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Cohomological field theories and four-manifold invariants

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Title
Cohomological field theories and four-manifold invariants
Name (type = personal)
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Nidaiev
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Iurii
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1988-
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Iurii Nidaiev
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author
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Moore
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Gregory W
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Gregory W Moore
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Advisory Committee
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chair
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Buckley
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Matthew R
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Matthew R Buckley
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Advisory Committee
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internal member
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Friedan
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Daniel
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Daniel Friedan
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Advisory Committee
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internal member
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Yuzbashyan
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Emil
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Emil Yuzbashyan
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Advisory Committee
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internal member
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Marino
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Marcos
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Marcos Marino
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Advisory Committee
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outside member
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Rutgers University
Role
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degree grantor
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School of Graduate Studies
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school
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Text
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theses
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2019
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2019-05
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2019
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English
Abstract (type = abstract)
Four-dimensional cohomological quantum field theories possess topological sectors of correlation functions that can be analyzed non-perturbatively on a general four-manifold. In this thesis, we explore various aspects of these topological models and their implications for smooth structure invariants of four-manifolds.
Cohomological field theories emerge when one considers topological twisting of ordinary quantum field theories with extended (N = 2 in the context of this thesis) supersymmetry. The global scalar supersymmetry of these theories allows one to use integrals/sums over their quantum vacua as a tool for their exact analysis. In the case of pure SU(2) N = 2 gauge theory this has lead to remarkable success of Witten’s field theory formulation of Donaldson invariants and discovery of Seiberg-Witten invariants which are the best presently available tool for distinguishing smooth structures on four-manifolds with fixed topological type. In chapter 3 of this thesis we analyze a new prescription for defining the integral over a Coulomb branch of vacua in Donaldson-Witten theory as well as discuss possible treatment of IR divergences associated with certain BRST-exact operators. Chapter 3 of the thesis is based on the work reported in [20] (arXiv:1901.03540 [hep-th]) and partly has been extracted from that paper.
On general grounds one expects that topological twisting of any N = 2 supersymmetric theory defines a smooth structure invariant. However, examples of Lagrangian theories strongly suggest that topological partition functions of Lagrangian theories are expressible through the classical cohomological invariants and Seiberg-Witten invariants. Therefore, the search for new 4-manifold invariants has to be restricted to so-called "non-Lagrangian" N = 2 theories. Though full non-Lagrangian theories are, at present, difficult to analyze due to their strongly-coupled nature and the lack of action principle, in chapter 4 we show how one can derive the topological partition function of a simplest non-trivial non-Lagrangian theory discovered by Argyres and Douglas and known as AD3 theory. We obtain a formula for the partition function of topologically twisted version of the AD3 theory on any compact, oriented, simply connected, four-manifolds without boundary and with b+2 > 0. The result can be, once again, expressed in terms of classical cohomological invariants and Seiberg-Witten invariants. We argue that our results hint at the existence of four-manifolds of new, presently unknown, type as well as narrow the search for new field theory invariants of four-manifolds to Non-Lagrangian superconformal points that admit Higgs branches. Chapter 4 of this thesis is based on the work reported in [40] (arXiv:1711.09257 [hep-th]) and partly has been extracted from that paper.
Finally, in chapter 5 we derive a twisted (0,2) two-dimensional model by putting the abelian low energy theory of single M5 brane described by the PST action on a direct product of a Riemann surface and a four-manifold. The resulting two dimensional topological model can potentially be used as a tool refining the u-plane integral to study topologically twisted N = 2 theories of class S.
Subject (authority = RUETD)
Topic
Physics and Astronomy
Subject (authority = LCSH)
Topic
Quantum field theory
Subject (authority = LCSH)
Topic
Four-manifolds (Topology)
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Rutgers University Electronic Theses and Dissertations
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ETD_9612
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1 online resource (viii, 113 pages) : illustrations
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Ph.D.
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Includes bibliographical references
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School of Graduate Studies Electronic Theses and Dissertations
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rucore10001600001
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Identifier (type = doi)
doi:10.7282/t3-gye7-jv86
Genre (authority = ExL-Esploro)
ETD doctoral
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The author owns the copyright to this work.
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Name
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Nidaiev
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Iurii
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2019-03-27 10:13:24
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Iurii Nidaiev
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Rutgers University. School of Graduate Studies
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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