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Lambda: A Mathematica-package for operator product expansions in vertex algebras
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy. (Teoretisk fysik)
2011 (English)In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 182, no 2, 409-418 p.Article in journal (Refereed) Published
Abstract [en]

We give an introduction to the Mathematica package Lambda, designed for calculating λ-brackets in both vertex algebras, and in SUSY vertex algebras. This is equivalent to calculating operator product expansions in two-dimensional conformal field theory. The syntax of λ-brackets is reviewed, and some simple examples are shown, both in component notation, and in N = 1 superfield notation.

Place, publisher, year, edition, pages
2011. Vol. 182, no 2, 409-418 p.
Keyword [en]
Conformal field theory, Lambda-bracket, Mathematica, Operator product expansions, Supersymmetry, Symbolic computation, Vertex algebra
National Category
Other Physics Topics
URN: urn:nbn:se:uu:diva-125406DOI: 10.1016/j.cpc.2010.09.018ISI: 000285661600013OAI: oai:DiVA.org:uu-125406DiVA: diva2:319546
Available from: 2010-05-18 Created: 2010-05-18 Last updated: 2011-11-09Bibliographically 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.
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 867
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
urn:nbn:se:uu:diva-159918 (URN)978-91-554-8185-8 (ISBN)
Public defence
2011-11-25, Polhemsalen, Ångströmlaboratoriet, Uppsala, 10:15 (English)
Available from: 2011-11-02 Created: 2011-10-11 Last updated: 2011-11-10Bibliographically approved

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Ekstrand, Joel
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