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Going Round in Circles: From Sigma Models to Vertex Algebras and Back
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Gå runt i cirklar : Från sigmamodeller till vertexalgebror och tillbaka. (Swedish)
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 [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
Supervisors
Available from: 2011-11-02 Created: 2011-10-11 Last updated: 2011-11-10Bibliographically approved
List of papers
1. Courant-like brackets and loop spaces
Open this publication in new window or tab >>Courant-like brackets and loop spaces
2011 (English)In: Journal of High Energy Physics (JHEP), ISSN 1029-8479, E-ISSN 1126-6708, Vol. 2011, no 3, 074- p.Article in journal (Refereed) Published
Abstract [en]

We study the algebra of local functionals equipped with a Poisson bracket. We discuss the underlying algebraic structures related to a version of the Courant-Dorfman algebra. As a main illustration, we consider the functionals over the cotangent bundle of the superloop space over a smooth manifold. We present a number of examples of the Courant-like brackets arising from this analysis.

Keyword
Differential and Algebraic Geometry, Sigma Models, Field Theories in Lower Dimensions, Superspaces
National Category
Other Physics Topics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-99720 (URN)10.1007/JHEP03(2011)074 (DOI)000289295300002 ()UUITP-09/09 (Local ID)UUITP-09/09 (Archive number)UUITP-09/09 (OAI)
Available from: 2009-03-19 Created: 2009-03-18 Last updated: 2011-11-09Bibliographically approved
2. Non-linear sigma models via the chiral de Rham complex
Open this publication in new window or tab >>Non-linear sigma models via the chiral de Rham complex
2009 (English)In: Advances in Theoretical and Mathematical Physics, ISSN 1095-0761, E-ISSN 1095-0753, Vol. 13, no 4, 1221-1254 p.Article in journal (Refereed) Published
Abstract [en]

We propose a physical interpretation of the chiral de Rham complex as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model. We show that the chiral de Rham complex on a Calabi-Yau manifold carries all information about the classical dynamics of the sigma model. Physically, this provides an opera- tor realization of the non-linear sigma model. Mathematically, the idea suggests the use of Hamiltonian flow equations within the vertex algebra formalism with the pos- sibility to incorporate both left and right moving sectors within one mathematical framework.

National Category
Other Physics Topics
Identifiers
urn:nbn:se:uu:diva-125405 (URN)000279344400006 ()
Available from: 2010-05-18 Created: 2010-05-18 Last updated: 2017-12-12Bibliographically approved
3. Chiral de Rham complex on Riemannian manifolds and special holonomy
Open this publication in new window or tab >>Chiral de Rham complex on Riemannian manifolds and special holonomy
2013 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 318, no 3, 575-613 p.Article in journal (Refereed) Published
Abstract [en]

Interpreting the chiral de Rham complex (CDR) as a formal Hamiltonian quantization of the supersymmetric non-linear sigma model, we suggest a setup for the study of CDR on manifolds with special holonomy. We discuss classical and partial quantum results. As a concrete example, we construct two commuting copies of the Odake algebra (an extension of the N=2 superconformal algebra) on the space of global sections of CDR of a Calabi-Yau 3-fold. This is the first example of such a vertex subalgebra which is non-linearly generated by a finite number of superfields.

National Category
Other Physics Topics
Research subject
Physics and Astronomy specializing in Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-151925 (URN)10.1007/s00220-013-1659-4 (DOI)000315738300001 ()
Available from: 2011-04-19 Created: 2011-04-19 Last updated: 2017-12-11Bibliographically approved
4. Lambda: A Mathematica-package for operator product expansions in vertex algebras
Open this publication in new window or tab >>Lambda: A Mathematica-package for operator product expansions in vertex algebras
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.

Keyword
Conformal field theory, Lambda-bracket, Mathematica, Operator product expansions, Supersymmetry, Symbolic computation, Vertex algebra
National Category
Other Physics Topics
Identifiers
urn:nbn:se:uu:diva-125406 (URN)10.1016/j.cpc.2010.09.018 (DOI)000285661600013 ()
Available from: 2010-05-18 Created: 2010-05-18 Last updated: 2017-12-12Bibliographically approved
5. Sheaves of N=2 supersymmetric vertex algebras on Poisson manifolds
Open this publication in new window or tab >>Sheaves of N=2 supersymmetric vertex algebras on Poisson manifolds
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. 

Keyword
Vertex algebra, SUSY vertex algebra, Poisson vertex algebra, Poisson geometry
National Category
Other Physics Topics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-159002 (URN)10.1016/j.geomphys.2012.07.003 (DOI)000309083500012 ()
Funder
Swedish Research Council, 621-2008-4273
Available from: 2011-09-20 Created: 2011-09-20 Last updated: 2017-12-08Bibliographically approved

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