Investigations in Calabi-Yau modularity and mirror symmetry
<p>This thesis lays out a number of different projects, linked by the common thread of Calabi-Yau manifolds.</p> <p>We produce tables of instanton numbers for various multiparameter Calabi-Yau manifolds. Studying those tables reveals, in some cases, a Coxeter group of symmetries t...
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| Other Authors: | |
| Format: | Thesis |
| Language: | English |
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2023
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| _version_ | 1797111587764436992 |
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| author | McGovern, J |
| author2 | Candelas, P |
| author_facet | Candelas, P McGovern, J |
| author_sort | McGovern, J |
| collection | OXFORD |
| description | <p>This thesis lays out a number of different projects, linked by the common thread of Calabi-Yau manifolds.</p>
<p>We produce tables of instanton numbers for various multiparameter Calabi-Yau manifolds. Studying those tables reveals, in some cases, a Coxeter group of symmetries that act on sets of instanton numbers. Instanton numbers are constant over the orbits of group actions on the curve classes, and these groups can be infinite.</p>
<p>We investigate to what extent this symmetry can be used to constrain the holomorphic ambiguity that arises in higher genus topological string free energy computations. In the example we study, at genus 2, combining this Coxeter constraint with the set of vanishing numbers fixes the holomorphic ambiguity.</p>
<p>A new class of solutions is provided to the supersymmetric flux vacuum equations, which have been conjectured elsewhere to give weight-two modular manifolds. Another search, complementary to those that have already been carried out, is made for rank-two attractors in the AESZ list. Two novel examples are found, both of which belong to moduli spaces with two points of maximal unipotent monodromy each. For one of these operators, the additional MUM point corresponds to another derived equivalent geometry. For the other operator, we compute nonintegral values of the triple intersection number and genus 0 instanton numbers, and so a geometric interpretation is lacking for the second MUM point.</p>
<p>We provide several instances of summation identities that express a ratio of critical L-values as an infinite sum, whose terms contain the Gromov-Witten invariants of a Calabi-Yau manifold Y . In one example there is no manifold Y, only an operator, but a set of invariants can nonetheless be computed for this and a summation identity is found. The L-functions are associated to a modular manifold X which is mirror to Y . These sums can be divergent, but Pade resummation cures this.</p>
<p>In addition to these results, we briefly review informing aspects of Calabi-Yau geometry, black holes in 4d <em>N</em> = 2 supergravity, flux compactifications, topological string theory, and number theory.</p> |
| first_indexed | 2024-03-07T08:10:56Z |
| format | Thesis |
| id | oxford-uuid:509cf718-3cac-4ed1-a2d1-d5c5a94f88aa |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:10:56Z |
| publishDate | 2023 |
| record_format | dspace |
| spelling | oxford-uuid:509cf718-3cac-4ed1-a2d1-d5c5a94f88aa2023-11-30T16:35:29ZInvestigations in Calabi-Yau modularity and mirror symmetryThesishttp://purl.org/coar/resource_type/c_db06uuid:509cf718-3cac-4ed1-a2d1-d5c5a94f88aaMathematical physicsEnglishHyrax Deposit2023McGovern, JCandelas, Pde la Ossa, XKuusela , PPanzer, EKlemm, A<p>This thesis lays out a number of different projects, linked by the common thread of Calabi-Yau manifolds.</p> <p>We produce tables of instanton numbers for various multiparameter Calabi-Yau manifolds. Studying those tables reveals, in some cases, a Coxeter group of symmetries that act on sets of instanton numbers. Instanton numbers are constant over the orbits of group actions on the curve classes, and these groups can be infinite.</p> <p>We investigate to what extent this symmetry can be used to constrain the holomorphic ambiguity that arises in higher genus topological string free energy computations. In the example we study, at genus 2, combining this Coxeter constraint with the set of vanishing numbers fixes the holomorphic ambiguity.</p> <p>A new class of solutions is provided to the supersymmetric flux vacuum equations, which have been conjectured elsewhere to give weight-two modular manifolds. Another search, complementary to those that have already been carried out, is made for rank-two attractors in the AESZ list. Two novel examples are found, both of which belong to moduli spaces with two points of maximal unipotent monodromy each. For one of these operators, the additional MUM point corresponds to another derived equivalent geometry. For the other operator, we compute nonintegral values of the triple intersection number and genus 0 instanton numbers, and so a geometric interpretation is lacking for the second MUM point.</p> <p>We provide several instances of summation identities that express a ratio of critical L-values as an infinite sum, whose terms contain the Gromov-Witten invariants of a Calabi-Yau manifold Y . In one example there is no manifold Y, only an operator, but a set of invariants can nonetheless be computed for this and a summation identity is found. The L-functions are associated to a modular manifold X which is mirror to Y . These sums can be divergent, but Pade resummation cures this.</p> <p>In addition to these results, we briefly review informing aspects of Calabi-Yau geometry, black holes in 4d <em>N</em> = 2 supergravity, flux compactifications, topological string theory, and number theory.</p> |
| spellingShingle | Mathematical physics McGovern, J Investigations in Calabi-Yau modularity and mirror symmetry |
| title | Investigations in Calabi-Yau modularity and mirror symmetry |
| title_full | Investigations in Calabi-Yau modularity and mirror symmetry |
| title_fullStr | Investigations in Calabi-Yau modularity and mirror symmetry |
| title_full_unstemmed | Investigations in Calabi-Yau modularity and mirror symmetry |
| title_short | Investigations in Calabi-Yau modularity and mirror symmetry |
| title_sort | investigations in calabi yau modularity and mirror symmetry |
| topic | Mathematical physics |
| work_keys_str_mv | AT mcgovernj investigationsincalabiyaumodularityandmirrorsymmetry |