Short star products in (Poisson) vertex algebras
Lead Research Organisation:
University of Glasgow
Department Name: School of Mathematics & Statistics
Abstract
Short star products were recently introduced by Etingof and Stryker, following works form physicists Beem, Peelaers and Rastelli in (super)conformal field theory, for filtered quantisations of Poisson algebras. The goal of the project is to extend this notion to the (Poisson) vertex algebra setting. Interesting applications in representation theory.
Organisations
People |
ORCID iD |
| Simone Castellan (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/R513222/1 | 01/10/2018 | 30/09/2023 | |||
| 2469548 | Studentship | EP/R513222/1 | 01/10/2020 | 01/10/2024 | Simone Castellan |
| EP/T517896/1 | 01/10/2020 | 30/09/2025 | |||
| 2469548 | Studentship | EP/T517896/1 | 01/10/2020 | 01/10/2024 | Simone Castellan |
| Description | The goal of the research project was to generalise the notion of a star product, which gives a Poisson algebra the structure of an associative algebra that is a quantization of the original algebra, to the context of Poisson vertex algebras. Poisson vertex algebras are know to have vertex algebras as quantizations, so the setting shares many similarities. I managed to compute some close formulae in some classes of interesting examples, such as free boson, affine and Virasoro vertex algebras. I constructed the formulae for two types of star product, associated to different quantizations. The former uses the quantization to the normal ordered product, the latter uses the Zhu (-1)-product. This is of interest because it works well with the Zhu quotient. The Zhu quotient associates to any Poisson vertex algebra a classical Poisson algebra, and to any Vertex algebra an associative algebra, giving us back the classical framework. Seen in the quotient, these star product reduces to classical star product for Poisson algebras. |
| Exploitation Route | These formulae could be useful to make computations easier, and help further research in the field of vertex algebras. Vertex algebras are known to be computationally very hard, so being able to explicitly relate them to Poisson vertex algebras (which are significantly easier) could be a big help. There are also possible applications to physics. Classical star product (like the Moyal-Weyl product) where first introduced by physicists for the so called "phace space formulation of quantum mechanics". From a mathematical point of view, this mean being able to work in a nicer Poisson algebra rather than in a non commutative associative algebra. Vertex algebras are one of the various mathematical tools being developed to better understand quantum field theory, which is still not mathematical precise. Given the mathematical analogy to classical star product, this could lead to similar applications in quantum field theory. |
| Sectors | Other |