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Three Results in Low-Dimensional Gau...

Mickler, Ryan.

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Three Results in Low-Dimensional Gauge Theories.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Three Results in Low-Dimensional Gauge Theories./
作者:	Mickler, Ryan.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2019,
面頁冊數:	117 p.
附註:	Source: Dissertations Abstracts International, Volume: 80-11, Section: B.
Contained By:	Dissertations Abstracts International80-11B.
標題:	Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=13862455
ISBN:	9781392112397

Three Results in Low-Dimensional Gauge Theories.
Mickler, Ryan.

Three Results in Low-Dimensional Gauge Theories. - Ann Arbor : ProQuest Dissertations & Theses, 2019 - 117 p.

Source: Dissertations Abstracts International, Volume: 80-11, Section: B.

Thesis (Ph.D.)--Northeastern University, 2019.

This item must not be sold to any third party vendors.

In this thesis we present three results that all concern properties of gauge theories in two and three dimensions. In Chapter 2, we consider Chern-Simons theory on a 3-manifold M that is the total space of a circle bundle over a 2d base Σ. We show that this theory is equivalent to a new 2d TQFT on the base, which we call Caloron BF theory, that can be obtained by an appropriate type of push-forward. This is a gauge theory on a bundle with structure group given by the full affine level k central extension of the loop group LG. The space of fields of this 2d theory is naturally symplectic, and this provides a new formulation of a result of Beasley-Witten about the equivariant localization of the Chern-Simons path integral. The main tool that we employ is the Caloron correspondence, originally due to Murray-Garland, that relates the space of gauge fields on M with a certain enlarged space of connections on an equivariant version of the loop space of the G-bundle. We show that the symplectic structure that Beasley-Witten found is related to a looped version of the Atiyah-Bott construction in 2-dimensional Yang-Mills theory. We also show that Wilson loops that wrap a single circle fiber are also described very naturally in this framework. In Chapter 3, we introduce a new class of supersymmetric Wilson loops in two dimensional gauge-theories with N=(0,2) supersymmetry. An essential component of our construction is the existence of an anomalous global symmetry in the theory. We derive a version of the chiral anomaly equation in N=(0,2) superspace. Using the supersymmetric anomaly equation, we show that these new loop operators preserve a single supercharge. By encircling local operators, these loop operators act as endomorphisms of the associated chiral algebra. As a special case, we study these 1/4-BPS line operators in the N=(2,2) supersymmetric gauged linear sigma model. For the CPN model, we show that that Wilson linear are described semi-classically as supersymmetric domain walls interpolating between the (N+1) massive vacua, and these Wilson lines act by automorphisms on the quantum cohomology ring. In Chapter 4, we summarize our proof of a conjecture of Mathai-Wu concerning the analytic torsion for elliptic complexes that are graded by Z 2. Motivated by topological T-duality, Mathai-Wu studied the complex of forms on an odd-dimensional manifold equipped with the twisted differential dH = d+H, where H is a closed odd-dimensional form. We show that the Ray-Singer metric on this twisted determinant line is equal to the untwisted Ray-Singer metric when the determinant lines are identified using a canonical isomorphism. We also study another analytical invariant of the twisted differential, the derived Euler characteristic χ'(dH), as defined by Bismut and Zhang.

ISBN: 9781392112397Subjects--Topical Terms:

515831
Mathematics.

Three Results in Low-Dimensional Gauge Theories.
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520 $a In this thesis we present three results that all concern properties of gauge theories in two and three dimensions. In Chapter 2, we consider Chern-Simons theory on a 3-manifold M that is the total space of a circle bundle over a 2d base Σ. We show that this theory is equivalent to a new 2d TQFT on the base, which we call Caloron BF theory, that can be obtained by an appropriate type of push-forward. This is a gauge theory on a bundle with structure group given by the full affine level k central extension of the loop group LG. The space of fields of this 2d theory is naturally symplectic, and this provides a new formulation of a result of Beasley-Witten about the equivariant localization of the Chern-Simons path integral. The main tool that we employ is the Caloron correspondence, originally due to Murray-Garland, that relates the space of gauge fields on M with a certain enlarged space of connections on an equivariant version of the loop space of the G-bundle. We show that the symplectic structure that Beasley-Witten found is related to a looped version of the Atiyah-Bott construction in 2-dimensional Yang-Mills theory. We also show that Wilson loops that wrap a single circle fiber are also described very naturally in this framework. In Chapter 3, we introduce a new class of supersymmetric Wilson loops in two dimensional gauge-theories with N=(0,2) supersymmetry. An essential component of our construction is the existence of an anomalous global symmetry in the theory. We derive a version of the chiral anomaly equation in N=(0,2) superspace. Using the supersymmetric anomaly equation, we show that these new loop operators preserve a single supercharge. By encircling local operators, these loop operators act as endomorphisms of the associated chiral algebra. As a special case, we study these 1/4-BPS line operators in the N=(2,2) supersymmetric gauged linear sigma model. For the CPN model, we show that that Wilson linear are described semi-classically as supersymmetric domain walls interpolating between the (N+1) massive vacua, and these Wilson lines act by automorphisms on the quantum cohomology ring. In Chapter 4, we summarize our proof of a conjecture of Mathai-Wu concerning the analytic torsion for elliptic complexes that are graded by Z 2. Motivated by topological T-duality, Mathai-Wu studied the complex of forms on an odd-dimensional manifold equipped with the twisted differential dH = d+H, where H is a closed odd-dimensional form. We show that the Ray-Singer metric on this twisted determinant line is equal to the untwisted Ray-Singer metric when the determinant lines are identified using a canonical isomorphism. We also study another analytical invariant of the twisted differential, the derived Euler characteristic χ'(dH), as defined by Bismut and Zhang.
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