DOCSLIB.ORG

Generalized Modular Forms Representable As Eta Products

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generalized Modular Forms Representable As Eta Products Modular Forms Representable As Eta Products A Dissertation Submitted to the Temple University Graduate Board in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY by Wissam Raji August, 2006 iii c by Wissam Raji August, 2006 All Rights Reserved iv ABSTRACT Modular Forms Representable As Eta Products Wissam Raji DOCTOR OF PHILOSOPHY Temple University, August, 2006 Professor Marvin Knopp, Chair In this dissertation, we discuss modular forms that are representable as eta products and generalized eta products . Eta products appear in many areas of mathematics in which algebra and analysis overlap. M. Newman [15, 16] published a pair of well-known papers aimed at using eta-product to construct forms on the group Γ0(n) with the trivial multiplier system. Our work here divides into three related areas. The first builds upon the work of Siegel [23] and Rademacher [19] to derive modular transformation laws for functions defined as eta products (and related products). The second continues work of Kohnen and Mason [9] that shows that, under suitable conditions, a generalized modular form is an eta product or generalized eta product and thus a classical modular form. The third part of the dissertation applies generalized eta-products to rederive some arithmetic identities of H. Farkas [5, 6]. v ACKNOWLEDGEMENTS I would like to thank all those who have helped in the completion of my thesis. To God, the beginning and the end, for all His inspiration and help in the most difficult days of my life, and for all the people listed below. To my advisor, Professor Marvin Knopp, a great teacher and inspirer whose support and guidance were crucial for the completion of my work. He taught me how to be a good researcher and a passionate teacher. I am honored to be the student of such a great mathematician. To Professor Pavel Guerzhoy, who provided his ideas in the last two sections of my thesis and especially his key idea in the fourth chapter. His ideas and techniques helped me to attack several problems successfully. To Professor Boris Datskovsky, for serving on my examining committee, for being a great teacher, a wonderful problem solver who amazed me all the time by his great abilities. To Professor Omar Hijab, for serving on my examining committee. Dr. Hijab is a mathematician who taught me how to be passionate about the subject. His support for me all over the years was very helpful in developing my intuition. To Professor Wladimir Pribitkin, for serving on my examining committee, for being very helpful in the whole process of the defense. To Professor Sinai Robins, a number theorist at Temple University whose comments helped in the development of several theorems in my thesis, for his continuous support and encouragement throughout the years. To Jose Gimenez and Karen Taylor, my mathematical siblings, for their support and advice throughout my thesis. To my friend Marilena Downing, whose encouragement and friendship were valuable to me. To Professor Bassam Shayya, my thesis advisor at the American University of Beirut for his confidence that I will make it one day and for his continuous vi encouragement to take further and further steps. To Professor Kamal Khuri-Makdisi, a brilliant number theorist whose guid- ance and support throughout my graduate studies gave me the confidence to work harder. He was of great help in developing my mathematical intuition during my Master’s degree study at the American University of Beirut. To Nancy Abdel-Halim, my real love who stood all the years of my graduate work by my side. She helped me in taking great decisions in my life which will have great effects on my future. To my family, for their infinite love and support throughout the years of my life. In particular, to my Dad Victor, my Mom Nadia and my brothers George and Fadi. vii To my family, Victor, Nadia, George, Fadi and Nancy. With love and admiration viii TABLE OF CONTENTS ABSTRACT iv ACKNOWLEDGEMENT v DEDICATION vii 1 Introduction 1 1.1 Basic Definitions . 1 2 Transformation Laws Of Classes Of Functions 5 2.1 Transformation Law of Jacobi θ3(w, τ) ............. 5 2.1.1 The Jacobi function θ3(τ) . 11 2.1.2 The Function w(z, τ)................... 11 2.2 Transformation Laws of a Class of Eta Products . 13 2.2.1 The Transformation law of g1(τ) under Γ0(n) . 14 2.2.2 A Special Case of g1(τ).................. 20 2.2.3 Another Special Case of g1(τ) . 21 2.3 Comments on Generalizing the Proof of Section (2.1) . 24 3 Generalized Modular Forms Representable As Eta-Products 26 3.1 Introduction . 26 3.2 GMF’s on Γ0(N) Representable as Generalized Eta-Products . 29 3.3 GMF’s on Γ1(N) Representable as Eta-Products . 33 3.4 GMF’s on Γ(N) Representable as Eta-Products . 37 4 Arithmetic Identities 40 4.1 Introduction . 40 4.2 Arithmetic Identities Modulo 3 . 41 4.3 Arithmetic Identities modulo 7 . 43 4.4 Arithmetic Identities Modulo 4 . 45 REFERENCES 48 1 CHAPTER 1 Introduction 1.1 Basic Definitions By SL2(Z) we mean the group of 2x2 matrices with integral entries and determinant 1. We call SL2(Z) the modular group Γ(1). We define the action of an element A ∈ SL2(Z) on the upper half plane H by az + b Az = , cz + d ! a b where A = . For a positive integer N, we define a subgroup of Γ(1); c d ( ! ) a b Γ(N) = : a, b, c, d ∈ Z, b ≡ c ≡ 0 mod N, a ≡ d ≡ 1 mod N, ad − bc = 1 . c d We call this subgroup the principal congruence subgroup of level N. For any other subgroup Γ ⊂ Γ(1), if Γ(N) ⊂ Γ for some N ∈ Z, then we call Γ a congruence subgroup. Definition 1.1 Let Γ be a subgroup of Γ(1). A fundamental region for Γ is an open subset R of H such that 1. no two distinct points of R are equivalent with respect to Γ, and 2. every point of H is equivalent to some point in the closure of R. 2 ! 1 1 Proposition 1.1 The full modular group is generated by S = and 0 1 ! 0 −1 T = 1 0 Definition 1.2 We say that ν is a multiplier system for the group Γ and weight k provided ν(M), M ∈ Γ, is a complex-valued function of absolute value 1, satisfying the following equation k k k ν(M1M2)(c3τ + d3) = ν(M1)ν(M2)(c1M2τ + d1) (c2τ + d2) , ! ! ! ∗ ∗ ∗ ∗ ∗ ∗ where M1 = , M2 = and M3 = M1M2 = . c1 d1 c2 d2 c3 d3 Definition 1.3 Let R be a fundamental region of Γ. A parabolic point (or parabolic vertex or parabolic cusp) of Γ is any real point q, or q = ∞, such that q ∈ closure(R), in the topology of the Riemann sphere. Definition 1.4 Suppose Γ ⊂ Γ(1) such that [Γ(1) : Γ] = µ. Let A1,A2, ..., Aµ be a set of right coset representatives of Γ in Γ(1). The width of Γ at qj = ∞ is the smallest positive integer λ such that Sλ ∈ Γ. Also, the width of Γ at any other cusp qj = Aj(∞) is the smallest positive integer λ such that λ −1 AjS Aj ∈ Γ. Definition 1.5 Let k ∈ R and ν(M) a multiplier system for Γ and of weight k. A function F (τ) defined and meromorphic in H is a modular form (MF) of weight k, with multiplier system (MS) ν, with respect to Γ, provided 1. F (Mτ) = ν(M)(cτ + d)kF (τ) for every M ∈ Γ; 2. The Fourier expansion of F at every cusp qj has the form ∞ −1 X 2πi(n+κj )(A τ)/λj F (τ) = σj(τ) an(j)e j , n=−n0 3 where σj(τ) = 1 if qj = ∞ −k σj(τ) = (τ − qj) if qj is finite. Here λj is the width at qj and 0 ≤ κj < 1 is determined by ν(ASλj A−1) = e2πiκj , if qj is finite and ν(Sλj ) = e2πiκj , if qj is infinity. If the first nonzero an(j) occurs for n = −n0 < 0, we say F has a pole at qj of order n0 − κj. If the first nonzero an(j) occurs for n = n0 ≥ 0, we say F is regular at qj with a zero of order n0 + κj. To decide whether a given function is a modular form on Γ, it is essential to determine how this function transforms under the action of Γ. With respect to the full modular group, it will be enough to determine how the function transforms under the generators S and T . Usually, it is easier to see how the function transforms under the action of S. In Chapter 2 of this thesis, we de- termine the transformation law of θ3(w, τ) under the action of T using Siegel’s method [23]. Notice that θ3(w, τ) is not a modular form but θ3(0, τ) = θ3(τ) 1 is. We will see in the (2.1.1) that θ3(τ) is a modular form of weight 2 . We then generalize Siegel’s method to determine the transformation laws for an entire class of modular forms under Γ0(N). Here Γ0(N) is a congruence subgroup to be defined later. This class of functions is a product of eta functions with very important properties to be used in the later chapters. We then impose some conditions to derive a class of functions which is invariant under Γ0(N).
Load more

Recommended publications
  • Modular Dedekind Symbols Associated to Fuchsian Groups and Higher-Order Eisenstein Series
    Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series Jay Jorgenson,∗ Cormac O’Sullivan†and Lejla Smajlovi´c May 20, 2018 Abstract Let E(z,s) be the non-holomorphic Eisenstein series for the modular group SL(2, Z). The classical Kro- necker limit formula shows that the second term in the Laurent expansion at s = 1 of E(z,s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressedits multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups Γ. The analogue of the eta function has a multiplier system that may be expressed in terms of a map S : Γ → R which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from Γ to C produces higher-order Eisenstein series. These series share many of the properties of E(z,s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher- order Dedekind symbols S∗ :Γ → R. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups Γ0(N) are rational. 1 Introduction 1.1 Kronecker limit functions and Dedekind sums Write elements of the upper half plane H as z = x + iy with y > 0. The non-holomorphic Eisenstein series E (s,z), associated to the full modular group SL(2, Z), may be given as ∞ 1 ys E (z,s) := .
    [Show full text]
  • Torus N-Point Functions for $\Mathbb {R} $-Graded Vertex Operator
    Torus n-Point Functions for R-graded Vertex Operator Superalgebras and Continuous Fermion Orbifolds Geoffrey Mason∗ Department of Mathematics, University of California Santa Cruz, CA 95064, U.S.A. Michael P. Tuite, Alexander Zuevsky† Department of Mathematical Physics, National University of Ireland, Galway, Ireland. October 23, 2018 Abstract We consider genus one n-point functions for a vertex operator su- peralgebra with a real grading. We compute all n-point functions for rank one and rank two fermion vertex operator superalgebras. In the arXiv:0708.0640v1 [math.QA] 4 Aug 2007 rank two fermion case, we obtain all orbifold n-point functions for a twisted module associated with a continuous automorphism generated by a Heisenberg bosonic state. The modular properties of these orb- ifold n-point functions are given and we describe a generalization of Fay’s trisecant identity for elliptic functions. ∗Partial support provided by NSF, NSA and the Committee on Research, University of California, Santa Cruz †Supported by a Science Foundation Ireland Frontiers of Research Grant, and by Max- Planck Institut f¨ur Mathematik, Bonn 1 1 Introduction This paper is one of a series devoted to the study of n-point functions for vertex operator algebras on Riemann surfaces of genus one, two and higher [T], [MT1], [MT2], [MT3]. One may define n-point functions at genus one following Zhu [Z], and use these functions together with various sewing pro- cedures to define n-point functions at successively higher genera [T], [MT2], [MT3]. In this paper we consider the genus one n-point functions for a Vertex Operator Superalgebra (VOSA) V with a real grading (i.e.
    [Show full text]
  • Mathematical Constants and Sequences
    Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) ..
    [Show full text]
  • Modular Forms in String Theory and Moonshine
    (Mock) Modular Forms in String Theory and Moonshine Miranda C. N. Cheng Korteweg-de-Vries Institute of Mathematics and Institute of Physics, University of Amsterdam, Amsterdam, the Netherlands Abstract Lecture notes for the Asian Winter School at OIST, Jan 2016. Contents 1 Lecture 1: 2d CFT and Modular Objects 1 1.1 Partition Function of a 2d CFT . .1 1.2 Modular Forms . .4 1.3 Example: The Free Boson . .8 1.4 Example: Ising Model . 10 1.5 N = 2 SCA, Elliptic Genus, and Jacobi Forms . 11 1.6 Symmetries and Twined Functions . 16 1.7 Orbifolding . 18 2 Lecture 2: Moonshine and Physics 22 2.1 Monstrous Moonshine . 22 2.2 M24 Moonshine . 24 2.3 Mock Modular Forms . 27 2.4 Umbral Moonshine . 31 2.5 Moonshine and String Theory . 33 2.6 Other Moonshine . 35 References 35 1 1 Lecture 1: 2d CFT and Modular Objects We assume basic knowledge of 2d CFTs. 1.1 Partition Function of a 2d CFT For the convenience of discussion we focus on theories with a Lagrangian description and in particular have a description as sigma models. This in- cludes, for instance, non-linear sigma models on Calabi{Yau manifolds and WZW models. The general lessons we draw are however applicable to generic 2d CFTs. An important restriction though is that the CFT has a discrete spectrum. What are we quantising? Hence, the Hilbert space V is obtained by quantising LM = the free loop space of maps S1 ! M. Recall that in the usual radial quantisation of 2d CFTs we consider a plane with 2 special points: the point of origin and that of infinity.
    [Show full text]
  • Generalized Jacobi Theta Functions, Macdonald's
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ScholarBank@NUS GENERALIZED JACOBI THETA FUNCTIONS, MACDONALD’S IDENTITIES AND POWERS OF DEDEKIND’S ETA FUNCTION TOH PEE CHOON (B.Sc.(Hons.), NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2007 Acknowledgements I would like to thank my thesis advisor Professor Chan Heng Huat, for his invalu- able guidance throughout these four years. Without his dedication and his brilliant insights, I would not have been able to complete this thesis. I would also like to thank Professor Shaun Cooper, whom I regard as my unofficial advisor. He taught me everything that I know about the Macdonald identities, and helped to improve and refine most of the work that I have done for this thesis. Lastly, I would like to thank Professor Liu Zhi-Guo because I first learnt about the Jacobi theta functions from his excellent research papers. Toh Pee Choon 2007 ii Contents Acknowledgements ii Summary v List of Tables vii 1 Jacobi theta functions 1 1.1 Classical Jacobi theta functions . 2 1.2 Generalized Jacobi theta functions . 6 2 Powers of Dedekind’s eta function 9 2.1 Theorems of Ramanujan, Newman and Serre . 9 2.2 The eighth power of η(τ)........................ 13 2.3 The tenth power of η(τ) ........................ 20 2.4 The fourteenth power of η(τ) ..................... 23 2.5 Modular form identities . 25 iii Contents iv 2.6 The twenty-sixth power of η(τ) ...................
    [Show full text]
  • On a Lattice Generalisation of the Logarithm and a Deformation of the Dedekind Eta Function
    Bull. Aust. Math. Soc. 102 (2020), 118–125 doi:10.1017/S000497272000012X ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE DEDEKIND ETA FUNCTION LAURENT BETERMIN´ (Received 14 September 2019; accepted 14 January 2020; first published online 20 February 2020) Abstract (m) We consider a deformation EL;Λ(it) of the Dedekind eta function depending on two d-dimensional simple lattices (L; Λ) and two parameters (m; t) 2 (0; 1), initially proposed by Terry Gannon. We show that the (m) minimisers of the lattice theta function are the maximisers of EL;Λ(it) in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms. 2010 Mathematics subject classification: primary 33E20; secondary 11F20, 49K30. Keywords and phrases: Dedekind eta function, theta function, lattice, optimisation. 1. Introduction and setting Many mathematical models from physics are written in terms of special functions whose properties give fundamental information about the system (see, for example, [16]). For example, properties of the Jacobi theta function and the Dedekind eta function defined for =(τ) > 0 by X −iπk2τ 1=24 Y n 2iπτ θ3(τ):= e and η(τ):= q (1 − q ); q = e ; (1.1) k2Z n2N have been widely used to identify ground states of periodic systems (see, for example, [4, 10, 13, 19]). Generalisations and deformations of these special functions which arise in more complex physical systems are also of great interest.
    [Show full text]
  • Eta Products, BPS States and K3 Surfaces
    Eta Products, BPS States and K3 Surfaces Yang-Hui He1 & John McKay2 1 Department of Mathematics, City University, London, EC1V 0HB, UK and School of Physics, NanKai University, Tianjin, 300071, P.R. China and Merton College, University of Oxford, OX14JD, UK [email protected] 2 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada [email protected] Abstract Inspired by the multiplicative nature of the Ramanujan modular discriminant, ∆, we consider physical realizations of certain multiplicative products over the Dedekind eta-function in two parallel directions: the generating function of BPS states in certain heterotic orbifolds and elliptic K3 surfaces associated to congruence subgroups of the modular group. We show that they are, after string duality to type II, the same K3 arXiv:1308.5233v3 [hep-th] 8 Jan 2014 surfaces admitting Nikulin automorphisms. In due course, we will present identities arising from q-expansions as well as relations to the sporadic Mathieu group M24. 1 Contents 1 Introduction and Motivation 3 1.1 Nomenclature . .5 2 Eta Products and Partition Functions 7 2.1 Bosonic String Oscillators . .7 2.2 Eta Products . .9 2.3 Partition Functions and K3 Surfaces . 10 2.4 Counting 1/2-BPS States . 11 3 K3 Surfaces and Congruence Groups 13 3.1 Extremal K3 Surfaces . 13 3.2 Modular Subgroups and Coset Graphs . 14 3.3 Summary . 17 3.4 Beyond Extremality . 20 3.5 Elliptic Curves . 21 4 Monsieur Mathieu 24 5 A Plethystic Outlook 28 6 Conclusions and Prospects 30 A Further Salient Features of Eta 33 2 A.1 Modularity .
    [Show full text]
  • An Addition Formula for the Jacobian Theta Function 2
    AN ADDITION FORMULA FOR THE JACOBIAN THETA FUNCTION WITH APPLICATIONS BING HE AND HONGCUN ZHAI Abstract. Liu [Adv. Math. 212(1) (2007), 389–406] established an addition formula for the Jacobian theta function by using the theory of elliptic functions. From this addition formula he obtained the Ramanujan cubic theta function identity, Winquist’s identity, a theta function identities with five parameters, and many other interesting theta function identities. In this paper we will give an addition formula for the Jacobian theta function which is equivalent to Liu’s addition formula. Based on this formula we deduce some known theta function identities as well as new identities. From these identities we shall establish certain new series expansions for η2(q) and η6(q), where η(q) is the Dedekind eta function, give new proofs for Jacobi’s two, four and eight squares theorems and confirm several q-trigonometric identities conjectured by W. Gosper. Another conjectured identity on the constant Πq is also settled. The series expansions for η6(q) led to new proofs of Ramanujan’s congruence p(7n + 5) ≡ 0(mod7). 1. Introduction Throughout this paper we take q = exp(πiτ), where Im τ > 0. To carry out our study we need the definitions of the Jacobian theta functions. Definition. Jacobian theta functions θ (z τ) for j =1, 2, 3, 4 are defined as [23, 31] j | 1 ∞ k k(k+1) (2k+1)zi 1 ∞ k k(k+1) θ (z τ)= iq 4 ( 1) q e =2q 4 ( 1) q sin(2k + 1)z, 1 | − − − k= k=0 X−∞ X ∞ ∞ 1 k(k+1) (2k+1)zi 1 k(k+1) θ (z τ)= q 4 q e =2q 4 q cos(2k + 1)z, 2 | k= k=0 X−∞ X ∞ ∞ k2 2kzi k2 θ3(z τ)= q e =1+2 q cos2kz, arXiv:1804.00580v3 [math.NT] 19 Jun 2018 | k= k=1 X−∞ X ∞ 2 ∞ 2 θ (z τ)= ( 1)kqk e2kzi =1+2 ( 1)kqk cos2kz.
    [Show full text]
  • Partitions and Rademacher's Exact Formula
    Partitions and Rademacher's Exact Formula Anurag Sahay KVPY Reg. No. SX - 11011010 Indian Institute of Technology, Kanpur Rijul Saini KVPY Reg. No. SA - 11010155 Chennai Mathematical Institute 22 June - 22 July, 2012 Certificate This is to certify that the project report titled `Partitions and Rademacher's Exact Formula' submitted by Anurag Sahay and Rijul Saini for their KVPY Summer Project is a record of bonafide work carried out by them under my supervision and guidance. Prof. Amitabha Tripathi Associate Professor Department of Mathematics Indian Institute of Technology, Delhi About the Project This document is a record of the topics read by in the Theory of Partitions from [1] and [2], as part of their Summer Project 2012, for the KVPY Fellow- ship, funded by the Department of Science and Technology (Govt. of India). This `reading project' was done under the supervision of Prof. Amitabha Tripathi, from 22nd June, 2012 to 22nd July, 2012, and was geared towards first gaining a broad overview of partition theory, and then, eventually, un- derstanding the proof of Rademacher's Exact Formula for the unrestricted partition function p(n). The purpose of this project was to provide some exposure to an advanced field of Number Theory by reading a comparitively more esoteric and involved proof than one would normally encounter in their undergraduate curriculum. Contents 1 Introduction 3 2 Preliminary Results in Partition Theory 4 2.1 Introduction . 4 2.2 Major Results . 5 2.3 Ramanujan's Partition Identities . 8 3 Rademacher's Exact Formula 9 3.1 Introduction . 9 3.2 Dedekind's Functional Equation .
    [Show full text]
  • Magnetized Riemann Surface of Higher Genus and Eta Quotients Of
    WU-HEP-20-08 Magnetized Riemann Surface of Higher Genus and Eta Quotients of Semiprime Level 1, Masaki Honda ∗ 1 Department of Physics, Waseda University, Tokyo 169-8555, Japan Abstract We study the zero mode solutions of a Dirac operator on a magnetized Riemann surface of higher genus. In this paper, we define a Riemann surface of higher genus as a quotient manifold of the Poincar´eupper half-plane by a congruence subgroup, especially Γ0(N). We present a method to construct basis of cusp forms since the zero mode solutions should be cusp forms. To confirm our method, we select a congruence subgroup of semiprime level and show the demonstration to some lower weights. In addition, we discuss Yukawa couplings and matrix regularization as applications. arXiv:2008.11461v1 [hep-th] 26 Aug 2020 ∗E-mail address: yakkuru [email protected] Contents 1 Introduction 1 2 Preliminaries 4 2.1 Poincar´eupperhalf-plane . 4 2.2 Automorphicformsandcuspforms . 5 2.3 Dedekind η-functionandetaquotients . 6 3 Dirac operator on Γ H˜ in gauge background 7 \ 3.1 Zweibein and spin connection . 8 3.2 Selectionofgaugebackground . 8 3.3 Diracoperator.................................. 9 4 Construction of zero mode solutions 11 4.1 Construction .................................. 11 4.2 Demonstration ................................. 14 5 Applications 16 5.1 Extradimensionalmodels ........................... 17 5.2 Matrix regularization . 17 6 Conclusion and discussion 18 A Explicit q-expansionsandcoefficients . 19 1 Introduction The standard model (SM) is a successful model to explain the results of high energy experiments. However, there are several problems to be solved, e.g., the origin of the generations and the chiral structure.
    [Show full text]
  • Numerics of Classical Elliptic Functions, Elliptic Integrals and Modular Forms
    Numerics of classical elliptic functions, elliptic integrals and modular forms Fredrik Johansson LFANT, Inria Bordeaux & Institut de Mathematiques´ de Bordeaux KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory DESY, Zeuthen, Germany 24 October 2017 1 / 49 Introduction Elliptic functions I F(z + !1m + !2n) = F(z); m; n 2 Z I Can assume !1 = 1 and !2 = τ 2 H = fτ 2 C : Im(τ) > 0g Elliptic integrals R p I R(x; P(x))dx; inverses of elliptic functions Modular forms/functions on H aτ+b k a b I F( cτ+d ) = (cτ + d) F(τ) for c d 2 PSL2(Z) I Related to elliptic functions with fixed z and varying lattice parameter !2=!1 = τ 2 H Jacobi theta functions (quasi-elliptic functions) I Used to construct elliptic and modular functions 2 / 49 Numerical evaluation Lots of existing literature, software (Pari/GP,Sage, Maple, Mathematica, Matlab, Maxima, GSL, NAG, . ). This talk will mostly review standard techniques (and many techniques will be omitted). My goal: general purpose methods with I Rigorous error bounds I Arbitrary precision I Complex variables Implementations in the C library Arb (http://arblib.org/) 3 / 49 Why arbitrary precision? Applications: I Mitigating roundoff error for lengthy calculations I Surviving cancellation between exponentially large terms I High order numerical differentiation, extrapolation I Computing discrete data (integer coefficients) I Integer relation searches (LLL/PSLQ) I Heuristic equality testing Also: I Can increase precision if error bounds are too pessimistic Most interesting range: 10 − 105 digits. (Millions, billions...?) 4 / 49 Ball/interval arithmetic A real number in Arb is represented by a rigorous enclosure as a high-precision midpoint and a low-precision radius: [3:14159265358979323846264338328 ± 1:07 · 10−30] Complex numbers: [m1 ± r1] + [m2 ± r2]i.
    [Show full text]
  • Vassiliev Invariants and a Strange Identity Related to the Dedekind Eta-Function
    Topology 40 (2001) 945}960 Vassiliev invariants and a strange identity related to the Dedekind eta-function Don Zagier* Max-Planck-Institut fur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany Received 9 August 1999; accepted 15 November 1999 Abstract In this paper the `functiona F(q)" (1!q)(1!q)2(1!qL) is studied. The series does not L converge in any open set, but has well-de"ned values and derivatives of all orders when q is a root of unity. It " ` is shown that the coe$cients of its Taylor expansion at q 1are equal to the numbers " of regular linearized chord diagramsa as de"ned by Stoimenow and hence give an upper bound (the best currently known) for the number of linearly independent Vassiliev invariants of degree D. There are similar expansions at other roots of unity. The same values and derivatives of all orders at all roots of unity are obtained as the limiting value of the function ! (!1)L"6n#1"qL>L , the `derivative of order one-half '' of the LZ9 Dedekind eta-function, and also exhibit a kind of modular behavior which can be seen as an example of a generalization of the classical theory of periods of modular forms to the case of half-integral weight. Functions of a similar type also occurred in recent joint work with Lawrence in connection with the Witten}Reshetikhin}Turaev invariants of knots. 2001Elsevier Science Ltd. All rights reserved. MSC: 57M27; 05A30; 11F11; 11F20; 11F67 Keywords: Vassiliev invariants; Knots; Dedekind -function; q-Calculus; Modular forms; Periods of modular forms Introduction An important open problem of knot theory is the determination of the number <(D) of linearly independent Vassiliev invariants of degree D, or equivalently of chord diagrams of degree D modulo the 4-term relation.
    [Show full text]