the Lagrangian field theory.
A. A. Sharapov

### Weyl Group Orbit Functions in Image Processing

Abstract
Generalized Fourier transform and related convolution based on Weyl group orbit functions transform can be used in spatial image filtering. In this paper we use the family of E-orbit functions of A 2 to provide an example of such a filtering.

### Poisson and Fourier Transforms for Tensor Products and an Overalgebra

Abstract
For the group $$G = SL\left( {2,\,\mathbb{R}} \right),$$ we write explicitly operators intertwining irreducible finite-dimensional representations T k with tensor products $$T_l \otimes T_m$$ (we call them Poisson and Fourier transforms), they are differential operators, and also we write explicit formulae for compositions of these transforms with Lie operators of the overgroup $$G \times G$$

### Unipotent Flow on $$\text{SL}_2 \left( {\Bbb Z} \right)\backslash \text{SL}_2 \left( {\mathbb{R}} \right)$$ : From Dynamics to Elementary Number Theory

Abstract
In this paper we give a simple and elementary proof of a quantitative density result for unipotent flow on $$\text{SL}_2 \left( {\Bbb Z} \right)\backslash \text{SL}_2 \left( {\mathbb{R}} \right)$$.
Nikolay Moshchevitin

### Lie Superalgebras of Krichever–Novikov Type

Abstract
Classically, starting from the Witt and Virasoro algebra important examples of Lie superalgebras were constructed. In this write-up of a talk presented at the Białowieża meetings we report on results on Lie superalgebras of Krichever–Novikov type. These algebras are multi-point and higher genus equivalents of the classical algebras. The grading in the classical case is replaced by an almost-grading. It is induced by a splitting of the set of points, were poles are allowed, into two disjoint subsets. With respect to a fixed splitting, or equivalently with respect to a fixed almost-grading, it is shown that there is up to rescaling and equivalence a unique non-trivial central extension of the Lie superalgebra of Krichever–Novikov type. It is given explicitly.
Martin Schlichenmaier

### On n-ary Lie Algebras of Type (r, l)

Abstract
These notes are devoted to the multiple generalization of a Lie algebra introduced by A.M. Vinogradov and M.M. Vinogradov. We compare definitions of such algebras in the usual and invariant case. Furthermore, we show that there are no simple n-ary Lie algebras of type (n - 1, l) for l > 0.
E. G. Vishnyakova

### Factorization Method for (q, h)-Hahn Orthogonal Polynomials

Abstract
We investigate a version of factorization method on a sequence of Hilbert spaces related to (q, h)-ladder. As an example we present (q, h)-orthogonal polynomials.
Alina Dobrogowska, Grzegorz Jakimowicz

### Examples of Hamiltonian Systems on the Space of Deformed Skew-symmetric Matrices

Abstract
We consider a dual space to a Lie algebra of deformed skew-symmetric matrices equipped with two compatible Poisson brackets: the Lie–Poisson bracket and the frozen bracket.We obtain a bi-Hamiltonian integrable system. As an example we present a case of a system on a two-dimensional quadric which can be seen as a generalization of the Neumann system.
Alina Dobrogowska, Tomasz Goliński

### Matrix Beta-integrals: An Overview

Abstract
First examples of matrix beta-integrals were discovered in 1930–50s by Siegel and Hua and in the 60s Gindikin obtained multi-parametric series of such integrals. We discuss beta-integrals related to the symmetric spaces, their interpolation with respect to the dimension of a ground field, and adelic analogs; also we discuss beta-integrals related to flag spaces.
Yurii A. Neretin

### Differential Equations on Complex Manifolds

Abstract
We discuss linear partial differential equations with constant coefficients on complex manifold $$\mathbb{C}^n$$. Using the Sternin–Shatalov integral transform we solve complex Cauchy problem and consider two applications: describe the singularities of the solution of the Cauchy problem and solve a physical problem of sweeping the charge inside the domain (balayage inwards problem).
Anton Savin, Boris Sternin

### Asymptotic Properties of Solutions of Neutral Type Difference System with Delays

Abstract
We consider a three-dimensional nonlinear difference system with deviating arguments of the following form
$$\left\{ {\begin{array}{lll} \Delta \left( {x_n + p_n x_n - \tau } \right) = a_n f\left( {y_n - 1} \right) \hfill \cr \qquad\qquad\Delta y_n = b_n g\left( {w_{n - m} } \right), \hfill \cr \qquad\qquad\,\,\,\Delta w_n = \delta c_n h\left( {x_{n - k} } \right) \hfill \cr \end{array}} \right.$$
where the first equation of the system is a neutral type difference equation. First, the classification of nonoscillatory solutions of the considered system is presented. Next, we present the sufficient conditions for boundedness of a nonoscillatory solution. The obtained results are illustrated by examples.
Ewa Schmeidel, Joanna Zonenberg, Barbara Łupińska

### Orbits of Darboux Groupoid for Hyperbolic Operators of Order Three

Abstract
Darboux transformations are viewed as morphisms in a Darboux category. Darboux transformations of type I which we defined previously, make an important subgroupoid. We describe the orbits of this subgroupoid for hyperbolic operators of order three.
We consider the algebras of differential invariants for our operators. In particular, we show that the Darboux transformations of this class can be lifted to transformations of differential invariants (which we calculate explicitly).
Ekaterina Shemyakova

### From Nicolaus Copernicus’ Economic Law up to the Present Day Economic Disasters (Report of a Dilettante)

Abstract
The commercial phenomena described by an almost unknown N. Copernicus treatise on money prepared in the XVIth century on request of the Polish king Sigismund I. the Old turn out relevant for the present day economic situation.
Bogdan Mielnik

### Economics – Physics of Social Sciences or Art?

Economic talk at the XXXIII Workshop on Geometric Methods in Physics
Abstract
Some questions about the nature of economics are raised, e.g., the one whether economics can give us universal laws describing the workings of the market. The discussion here presented refers also to the debate on the state of the economic theory in the wake of the recent global financial crises. The paper concludes that economics is unable to give us such explanations of real economic processes that do not need further investigations. Also, the author claims that due to the extensive use of metaphors and economists’ creativity in modeling economic phenomena economics is close to art, and thus one can even talk about the beauty of the science of Adam Smith.
Lukasz Hardt
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