Werner's Measure on Self-Avoiding Loops and Representations of the Virasoro Algebra

Persistent Link:
http://hdl.handle.net/10150/577250
Title:
Werner's Measure on Self-Avoiding Loops and Representations of the Virasoro Algebra
Author:
Chávez, Ángel A.
Issue Date:
2015
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
Werner has proven the existence and essential uniqueness of a conformally invariant family of locally-finite measures on self-avoiding loops on Riemann surfaces. The measures can be thought of as self-avoiding loop analogues of Schramm-Loewner evolution with parameter κ=8/3. This family is determined by a single measure on (normalized) holomorphic univalent functions on the unit disk. We will devise an algorithm for calculating moments of their Taylor coefficients. And in special cases, we can present closed-form solutions. Essentially, our algorithm arises as a consequence of non-degeneracy for a newly-realized family of highest-weight representations of the Virasoro algebra (we provide an explicit isomorphism between these representations and those constructed by Kirillov and Yuriev). Moreover, our algorithm leads to an alternate proof of essential uniqueness of Werner's family, as first seen in the author's joint work with Douglas Pickrell. Kontsevich and Suhov have conjectured the existence and essential uniqueness of a one-parameter deformation of Werner's family to a family of measures having values in powers of determinant line bundles (the deformation parameter is given by the real parameter κ satisfying 0 ≤ κ ≤ 4). Benoist and Dubédat recently proved the existence part of this conjecture for κ=2. We will provide an outline of how the argument for the Werner case can be adapted to prove the uniqueness part of Kontsevich and Suhov's conjecture.
Type:
text; Electronic Dissertation
Keywords:
Mathematics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Pickrell, Doug

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleWerner's Measure on Self-Avoiding Loops and Representations of the Virasoro Algebraen_US
dc.creatorChávez, Ángel A.en
dc.contributor.authorChávez, Ángel A.en
dc.date.issued2015en
dc.publisherThe University of Arizona.en
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en
dc.description.abstractWerner has proven the existence and essential uniqueness of a conformally invariant family of locally-finite measures on self-avoiding loops on Riemann surfaces. The measures can be thought of as self-avoiding loop analogues of Schramm-Loewner evolution with parameter κ=8/3. This family is determined by a single measure on (normalized) holomorphic univalent functions on the unit disk. We will devise an algorithm for calculating moments of their Taylor coefficients. And in special cases, we can present closed-form solutions. Essentially, our algorithm arises as a consequence of non-degeneracy for a newly-realized family of highest-weight representations of the Virasoro algebra (we provide an explicit isomorphism between these representations and those constructed by Kirillov and Yuriev). Moreover, our algorithm leads to an alternate proof of essential uniqueness of Werner's family, as first seen in the author's joint work with Douglas Pickrell. Kontsevich and Suhov have conjectured the existence and essential uniqueness of a one-parameter deformation of Werner's family to a family of measures having values in powers of determinant line bundles (the deformation parameter is given by the real parameter κ satisfying 0 ≤ κ ≤ 4). Benoist and Dubédat recently proved the existence part of this conjecture for κ=2. We will provide an outline of how the argument for the Werner case can be adapted to prove the uniqueness part of Kontsevich and Suhov's conjecture.en
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectMathematicsen
thesis.degree.namePh.D.en
thesis.degree.leveldoctoralen
thesis.degree.disciplineGraduate Collegeen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Arizonaen
dc.contributor.advisorPickrell, Dougen
dc.contributor.committeememberPickrell, Dougen
dc.contributor.committeememberFaris, Williamen
dc.contributor.committeememberGlickenstein, Daviden
dc.contributor.committeememberKennedy, Tomen
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