Chiara Esposito

A high order q-Difference equation for q-Hahn multiple orthogonal polynomials

A high-order linear q-difference equation with polynomial coefficients having q-Hahn multiple orthogonal polynomials as eigenfunctions is given. The order of the equation coincides with the number of orthogonality conditions that these polynomials satisfy. Some limiting situations when are studied. Indeed, the difference equation for Hahn multiple orthogonal polynomials given in Lee [J. Approx. Theory (2007), doi: 10.1016/j.jat.2007.06.002] is obtained as a limiting case.

Reduction of pre-Hamiltonian actions

Abstract We prove a reduction theorem for the tangent bundle of a Poisson manifold ( M , π ) endowed with a pre-Hamiltonian action of a Poisson–Lie group ( G , π G ) . In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden–Ratiu reduction of M . If the manifold M is symplectic and simply connected, the reduced tangent bundle is integrable and its integral symplectic groupoid is the Marsden–Weinstein reduction of the pair groupoid M × M .

Rigidity of infinitesimal momentum maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.

Obstructions for twist star products

In this short note, we point out that not every star product is induced by a Drinfel’d twist by showing that not every Poisson structure is induced by a classical r-matrix. Examples include the higher genus symplectic Pretzel surfaces and the symplectic sphere

Deformation Quantization and Formality Theory

{\mathbb {S}}^2

Kontsevich’s Formula and Globalization

S2.

Formality Theory: From Poisson structures to Deformation Quantization

The philosophy of deformation was proposed by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer in the seventies and since then, many developments occurred. Deformation quantization is based on such a philosophy in order to provide a mathematical procedure to pass from classical mechanics to quantum mechanics. Basically, it consists in deforming the pointwise product of functions to get a non-commutative one, which encodes the quantum mechanics behaviour. In formal deformation quantization, the non-commutative product (also said, star product) is given by a formal deformation of the pointwise product, i.e. by a formal power series in the deformation parameter which physically play the role of Planck’s constant ̵ h. From a physical point of view this is clearly not sufficient to provide a reasonable quantum mechanical description and hence one needs to overcome the formal power series aspects in some way. One option is strict deformation quantization, which produces quantum algebras not just in the space of formal power series but in terms of C∗-algebras, as suggested by Rieffel, with e.g. a continuous dependence on ̵ h. There are several other options in between continuous and formal dependence on ̵ h like analytic or smooth deformations. The Oberwolfach workshop Deformation quantization: between formal to strict consolidated, continued, and extended these research activities with a focus on the study of the connection between formal and strict deformation quantization in their various flavours and their applications in particular those in quantum physics and non-commutative geometry. It brought together specialists in differential geometry, operator algebras, non-commutative geometry, and quantum field theory with research interests in the mentioned quantization procedures. The aim of the workshop was to develop a coherent viewpoint of the many recent diverse developments in the field and to initiate new lines of research. 572 Oberwolfach Report 11/2015 Mathematics Subject Classification (2010): 53D55, 14D15. Introduction by the Organisers Formal deformation quantization as introduced by Bayen et al. has reached by now a very satisfying state: with the highly non-trivial formality theorem of Kontsevich the questions on existence as well as on classification of formal star products on general Poisson manifolds have been settled and answered in the positive in 1997. Alternative approaches to the globalization of Kontsevich’s result were also obtained by Cattaneo, Felder, and Tomassini in 2002 as well as by Dolgushev 2005. Before, the symplectic case was investigated by various groups. Here the existence of star products was shown by Lecomte and DeWilde already 1983, later independently by Fedosov in 1986 and by Omori, Maeda, and Yoshioka in 1991. The classification of star products in the symplectic case was obtained by Nest and Tsygan in 1995 and independently by Deligne in 1995 and Bertelson, Cahen, and Gutt in 1997. The representation theory of the deformed algebras, which is crucial for a physical application, has been investigated in detail by many people: among other things, the full classification of the star product algebras up to Morita equivalence was obtained by Waldmann, Bursztyn and Dolgushev in 2012. For a physical interpretation of the star product algebras as observable algebras of a quantized physical system, the formal parameter has to be identified with Planck’s constant h̵. Hence a convergence of the formal series in h̵ is crucial. In the early era of deformation quantization the formal star products have been constructed by means of asymptotic expansions of other quantizations like Berezin-Toeplitz quantizations on quantizable Kahler manifolds or symbol calculus quantizations on cotangent bundles. Beside producing rather explicit examples like the constructions of Cahen, Gutt and Rawnsley case as well as Karabegov in the Kahler case or Bordemann, Neumaier, Pflaum and Waldmann in the cotangent bundle case, the good understanding of the formal star products also led to interesting results on the convergent origins: here the computations of characteristic classes by Karabegov and Schlichenmaier or the index theorems of Fedosov as well as Nest and Tsygan should be mentioned. For the whole world beyond smooth Poisson manifolds the works of Pflaum, Posthuma, and Tang show first deep results on deformation quantization also in this case. On a more analytic oriented approach based on a C∗-algebraic formulation using continuous fields of C∗-algebras, Rieffel showed how an action of R on a C∗-algebra can be used to deform this C∗-algebra in a continuous way. Applied to the bounded continuous functions on a manifold, this ultimately leads again to a formal star product by an asymptotic expansion of the continuous deformation for h̵ Ð→ 0, at least on sufficiently smooth vectors of the action. Ever since, Rieffel’s paradigma of deformation by group actions was studied in many contexts and substantially extended recently to other (non-abelian) Lie groups than R by Bieliavsky and Gayral and coworkers. On a more abstract level, Natsume, Nest, and Peter considered symplectic manifolds with a topological condition (trivial Mini-Workshop: Deformation quantization: between formal to strict 573 second fundamental group) and showed that a strict quantization always exists, based on the usage of Darboux charts and a Čech cohomological argument. The relation between formal and strict deformation quantization has been subject of several studies, but there still remain deep open questions. Since the approach of formal deformation quantization is universal, as proved by Kontsevich, it is natural to try to find the way back: from the easy formal situation to the more complicated convergent one. Since the above mentioned quantization schemes all use particular geometric features, one can hope to recover not only a convergent quantization as required by physics, but also interesting information about the underlying geometry. There are only few examples where this way backwards was investigated: in the flat case, Beiser, Rmer and Waldmann considered the convergence of the Wick star product on C and recovered the full symmetry, coherent states, and the Bargmann-Fock representation from the convergence conditions. While this example is still geometrically rather trivial, it already shows a rich structure beyond the locally multiplicatively convex theory. It can be extended to infinite dimensions in a rather conceptual way as recently shown by Waldmann. The relations to the approches of Dito’s star products on Hilbert spaces still remain to be investigated. Later, Beiser and Waldmann considered a Wick-type star product on the Poincaré disk. Here the underlying geometry is topologically still trivial but enjoys a curved Kahler structure. Again, in this example the full symmetry of the problem is recovered and the foundations of a representation theory to establish the relations with the Berezin-Toeplitz quantizations are formed. Bieliavsky, Detournay, and Spindel gave a deformation of the Poincaré disk in a C∗-algebraic approach thus complementing the picture from the other side. However, the precise relations between the different versions of convergence remain unclear. Even though these examples seem to be isolated at the moment,they can be seen as a proof of concept that investigating the convergence of formal star products gives both physically relevant and manageable observable algebras and interesting information about the underlying geometry. Understanding the analytic aspects of deformation quantization has led to many non-trivial and surprising applications beyond the field of deformation quantization itself . Here we only want to mention a few: the works of Anderson and coworkers on the mapping class group where the results of Bordemann, Meinrenken, and Schlichenmaier on the asymptotic properties of Berezin-Toeplitz quantization enter in a crucial way. The works of Lechner show how one can use Rieffel’s deformations to construction new examples of quantum field theories as deformations of free theories. In some sense they can be seen as quantum field theories on a non-commutative Minkowski spacetime. Quantum deformations of classical geometries lead to interesting spaces in non-commutative geometry, here the quantum spheres of Connes and Landi provide a non-trivial and rich class where concepts of non-commutative geometry can be tested explicitly. Still many questions remain open: first, the above mentioned examples have to be investigated further to understand their relations and connections. Moreover, the quest for convergence of star products in order to produce (ultimately) a 574 Oberwolfach Report 11/2015 continuous field of C∗-algebras has to be extended beyond the above examples. Here one can think of other types of algebras between the formal power series on the one hand and the C∗-algebras on the other hand: in particular locally convex algebras and also bornological algebras may provide a good bridge. Here the techniques developed by Meyer on bornological algebras will play a crucial role. The overall goal of the workshop was to develop a coherent viewpoint of the many recent developments on the analytic aspects of deformation quantization as described above with particular emphasis on the connection between formal and strict and their potential applications in physics. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Mini-Workshop: Deformation quantization: between formal to strict 575 Mini-Workshop: Deformation quantization: between formal to strict

A universal construction of universal deformation formulas, Drinfeld twists and their positivity

This chapter will be devoted to the theory of formal deformation quantization. We will recall the description of a quantum physical system in terms of a non-commutative algebra of operators (quantum observables) on a Hilbert space.