Deformation quantization of algebraic varieties pdf

In the last decade, the development of new ideas in quantum theory, including geometric and deformation quantization, the nonabelian berrys geometric factor, super and brst symmetries, noncommutativity, has called into play the geometric techniques based on the deep interplay between algebra, differential geometry and topology. The notes also contain four appendices, and a bibliography. Abelian deformations of ordinary algebras of functions are studied. Marc rieffel, quantization and c c\astalgebras, contemporary mathematics vol. Jorgen ellegaard andersen centre for quantum geometry of moduli spaces aarhus university.

This was put on a firm basis by foundational work of kunihiko kodaira and donald c. Recommend this journal email your librarian or administrator to recommend adding this journal to your organisations collection. Download pdf complex manifolds and deformation of complex. W e have an algebr aic star pro duct on m or m c if the bilinear op erators b i in 2 are algebraic operator s. Deformation quantization strasbourg, 2001, irma lectures in mathematics and theoretical physics vol. As we already mentioned, kashiwara was the first to consider stacky quantizations.

Let x be a smooth affine algebraic variety over a field k of characteristic 0, and let r be a complete parameter kalgebra e. Each contribution in this volume expands on the ideas of gerstenhaber and stasheff. A direct application of the formality theorem to an. Pdf twisted deformation quantization of algebraic varieties. Michael penkava, pol vanhaecke, deformation quantization of polynomial poisson algebras, journal of algebra 227, 365n393 2000 arxiv. Shifted poisson structures and deformation quantization. Proceedings of the meeting of theoretical physicists and mathematicians, strasbourg, may 31 june 2, 2001 rencontre entre physiciens theoriciens et mathematiciens, strasbourg, 31 mai 2 juin 2001 pp. A frobeniusprojective structure is an analogue of a complex projective structure on a riemann surface, which was introduced by y. In the papers i, ii and iii, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Pdf quantization on algebraic curves with frobenius.

Shifted poisson structures and deformation quantization d. Deformation quantization in algebraic geometry sciencedirect. The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties. The notion of strict c c\ast algebraic deformation quantization was introduced in. Download fulltext pdf download fulltext pdf twisted deformation quantization of algebraic varieties survey article pdf available january 2008 with 37 reads. C c which makes c into a lie algebra, and is a biderivation i. In 35 i proposed a way to use results of 33 in the case of algebraic varieties. Fioresi and others published on the deformation quantization of affine algebraic varieties find, read and.

This result extends work of kontsevich, and our own earlier work, on deformation quantization of algebraic varieties. Deformation quantization and geometric quantization of abelian varieties c e n t r e f o r q u a n t u m g e o m e t r y o f m o d u l i s p a c e s q g m progress report jakob lindblad blaavand advisor. It incorporates some recent progress on the subject. The precise statement about twisted deformation quantization of an algebraic variet y appears for the. Notes on deformation quantization shilin yu abstract. I am assuming some knowledge of deformation quantization, algebraic geometry and stacks. I talked about twisted deformation quantization in the past ye4.

The role of harrison cohomology in classifying such deformations is illustrated in the context of simple examples chosen for their relevance to physics. It is made up mainly from the material in referativnyi zhurnal matematika during 19651973. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic geometry. Thus, we introduce a new notion of a semiformal deformation, a replacement in algebraic. Amnon yekutieli, deformation quantization in algebraic geometry, advances in mathematics 198 2005, 383432. Twisted deformation quantization of algebraic varieties amnonyekutieli dedicatedtorinayekutieli,mymother,onher80thbirthday abstract. However this is a new lecture, written especially for this workshop. In 36, i proposed a way to use the results of 34 in the case of algebraic varieties. It seems that for rational poisson varieties, deformation quantization is truly canonical in a very strong sense. The quantization procedure i am considering isthe berezintoeplitz quantization, resp. Harrison cohomology and abelian deformation quantization on algebraic varieties.

Recall that a poisson bracket on c is a kbilinear function. Deformation quantization in algebraic geometry core. Deformation quantization of algebraic varieties springerlink. Deformation quantization of algebraic varieties, letters in. Twisted deformation quantization of algebraic varieties arxiv.

It seems that for rational poisson varieties deformation quantization is truly canonical in a very strong sense. Quantum mirrors of log calabiyau surfaces and highergenus. Twisted deformation quantization of algebraic varieties core. Derived algebraic geometry and deformation quantization.

Using the universal deformation formulas of kontsevich, and an algebrogeometric approach to the bundle of formal coordinate systems over a smooth variety x, we prove existence of deformation quantization of the sheaf ox assuming the vanishing of certain cohomologies. In the present paper, we study the relationship between deformation quantizations and frobeniusprojective structures defined on an algebraic curve in positive characteristic. A direct application of the formality theorem to an algebraic poisson manifold gives a canonical sheaf of categories deforming coherent sheaves. Twisted deformation quantization of algebraic varieties. Geometric and algebraic topological methods in quantum mechanics. These higher algebraic structures provide a common language essential in the study of deformation quantization, theory of algebroids and groupoids, symplectic field theory, and much more.

Spencer, after deformation techniques had received a great deal of more tentative application in the italian school of algebraic geometry. Topics in computational algebraic geometry and deformation. Fioresi and others published on the deformation quantization of affine algebraic varieties find, read and cite all the research you need on researchgate. Harrison cohomology and abelian deformation quantization on. We study deformation quantization of poisson algebraic varieties. Introduction deformation quantization is a term coined by moshe flato, who suggested that any nontrivial associative deformation of an algebra of functions should be interpreted as a kind of quantization.

We prove that there is a twisted quantization operation from twisted poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. It is well known that harrison cohomology is trivial on smooth manifolds and that, consequently, abelian. The paper is devoted to peculiarities of the deformation quantization in the algebrogeometric context. The precise statement about twisted deformation quantization of an algebraic variety appears for the first time in this paper this is theorem 0. Jan 19, 2007 fronsdal c harrison cohomology and abelian deformation quantization on algebraic varieties. Such an additional structure has some equivalent objects, e. A quantization of a 0 is an associative algebra ar, together with an isomorphism ama. Some background on deformation quantization let kbe a. Twisted deformation quantization notes are available online.

This quantization procedure is adapted to the complex structure which isa requirement for the fact that it can be formulated in terms ofcomplex algebraic geometry. You might also be interested by the very inspiring paper operads and motives in deformation quantization. Pdf on the deformation quantization of affine algebraic. Deformation quantization g and geometric quantization t c of. Twisted deformation quantization of algebraic varieties article pdf available in advances in mathematics 268 january 2015 with 81 reads how we measure reads. Oct 07, 2004 deformation quantization of algebraic varieties kontsevich, maxim 20041007 00. A special case of this, which has been known for a long, is the notion of oneparameter deformation. Kontsevich, deformation quantization of algebraic varieties, lett. Notes on deformation quantization contents deformation theory.