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Twisting and Gluing: On Topological Field Theories, Sigma Models and Vertex Algebras
Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Theoretical Physics.
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 [en]
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: urn:nbn:se:uu:diva-173225ISBN: 978-91-554-8379-1 (print)OAI: oai:DiVA.org:uu-173225DiVA: diva2:517246
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
List of papers
1. Cohomological localization of Chern-Simons theory
Open this publication in new window or tab >>Cohomological localization of Chern-Simons theory
2011 (English)In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, no 8, 8- p.Article in journal (Refereed) Published
Abstract [en]

We generalize the framework introduced by Kapustin et al. for doing path integral localization in Chern-Simons theory to work on any Seifert manifold. This is done by topologically twisting the supersymmetric theory considered by Kapustin et al., after which the theory takes a cohomological form. We also consider Wilson loops which wrap the fiber directions and compute their expectation values. We discuss the relation with other approaches to exact path integral calculations in Chern-Simons theory.

Keyword
Chern-Simons Theories, Topological Field Theories, Supersymmetric gauge theory
National Category
Physical Sciences
Identifiers
urn:nbn:se:uu:diva-160219 (URN)10.1007/JHEP08(2011)008 (DOI)000294901400074 ()
Available from: 2011-10-18 Created: 2011-10-18 Last updated: 2017-12-08Bibliographically approved
2. Twisted supersymmetric 5D Yang-Mills theory and contact geometry
Open this publication in new window or tab >>Twisted supersymmetric 5D Yang-Mills theory and contact geometry
2012 (English)In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, no 5, 125- p.Article in journal (Refereed) Published
Abstract [en]

We extend the localization calculation of the 3D Chern-Simons partition function over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N = 1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five-dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on S-5 for the twisted supersymmetric Yang-Mills theory with different Chern-Simons couplings. The final answer is given in terms of a matrix model. Our construction admits generalizations to higher dimensional contact manifolds. This work is inspired by the work of Baulieu-Losev-Nekrasov from the mid 90's, and in a way it is covariantization of their ideas for a contact manifold.

Keyword
Topological Field Theories, Chern-Simons Theories, Supersymmetric gauge theory
National Category
Other Physics Topics
Research subject
Theoretical Physics
Identifiers
urn:nbn:se:uu:diva-173223 (URN)10.1007/JHEP05(2012)125 (DOI)000305238600045 ()
Available from: 2012-04-20 Created: 2012-04-20 Last updated: 2017-12-07Bibliographically approved
3. 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
4. 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

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