Going Round in Circles: From Sigma Models to Vertex Algebras and Back
2011 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
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.
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
Other Physics Topics
Research subject Theoretical Physics
IdentifiersURN: urn:nbn:se:uu:diva-159918ISBN: 978-91-554-8185-8OAI: oai:DiVA.org:uu-159918DiVA: diva2:447665
2011-11-25, Polhemsalen, Ångströmlaboratoriet, Uppsala, 10:15 (English)
Alekseev, Anton, Professor
List of papers