Chiralization of star-products and the isomorphisms of deformations and quantizations of Kleinian singularities

Castellan, Simone (2024) Chiralization of star-products and the isomorphisms of deformations and quantizations of Kleinian singularities. PhD thesis, University of Glasgow.

Abstract

This thesis contains two directions of research, both related to the quantizations and deformations of Poisson structures.

In the first part, we study the chiralization of star-products, a problem related to the quantization of Poisson vertex algebras. For a Poisson algebra A, a star product is a new product ⋆ such that the associative algebra A = (A, ⋆) is a quantization of A. Famous examples are the Moyal-Weyl and Gutt star-products. Poisson vertex algebras are the chiralization of Poisson algebras and their quantizations are vertex algebras, which are the chiralization of associative algebras. A star-deformation of a Poisson vertex algebra V is a vertex algebra structure Y⋆ such that the vertex algebra V = (V, Y⋆) is a quantization of V. Star-deformations can thus be seen as the chiral analogue of a star-product, although they are not compatible with the Zhu functors. To solve this, we introduce and study the algebraic structure of ℏ-deformed vertex algebras, which is closely related to the Zhu functor. A chiral star-product is then defined as a deformation of a Poisson vertex algebra into an ℏ-vertex algebra. We show that chiral star-products commute with the Zhu functor, giving back a star-product on the corresponding Poisson algebra. By putting ℏ = 0, we obtain a star-deformation. We study the problem of constructing chiral star-products and we provide explicit formulae in some important examples, including when V is a free-field vertex algebra, the affine vertex algebra, or the Virasoro vertex algebra. In particular, these formulae give the chiralization of the Moyal-Weyl and Gutt star-products. Additionally, we provide a new, more natural proof of the associativity of the Zhu algebra using the formalism of ℏ-vertex algebras.

In the second part, we deal with the algebra of functions on Kleinian singularities. It is known that, in this case, the parameter space of filtered Poisson deformations and the parameter space of non-commutative quantizations coincide. We consider all possible isomorphisms between the various deformations (as Poisson algebras) and all isomorphisms between the quantizations (as associative algebras); these form two groupoids, which we denote PIso and Iso. We prove that, for a Kleinian singularity of type A or D, the groupoids Iso and PIso are isomorphic. In particular, the group of automorphisms of the deformation and the quantization corresponding to the same deformation parameter are isomorphic. Furthermore, we describe the groups of automorphisms as abstract groups: for type A they have an amalgamated free product structure, for type D they are subgroups of the group of Dynkin diagram automorphisms. For type D we additionally compute all the possible isomorphisms between deformations as affine varieties.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the Engineering and Physical Sciences Research Council (EPSRC).
• Subjects: H Social Sciences > HA Statistics; Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: Engineering and Physical Sciences Research Council (EPSRC)
• Supervisor's Name: Bellamy, Professor Gwyn and Valeri, Dr. Daniele
• Date of Award: 2024
• Depositing User: Theses Team
• Unique ID: glathesis:2024-84702
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 15 Nov 2024 12:20
• Last Modified: 15 Nov 2024 14:18
• Thesis DOI: 10.5525/gla.thesis.84702
• URI: https://theses.gla.ac.uk/id/eprint/84702