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Sheaves of N=2 supersymmetric vertex algebras on Poisson manifolds
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
IMPA.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
2012 (English)In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 62, no 11, 2259-2278 p.Article in journal (Refereed) Published
Abstract [en]

We construct a sheaf of N = 2 vertex algebras naturally associated to any Poisson manifold. The relation of this sheaf to the chiral de Rham complex is discussed. We reprove the result about the existence of two commuting N = 2 superconformal structures on the space of sections of the chiral de Rham complex of a Calabi-Yau manifold, but now calculated in a manifest N = 2 formalism. We discuss how the semi-classical limit of this sheaf of N = 2 vertex algebras is related to the classical supersymmetric non-linear sigma model. 

Place, publisher, year, edition, pages
2012. Vol. 62, no 11, 2259-2278 p.
Keyword [en]
Vertex algebra, SUSY vertex algebra, Poisson vertex algebra, Poisson geometry
National Category
Other Physics Topics
Research subject
Theoretical Physics
Identifiers
URN: urn:nbn:se:uu:diva-159002DOI: 10.1016/j.geomphys.2012.07.003ISI: 000309083500012OAI: oai:DiVA.org:uu-159002DiVA: diva2:442084
Funder
Swedish Research Council, 621-2008-4273
Available from: 2011-09-20 Created: 2011-09-20 Last updated: 2017-12-08Bibliographically approved
In thesis
1. Going Round in Circles: From Sigma Models to Vertex Algebras and Back
Open this publication in new window or tab >>Going Round in Circles: From Sigma Models to Vertex Algebras and Back
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Gå runt i cirklar : Från sigmamodeller till vertexalgebror och tillbaka.
Abstract [en]

In this thesis, we investigate sigma models and algebraic structures emerging from a Hamiltonian description of their dynamics, both in a classical and in a quantum setup. More specifically, we derive the phase space structures together with the Hamiltonians for the bosonic two-dimensional non-linear sigma model, and also for the N=1 and N=2 supersymmetric models.

A convenient framework for describing these structures are Lie conformal algebras and Poisson vertex algebras. We review these concepts, and show that a Lie conformal algebra gives a weak Courant–Dorfman algebra. We further show that a Poisson vertex algebra generated by fields of conformal weight one and zero are in a one-to-one relationship with Courant–Dorfman algebras.

Vertex algebras are shown to be appropriate for describing the quantum dynamics of supersymmetric sigma models. We give two definitions of a vertex algebra, and we show that these definitions are equivalent. The second definition is given in terms of a λ-bracket and a normal ordered product, which makes computations straightforward. We also review the manifestly supersymmetric N=1 SUSY vertex algebra.

We also construct sheaves of N=1 and N=2 vertex algebras. We are specifically interested in the sheaf of N=1 vertex algebras referred to as the chiral de Rham complex. We argue that this sheaf can be interpreted as a formal quantization of the N=1 supersymmetric non-linear sigma model. We review different algebras of the chiral de Rham complex that one can associate to different manifolds. In particular, we investigate the case when the manifold is a six-dimensional Calabi–Yau manifold. The chiral de Rham complex then carries two commuting copies of the N=2 superconformal algebra with central charge c=9, as well as the Odake algebra, associated to the holomorphic volume form.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2011. i-viii, 85 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 867
Keyword
Chiral de Rham complex, Conformal field theory, Poisson vertex algebra, Sigma model, String theory, Vertex algebra
National Category
Other Physics Topics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-159918 (URN)978-91-554-8185-8 (ISBN)
Public defence
2011-11-25, Polhemsalen, Ångströmlaboratoriet, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2011-11-02 Created: 2011-10-11 Last updated: 2011-11-10Bibliographically approved

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Ekstrand, JoelZabzine, Maxim

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