Quantization on Curves: With an Appendix by Maxim Kontsevich

Authors: Frønsdal, Christian¹; Kontsevich, Maxim²

Source: Letters in Mathematical Physics, Volume 79, Number 2, February 2007 , pp. 109-129(21)

Publisher: Springer

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Abstract:

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. The Harrison component of Hochschild cohomology, vanishing on smooth manifolds, reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian *-products. This paper begins a study of abelian quantization on plane curves over <EquationSource Format="TEX"><![CDATA[$$mathbb{C}$$]]></EquationSource> , being algebraic varieties of the form <EquationSource Format="TEX"><![CDATA[$${mathbb{C}}^2/R$$]]></EquationSource> , where R is a polynomial in two variables; that is, abelian deformations of the coordinate algebra <EquationSource Format="TEX"><![CDATA[$$mathbb{C}[x,y]/(R$$]]></EquationSource> ). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co)homology and its Barr-Gerstenhaber-Schack decomposition. Homology is the same for all plane curves <EquationSource Format="TEX"><![CDATA[$$mathbb{C}[x,y]/R$$]]></EquationSource> , but the cohomology depends on the local algebra of the singularity of R at the origin. The Appendix, by Maxim Kontsevich, explains in modern mathematical language a way to calculate Hochschild and Harrison cohomology groups for algebras of functions on singular planar curves etc. based on Koszul resolutions.

Keywords: 53D55; 14A22; 16E40; 16S60; 81S10; quantization; deformation; Harrison cohomology; singular curves

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s11005-006-0137-8

Affiliations: 1: Email: fronsdal@physics.ucla.edu 2: Email: maxim@ihes.fr

Publication date: 2007-02-01