subject:(Jacobi form)

University of Arizona

1. Quinones, Jason. Mathematical Aspects of Field Theory: Nahm's Equations and Jacobi Forms .

Degree: 2020, University of Arizona

► This thesis consists of two projects wherein we explore some mathematical aspects of field theory. In the first project, we address Nahm's equations, which is… (more)

▼ This thesis consists of two projects wherein we explore some mathematical aspects of field theory. In the first project, we address Nahm's equations, which is an integrable system with a Lax pair. We consider boundary conditions on Nahm's equations that correspond to the Dirac multimonopole in Yang-Mills theory. The algebro-geometric integration method is to construct solutions via a linear flow in the Jacobian of the spectral curve associated to the Lax pair. We construct a frame of sections of this linear flow, which allows us to obtain exact solutions to Nahm's equations for arbitrary rank n. Nahm's equations with our boundary conditions correspond to the Dirac multimonopole via the ADHMN construction. The ADHMN construction requires us to find normalizable zero modes of Dirac operators. We again use the frame of sections of the linear flow on the Jacobian of the spectral curve to construct these normalizable zero modes. In the second project, we consider weak Jacobi forms of weight 0. The polar coefficients of such weak Jacobi form are known to uniquely determine the weak Jacobi form, and we improve on the number of polar coefficients that determine the weight 0 form. Weak Jacobi forms of weight 0 may be exponentially lifted to Siegel modular forms, which appear in the string-theory of black holes. In connection to this, the growth of a certain sum about a term q^a y^b in the Fourier-Jacobi expansion of the underlying weak Jacobi form is of interest to us. We discover that the weak Jacobi forms which are quotients of theta functions give us a large class of forms that are slow growing about their most polar term. Additionally, the characteristics of growth behavior for a weak Jacobi form about a term y^b are known, here we investigate growth behavior about an arbitrary q^a y^b term and find several analogues. Advisors/Committee Members: Keller, Christoph A (advisor), Glickenstein, David A. (committeemember), Cherkis, Sergey (committeemember), Ercolani, Nicholas (committeemember).

Subjects/Keywords: anti-self duality; holography; Jacobi form; monopole; Nahm transform; Nahm's equations

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Quinones, J. (2020). Mathematical Aspects of Field Theory: Nahm's Equations and Jacobi Forms . (Doctoral Dissertation). University of Arizona. Retrieved from http://hdl.handle.net/10150/645779

Chicago Manual of Style (16^th Edition):

Quinones, Jason. “Mathematical Aspects of Field Theory: Nahm's Equations and Jacobi Forms .” 2020. Doctoral Dissertation, University of Arizona. Accessed March 07, 2021. http://hdl.handle.net/10150/645779.

MLA Handbook (7^th Edition):

Quinones, Jason. “Mathematical Aspects of Field Theory: Nahm's Equations and Jacobi Forms .” 2020. Web. 07 Mar 2021.

Vancouver:

Quinones J. Mathematical Aspects of Field Theory: Nahm's Equations and Jacobi Forms . [Internet] [Doctoral dissertation]. University of Arizona; 2020. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/10150/645779.

Council of Science Editors:

Quinones J. Mathematical Aspects of Field Theory: Nahm's Equations and Jacobi Forms . [Doctoral Dissertation]. University of Arizona; 2020. Available from: http://hdl.handle.net/10150/645779

Kyoto University / 京都大学

2. Ren-He, Su. The Kohnen plus space for Hilbert-Siegel modular forms : ヒルベルトジーゲルモジュラー形式に関するコーネンプラス空間.

Degree: 博士(理学), 2016, Kyoto University / 京都大学

URL: http://hdl.handle.net/2433/215374 ; http://dx.doi.org/10.14989/doctor.k19548

新制・課程博士

甲第19548号

理博第4208号

Subjects/Keywords: automorphic form; Hilbert-Siegel modular form; Kohnen plus space; Jacobi form; Weil representation

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Ren-He, S. (2016). The Kohnen plus space for Hilbert-Siegel modular forms : ヒルベルトジーゲルモジュラー形式に関するコーネンプラス空間. (Thesis). Kyoto University / 京都大学. Retrieved from http://hdl.handle.net/2433/215374 ; http://dx.doi.org/10.14989/doctor.k19548

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Ren-He, Su. “The Kohnen plus space for Hilbert-Siegel modular forms : ヒルベルトジーゲルモジュラー形式に関するコーネンプラス空間.” 2016. Thesis, Kyoto University / 京都大学. Accessed March 07, 2021. http://hdl.handle.net/2433/215374 ; http://dx.doi.org/10.14989/doctor.k19548.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Ren-He, Su. “The Kohnen plus space for Hilbert-Siegel modular forms : ヒルベルトジーゲルモジュラー形式に関するコーネンプラス空間.” 2016. Web. 07 Mar 2021.

Vancouver:

Ren-He S. The Kohnen plus space for Hilbert-Siegel modular forms : ヒルベルトジーゲルモジュラー形式に関するコーネンプラス空間. [Internet] [Thesis]. Kyoto University / 京都大学; 2016. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/2433/215374 ; http://dx.doi.org/10.14989/doctor.k19548.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Ren-He S. The Kohnen plus space for Hilbert-Siegel modular forms : ヒルベルトジーゲルモジュラー形式に関するコーネンプラス空間. [Thesis]. Kyoto University / 京都大学; 2016. Available from: http://hdl.handle.net/2433/215374 ; http://dx.doi.org/10.14989/doctor.k19548

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Kyoto University

3. Ren-He, Su. The Kohnen plus space for Hilbert-Siegel modular forms .

Degree: 2016, Kyoto University

URL: http://hdl.handle.net/2433/215374

Subjects/Keywords: automorphic form; Hilbert-Siegel modular form; Kohnen plus space; Jacobi form; Weil representation

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Ren-He, S. (2016). The Kohnen plus space for Hilbert-Siegel modular forms . (Thesis). Kyoto University. Retrieved from http://hdl.handle.net/2433/215374

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Ren-He, Su. “The Kohnen plus space for Hilbert-Siegel modular forms .” 2016. Thesis, Kyoto University. Accessed March 07, 2021. http://hdl.handle.net/2433/215374.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Ren-He, Su. “The Kohnen plus space for Hilbert-Siegel modular forms .” 2016. Web. 07 Mar 2021.

Vancouver:

Ren-He S. The Kohnen plus space for Hilbert-Siegel modular forms . [Internet] [Thesis]. Kyoto University; 2016. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/2433/215374.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Ren-He S. The Kohnen plus space for Hilbert-Siegel modular forms . [Thesis]. Kyoto University; 2016. Available from: http://hdl.handle.net/2433/215374

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ)

4. Θεοφανίδης, Θεοχάρης. Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής.

Degree: 2011, Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ)

URL: http://hdl.handle.net/10442/hedi/27049

► J. de Dios Perez, F. G. Santos and Y. J. Suh in [29], studied real hypersurfaces of dimension greater than 3 in complex projective spaces,… (more)

Subjects/Keywords: Διαφορική γεωμετρία; Πολλαπλότητα Riemann; Μιγαδικός χώρος μορφής; Πραγματική υπερεπιφάνεια; Δομή σχεδόν επαφής; Τελεστής δομής Jacobi; Differential geometry; Riemannian manifolds; Complex space form; Real hypersurface; Almost contact structure; Jacobi structure operator

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Θεοφανίδης, . �. (2011). Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής. (Thesis). Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ). Retrieved from http://hdl.handle.net/10442/hedi/27049

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^th Edition):

Θεοφανίδης, Θεοχάρης. “Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής.” 2011. Thesis, Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ). Accessed March 07, 2021. http://hdl.handle.net/10442/hedi/27049.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^th Edition):

Θεοφανίδης, Θεοχάρης. “Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής.” 2011. Web. 07 Mar 2021.

Vancouver:

Θεοφανίδης �. Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής. [Internet] [Thesis]. Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ); 2011. [cited 2021 Mar 07]. Available from: http://hdl.handle.net/10442/hedi/27049.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Θεοφανίδης �. Μελέτη πραγματικών υπερεπιφανειών μη ευκλείδειων μιγαδικών χώρων μορφής. [Thesis]. Aristotle University Of Thessaloniki (AUTH); Αριστοτέλειο Πανεπιστήμιο Θεσσαλονίκης (ΑΠΘ); 2011. Available from: http://hdl.handle.net/10442/hedi/27049

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Universiteit Utrecht

5. Zwegers, S.P. Mock Theta Functions.

Degree: 2002, Universiteit Utrecht

URL: http://dspace.library.uu.nl:8080/handle/1874/878

► The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death.… (more)

▼ The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death. In this dissertation, I consider several of the examples that Ramanujan gave of mock theta functions, and relate them to real-analytic modular forms of weight 1/2. In Chapter 1, I consider a sum, which was also studied by Lerch. This Lerch sum transforms almost as a Jacobi form under substitutions in (upsilon, nu, tau ). I show that the transformation behaviour becomes that of a Jacobi form if we add a (relatively simple) correction term. This correction term is real-analytic in (upsilon, nu, tau) but not holomorphic. For special values of (upsilon, nu), we could call the Lerch sum (considered as a function of tau ) a mock theta function, although these examples were not considered by Ramanujan. In Chapter 2, I consider theta functions for indefinite quadratic forms. These indefinite theta functions are modified versions of theta functions introduced by Göttsche and Zagier. I find elliptic and modular transformation properties of these functions, similar to the properties of theta functions associated to positive definite quadratic forms. In the case of positive definite quadratic forms, the theta functions are holomorphic. The theta functions in the chapter are not holomorphic. By taking special values of certain parameters, we get most of the examples of mock theta functions given by Ramanujan. Andrews gave most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients. In Chapter 4, I use the results from Chapter 2 to get the modular transformation properties of the seventh-order mock v-functions and most of the fifth-order functions. The final result is that we can write each of these mock theta-functions as the sum of two functions ? and G, where: - ? is a real-analytic modular form of weight 1/2 and is an eigenfunction of the appropriate Casimir operator with eigenvalue 3/16 (this is also the eigenvalue of holomorphic modular forms of this weight); and - G is a theta series associated to a negative definite unary quadratic form. Moreover G is bounded as ? tends vertically to any rational limit. Many of the results of Chapter 4 could be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter 2. This means that I have actually given three approaches to proving modularity properties of the mock theta -functions.

Subjects/Keywords: Wiskunde en Informatica; mock theta function; indefinite theta function; indefinite quadratic form; theta series; Jacobi form; real-analytic modular form

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Zwegers, S. P. (2002). Mock Theta Functions. (Doctoral Dissertation). Universiteit Utrecht. Retrieved from http://dspace.library.uu.nl:8080/handle/1874/878

Chicago Manual of Style (16^th Edition):

Zwegers, S P. “Mock Theta Functions.” 2002. Doctoral Dissertation, Universiteit Utrecht. Accessed March 07, 2021. http://dspace.library.uu.nl:8080/handle/1874/878.

MLA Handbook (7^th Edition):

Zwegers, S P. “Mock Theta Functions.” 2002. Web. 07 Mar 2021.

Vancouver:

Zwegers SP. Mock Theta Functions. [Internet] [Doctoral dissertation]. Universiteit Utrecht; 2002. [cited 2021 Mar 07]. Available from: http://dspace.library.uu.nl:8080/handle/1874/878.

Council of Science Editors:

Zwegers SP. Mock Theta Functions. [Doctoral Dissertation]. Universiteit Utrecht; 2002. Available from: http://dspace.library.uu.nl:8080/handle/1874/878

6. Zwegers, S.P. Mock Theta Functions.

Degree: 2002, University Utrecht

URL: https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; 1874/878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878

► The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death.… (more)

▼ The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death. In this dissertation, I consider several of the examples that Ramanujan gave of mock theta functions, and relate them to real-analytic modular forms of weight 1/2. In Chapter 1, I consider a sum, which was also studied by Lerch. This Lerch sum transforms almost as a Jacobi form under substitutions in (upsilon, nu, tau ). I show that the transformation behaviour becomes that of a Jacobi form if we add a (relatively simple) correction term. This correction term is real-analytic in (upsilon, nu, tau) but not holomorphic. For special values of (upsilon, nu), we could call the Lerch sum (considered as a function of tau ) a mock theta function, although these examples were not considered by Ramanujan. In Chapter 2, I consider theta functions for indefinite quadratic forms. These indefinite theta functions are modified versions of theta functions introduced by Göttsche and Zagier. I find elliptic and modular transformation properties of these functions, similar to the properties of theta functions associated to positive definite quadratic forms. In the case of positive definite quadratic forms, the theta functions are holomorphic. The theta functions in the chapter are not holomorphic. By taking special values of certain parameters, we get most of the examples of mock theta functions given by Ramanujan. Andrews gave most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients. In Chapter 4, I use the results from Chapter 2 to get the modular transformation properties of the seventh-order mock v-functions and most of the fifth-order functions. The final result is that we can write each of these mock theta-functions as the sum of two functions ? and G, where: - ? is a real-analytic modular form of weight 1/2 and is an eigenfunction of the appropriate Casimir operator with eigenvalue 3/16 (this is also the eigenvalue of holomorphic modular forms of this weight); and - G is a theta series associated to a negative definite unary quadratic form. Moreover G is bounded as ? tends vertically to any rational limit. Many of the results of Chapter 4 could be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter 2. This means that I have actually given three approaches to proving modularity properties of the mock theta -functions.

Subjects/Keywords: mock theta function; indefinite theta function; indefinite quadratic form; theta series; Jacobi form; real-analytic modular form

Record Details Similar Records Cite « Share

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Zwegers, S. P. (2002). Mock Theta Functions. (Doctoral Dissertation). University Utrecht. Retrieved from https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; 1874/878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878

Chicago Manual of Style (16^th Edition):

Zwegers, S P. “Mock Theta Functions.” 2002. Doctoral Dissertation, University Utrecht. Accessed March 07, 2021. https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; 1874/878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878.

MLA Handbook (7^th Edition):

Zwegers, S P. “Mock Theta Functions.” 2002. Web. 07 Mar 2021.

Vancouver:

Zwegers SP. Mock Theta Functions. [Internet] [Doctoral dissertation]. University Utrecht; 2002. [cited 2021 Mar 07]. Available from: https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; 1874/878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878.

Council of Science Editors:

Zwegers SP. Mock Theta Functions. [Doctoral Dissertation]. University Utrecht; 2002. Available from: https://dspace.library.uu.nl/handle/1874/878 ; URN:NBN:NL:UI:10-1874-878 ; 1874/878 ; URN:NBN:NL:UI:10-1874-878 ; https://dspace.library.uu.nl/handle/1874/878

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