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Going Round in Circles: From Sigma Models to Vertex Algebras and BackPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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PrimeFaces.cw("AccordionPanel","widget_formSmash_responsibleOrgs",{id:"formSmash:responsibleOrgs",widgetVar:"widget_formSmash_responsibleOrgs",multiple:true}); 2011 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
##### Abstract [en]

##### 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 [en]

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: urn:nbn:se:uu:diva-159918ISBN: 978-91-554-8185-8 (print)OAI: oai:DiVA.org:uu-159918DiVA: diva2:447665
##### Public defence

2011-11-25, Polhemsalen, Ångströmlaboratoriet, Uppsala, 10:15 (English)
##### Opponent

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##### Supervisors

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt511",{id:"formSmash:j_idt511",widgetVar:"widget_formSmash_j_idt511",multiple:true});
#####

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Available from: 2011-11-02 Created: 2011-10-11 Last updated: 2011-11-10Bibliographically approved
##### List of papers

Gå runt i cirklar : Från sigmamodeller till vertexalgebror och tillbaka. (Swedish)

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.

1. Courant-like brackets and loop spaces$(function(){PrimeFaces.cw("OverlayPanel","overlay208658",{id:"formSmash:j_idt553:0:j_idt557",widgetVar:"overlay208658",target:"formSmash:j_idt553:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

2. Non-linear sigma models via the chiral de Rham complex$(function(){PrimeFaces.cw("OverlayPanel","overlay319544",{id:"formSmash:j_idt553:1:j_idt557",widgetVar:"overlay319544",target:"formSmash:j_idt553:1:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

3. Chiral de Rham complex on Riemannian manifolds and special holonomy$(function(){PrimeFaces.cw("OverlayPanel","overlay411667",{id:"formSmash:j_idt553:2:j_idt557",widgetVar:"overlay411667",target:"formSmash:j_idt553:2:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

4. Lambda: A Mathematica-package for operator product expansions in vertex algebras$(function(){PrimeFaces.cw("OverlayPanel","overlay319546",{id:"formSmash:j_idt553:3:j_idt557",widgetVar:"overlay319546",target:"formSmash:j_idt553:3:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

5. Sheaves of N=2 supersymmetric vertex algebras on Poisson manifolds$(function(){PrimeFaces.cw("OverlayPanel","overlay442084",{id:"formSmash:j_idt553:4:j_idt557",widgetVar:"overlay442084",target:"formSmash:j_idt553:4:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});