Mock Theta Functions

Abstract

In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms. In Chapter 1 we give results for Lerch sums (also called Appell functions, or generalized Lambert series). In Chapter 2 we consider indefinite theta functions of type (r-1,1). Chapter 3 deals with Fourier coefficients of meromorphic Jacobi forms. In Chapter 4 we use the results from Chapter 2 to give explicit results for 8 of the 10 fifth order mock theta functions and all 3 seventh order functions, that were originally defined by Ramanujan. The result is that we can find a correction term, which is a period integral of a weight 3/2 unary theta functions, such that if we add it to the mock theta function, we get a weight 1/2 real-analytic modular form, which is annihilated by the hyperbolic Laplacian.

Ph.D. thesis, Utrecht University, 2002. With Dutch title page and abstract

Cited by in corpus (151)

• Notes on the K3 Surface and the Mathieu group M_24
• Quantum Black Holes, Wall Crossing, and Mock Modular Forms
• Hecke-type double sums, Appell-Lerch sums, and mock theta functions (I)
• Note on Twisted Elliptic Genus of K3 Surface
• Logarithmic Conformal Field Theory: Beyond an Introduction
• The non-compact elliptic genus: mock or modular
• A Farey tale for N=4 dyons
• 3d Modularity
• A Twisted Non-compact Elliptic Genus
• D3-instantons, Mock Theta Series and Twistors
• K3 Surfaces, N=4 Dyons, and the Mathieu Group M24
• Rank differences for overpartitions
• Multiple D3-instantons and mock modular forms I
• On a modular property of N=2 superconformal theories in four dimensions
• Quantum Knot Invariant for Torus Link and Modular Forms
• Narain to Narnia
• Indefinite theta series and generalized error functions
• Asymptotics for rank and crank moments
• Mock Modular Mathieu Moonshine Modules
• Superspace conformal field theory
• The Largest Mathieu Group and (Mock) Automorphic Forms
• Heegner divisors, $L$-functions and harmonic weak Maass forms
• Exponential BPS graphs and D brane counting on toric Calabi-Yau threefolds: Part I
• Dyson's Ranks and Appell-Lerch Sums
• S-duality and refined BPS indices
• Eichler-Selberg Type Identities for Mixed Mock Modular Forms
• The Bailey chain and mock theta functions
• Supersymmetric $U(N)$ Chern-Simons-matter theory and phase transitions
• Lessons on Black Holes from the Elliptic Genus
• Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products
• Umbral Moonshine and the Niemeier Lattices
• Bilateral series and Ramanujan's radial limits
• Wall-crossing made smooth
• Vafa-Witten theory and iterated integrals of modular forms
• Single-centered black hole microstate degeneracies from instantons in supergravity
• A proof of the Thompson Moonshine Conjecture
• Renormalization and BRST symmetry in Donaldson-Witten theory
• Elliptic genera and real Jacobi forms
• On a completed generating function of locally harmonic Maass forms
• Quantum hypermultiplet moduli spaces in N=2 string vacua: a review
• Mixed Rademacher and BPS Black Holes
• Automorphic properties of generating functions for generalized odd rank moments and odd Durfee symbols
• Vafa-Witten invariants from modular anomaly
• Sums of class numbers and mixed mock modular forms
• Quantum geometry of elliptic Calabi-Yau manifolds
• Mock theta double sums
• BPS Saturated String Amplitudes: K3 Elliptic Genus and Igusa Cusp Form
• A double-sum Kronecker-type identity
• Parity of coefficients of mock theta functions
• Intersection cohomology of moduli spaces of sheaves on surfaces
• Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds
• Rank $N$ Vafa-Witten invariants, modularity and blow-up
• The Partition Function Of Argyres-Douglas Theory On A Four-Manifold
• Exceptional Algebra and Sporadic Groups at c=12
• Generalised Umbral Moonshine
• Representations of affine superalgebras and mock theta functions III
• Scaling Black Holes and Modularity
• Umbral Moonshine and K3 Surfaces
• The tenth order mock theta functions revisited
• Proof of the Umbral Moonshine Conjecture
• On string functions and double-sum formulas
• Eichler integrals and harmonic weak Maass forms
• Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials
• ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine
• Mordell integrals and Giveon-Kutasov duality
• An Elliptic Triptych
• N=2 Moonshine
• Compact Formulas for the Completed Mock Modular Forms
• Renormalization and quantum modular forms, part II: Mock theta functions
• Proofs of Some Conjectures of Chan on Appell-Lerch Sums
• Mocking the u-plane integral
• Squashed toric sigma models and mock modular forms
• Rademacher Sums and Rademacher Series
• Donaldson-Witten theory, surface operators and mock modular forms
• Eulerian series, zeta functions and the arithmetic of partitions
• Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition
• Polar harmonic Maaß forms and holomorphic projection
• The Eichler cohomology theorem for Jacobi forms
• Almost harmonic Maass forms and Kac-Wakimoto characters
• Refined Partition Functions for Open Superstrings with 4, 8 and 16 Supercharges
• Indefinite theta functions and black hole partition functions
• Eulerian series as modular forms revisited
• Mock Modular Index of M2-M5 Brane System
• H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series
• On the elliptic genera of manifolds of Spin(7) holonomy
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• Three Avatars of Mock Modularity
• L-functions for Meromorphic Modular Forms and Sum Rules in Conformal Field Theory
• Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion for K3 Surface
• Liouville Theory and Elliptic Genera
• Ramanujan and Quantum Black Holes
• Schur's partition theorem and mixed mock modular forms
• On Hecke-type double-sums and general string functions for the affine Lie algebra $A_1^{(1)}$
• Polar harmonic Maass forms and their applications
• Uncovering Ramanujan's "Lost" Notebook: An Oral History
• Singular invariants and coefficients of weak harmonic Maass forms of weight 5/2
• Generalized Donaldson-Thomas invariants on the local projective plane
• Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series
• The Bloch-Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms
• Universal mock theta functions and two-variable Hecke-Rogers identities
• On the Fourier coefficients of negative index meromorphic Jacobi forms
• Congruences for Ramanujan's f and ω functions via generalized Borcherds products
• Characters of (relatively) integrable modules over affine Lie superlagebras
• Rank-Crank type PDE's and non-holomorphic Jacobi forms
• A note on the $l$-fold Bailey Lemma and Mixed Mock Modular forms
• More on some Mock theta Double sums
• Completions and algebraic formulas for the coefficients of Ramanujan's mock theta functions
• On the Fourier coefficients of meromorphic Jacobi forms
• TASI Lectures on Moonshine
• A generalization of the Riemann-Siegel formula
• A characterization of modified mock theta functions
• Locally harmonic Maass forms and periods of meromorphic modular forms
• Mixed mock modular q-series
• Quantum Black Hole Entropy and Localization in Supergravity
• Locally harmonic Maass forms and the kernel of the Shintani lift
• Maass waveforms arising from sigma and related indefinite theta functions
• Harmonic Maass forms and periods
• Period Relations, Jacobi Forms and Eichler Integral
• Decompositions of index one Jacobi forms into $N=4$ characters and formulas for mock modular forms
• Computing the truncated theta function via Mordell integral
• Partition Identities for Ramanujan's Third Order Mock Theta Functions
• A family of mock theta functions of weights 1/2 and 3/2 and their congruence properties
• Properties of Lerch Sums and Ramanujan's Mock Theta Functions
• Zagier's weight $3/2$ mock modular form
• Professor Nambu, String Theory and Moonshine Phenomenon
• The $t$-analogs of string functions for $A_1^{(1)}$ and Hecke indefinite modular forms
• Theta liftings on weak Maass forms
• Representations of superconformal algebras and mock theta functions
• Domain walls and M2-branes partition functions: M-theory and ABJM Theory
• Radial limits of mock theta functions
• Indefinite theta functions for counting attractor backgrounds
• A general double sum identity, mock theta functions, and Bailey pairs
• k-Run Overpartitions and Mock Theta Functions
• Generalized modular forms including the weak Maass forms,the Ramanujan's Theta functions and the Tau function
• Hecke-type identities associated with definite quadratic forms
• Generating functions of planar polygons from homological mirror symmetry of elliptic curves
• On the modularity of certain functions from the Gromov-Witten theory of elliptic orbifolds
• Cohomological relation between Jacobi forms and skew-holomorphic Jacobi forms
• Rank-Crank type PDEs and generalized Lambert series identities
• On the irrationality of Ramanujan's mock theta functions and other q-series at an infinite number of points
• On certain explicit congruences for mock theta functions
• Shifted Convolution $L$-Series Values for Elliptic Curves
• Partial theta functions and mock modular forms as q-hypergeometric series
• Mock modular forms as $p$-adic modular forms
• Generating Functions of the Hurwitz Class Numbers Associated with Certain Mock Theta Functions
• Algebraicity of higher Green functions at a CM point
• Appell-Lerch Sums and N=2 Moduli
• Asymptotic formulas for stacks and unimodal sequences
• Asymptotic expansions, $L$-values and a new Quantum Modular Form
• Asymptotic distribution of odd-balanced unimodal sequences with rank congruent to a mod c
• Isomorphism of Skew-Holomorphic Harmonic Maass-Jacobi Forms and Certain Weak Harmonic Maass Forms