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Chiral de Rham complex on Riemannian manifolds and special holonomy
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
Department of Mathematics, University of California, Berkeley.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
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.

Place, publisher, year, edition, pages
2013. Vol. 318, no 3, 575-613 p.
National Category
Other Physics Topics
Research subject
Physics and Astronomy specializing in Theoretical Physics
Identifiers
URN: urn:nbn:se:uu:diva-151925DOI: 10.1007/s00220-013-1659-4ISI: 000315738300001OAI: oai:DiVA.org:uu-151925DiVA: diva2:411667
Available from: 2011-04-19 Created: 2011-04-19 Last updated: 2017-12-11Bibliographically 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
2. Twisting and Gluing: On Topological Field Theories, Sigma Models and Vertex Algebras
Open this publication in new window or tab >>Twisting and Gluing: On Topological Field Theories, Sigma Models and Vertex Algebras
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two parts, which can be read separately. In the first part we study aspects of topological field theories. We show how to topologically twist three-dimensional N=2 supersymmetric Chern-Simons theory using a contact structure on the underlying manifold. This gives us a formulation of Chern-Simons theory together with a set of auxiliary fields and an odd symmetry. For Seifert manifolds, we show how to use this odd symmetry to localize the path integral of Chern-Simons theory. The formulation of three-dimensional Chern-Simons theory using a contact structure admits natural generalizations to higher dimensions. We introduce and study these theories. The focus is on the five-dimensional theory, which can be understood as a topologically twisted version of N=1 supersymmetric Yang-Mills theory. When formulated on contact manifolds that are circle fibrations over a symplectic manifold, it localizes to contact instantons. For the theory on the five-sphere, we show that the perturbative part of the partition function is given by a matrix model.

In the second part of the thesis, we study supersymmetric sigma models in the Hamiltonian formalism, both in a classical and in a quantum mechanical setup. We argue that the so called Chiral de Rham complex, which is a sheaf of vertex algebras, is a natural framework to understand quantum aspects of supersymmetric sigma models in the Hamiltonian formalism. We show how a class of currents which generate symmetry algebras for the classical sigma model can be defined within the Chiral de Rham complex framework, and for a six-dimensional Calabi-Yau manifold we calculate the equal-time commutators between the currents and show that they generate the Odake algebra.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2012. 95 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 938
Keyword
Topological field theory, Chern-Simons theory, Contact geometry, Sigma models, Poisson vertex algebras, Vertex algebras, String theory
National Category
Other Physics Topics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-173225 (URN)978-91-554-8379-1 (ISBN)
Public defence
2012-06-08, Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
Opponent
Supervisors
Available from: 2012-05-16 Created: 2012-04-20 Last updated: 2012-08-01Bibliographically approved

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Ekstrand, JoelZabzine, Maxim

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