Theta Functions Research Articles

On computing high-dimensional Riemann theta functions

Riemann theta functions play a crucial role in the field of nonlinear Fourier analysis, where they are used to realize inverse nonlinear Fourier transforms for periodic signals. The practical applicability of this approach has however been limited since Riemann theta functions are multi-dimensional Fourier series whose computation suffers from the curse of dimensionality. In this paper, we investigate several new approaches to compute Riemann theta functions with the goal of unlocking their practical potential. Our first contributions are novel theoretical lower and upper bounds on the series truncation error. These bounds allow us to rule out several of the existing approaches for the high-dimension regime. We then propose to consider low-rank tensor and hyperbolic cross based techniques. We first examine a tensor-train based algorithm which utilizes the popular scaling and squaring approach. We show theoretically that this approach cannot break the curse of dimensionality. Finally, we investigate two other tensor-train based methods numerically and compare them to hyperbolic cross based methods. Using finite-genus solutions of the Korteweg–de Vries (KdV) and nonlinear Schrödinger equation (NLS) equations, we demonstrate the accuracy of the proposed algorithms. The tensor-train based algorithms are shown to work well for low genus solutions with real arguments but are limited by memory for higher genera. The hyperbolic cross based algorithm also achieves high accuracy for low genus solutions. Its novelty is the ability to feasibly compute moderately accurate solutions (a relative error of magnitude 0.01) for high dimensions (up to 60). It therefore enables the computation of complex inverse nonlinear Fourier transforms that were so far out of reach.

Vanishing Coefficients in Three Families of Products of Theta Functions. II.

Vanishing Coefficients in Three Families of Products of Theta Functions. II.

Identities involving mock theta functions and theta functions

In this paper, we first present new representations for several mock theta functions. In view of the sign flips on these identities, some interesting results involving theta functions are established by new Bailey pairs and Watson’s 8 ϕ 7 _8\phi _7 transformation formula. Moreover, the truncated forms for the theta series identities are also considered.

Narain CFTs and quantum codes at higher genus

Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of “isospectral theories” — CFTs with the same genus 1 partition function — and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The n = 4 theory conjectured to have the largest possible gap in Virasoro characters, the SO(8) WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.

Large gap asymptotics on annuli in the random normal matrix model

We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form exp(C1n2+C2nlogn+C3n+C4n+C5logn+C6+Fn+O(n-112)),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\exp \\Bigg ( C_{1} n^{2} + C_{2} n \\log n + C_{3} n + C_{4} \\sqrt{n} + C_{5}\\log n + C_{6} + {\\mathcal {F}}_{n} + \\mathcal {O}\\Big ( n^{-\\frac{1}{12}}\\Big )\\Bigg ), \\end{aligned}$$\\end{document}where n is the number of points of the process. We determine the constants C1,…,C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{6}$$\\end{document} explicitly, as well as the oscillatory term Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} which is of order 1. We also allow one annulus to be a disk, and one annulus to be unbounded. For the complex Ginibre point process, we improve on the best known results: (i) when the hole region is a disk, only C1,…,C4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},\\ldots ,C_{4}$$\\end{document} were previously known, (ii) when the hole region is an unbounded annulus, only C1,C2,C3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1},C_{2},C_{3}$$\\end{document} were previously known, and (iii) when the hole region is a regular annulus in the bulk, only C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} was previously known. For general values of our parameters, even C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{1}$$\\end{document} is new. A main discovery of this work is that Fn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {F}}_{n}$$\\end{document} is given in terms of the Jacobi theta function. As far as we know this is the first time this function appears in a large gap problem of a two-dimensional point process.

Families of identities involving universal mock theta functions

The purpose of this paper is to establish several families of identities involving universal mock theta functions and express them in terms of Appell–Lerch sums. The subsequent consequences of these results are a variety of interesting identities on theta series and classical mock theta functions.

Quadratic superlattices: A type of nonperiodic lattice with extended states

Deterministic nonperiodic lattices often exhibit properties intermediate between those of random and periodic crystals. Properties that have attracted interest include complex heirarchical structure factors, the breaking up of the band into a dense set of critical states and minigaps, and the interplay between extended and localized states. In one dimension, considerable attention has focused on nonperiodic lattices showing properties intermediate between periodic and random. We propose a one-dimensional nonperiodic lattice with lattice sites at ${0}^{2}d, {1}^{2}d, {2}^{2}d$, ... with length $d$ playing the r\^ole of the lattice constant. Here we show that the structure factor ${S}_{0}(k)$, which governs how waves are scattered and is reflected in many physical properties, is simply related to the Jacobi theta function ${\ensuremath{\vartheta}}_{3}(q)$. ${S}_{0}(k)$ is periodic in wave vector $k$ and is singular continuous, consisting of a dense set of peaks with $k$ given by rational-fraction multiples of $\frac{\ensuremath{\pi}}{d}$, though the scaling properties of ${S}_{0}(k)$ are the same as for periodic crystals. The electronic structure considered in a tight-binding model shows a bandlike spectrum, but broken by a dense set of minigaps near peak locations in ${S}_{0}(k)$. These lattices show novel properties such as the coexistence of a singular-continuous spectrum with extended states. Quadratic lattices may enable the physical realization of special functions of mathematical and physical importance as well as for the design of nonperiodic diffraction gratings, antenna arrays, ad coupled optical resonators to produce complex quasirandom behavior and are of interest to study quantum interactions in nonperiodic optical lattices.

Algebro-geometric quasi-periodic solutions to the Satsuma–Hirota hierarchy

In this paper, we study the theory of the resulting tetragonal curve and derive three kinds of Abel differentials, Baker–Akhiezer function and meromorphic function and so on, from which a systematic method is developed to construct algebro-geometric quasi-periodic solutions of soliton equations associated with the 4 × 4 matrix spectral problems. As an illustrative example, the Satsuma–Hirota hierarchy related to the 4 × 4 matrix spectral problem is obtained by utilizing the Lenard recursion equation and zero-curvature equation. Based on the characteristic polynomial of Lax matrix for the Satsuma–Hirota hierarchy, we introduce a tetragonal curve Kg of genus g and study the asymptotic properties of the Baker–Akhiezer function ψ1 and the meromorphic function ϕ near the infinite points on the tetragonal curve. The straightening out of various flows is exactly given through the Abel map and Abel–Jacobi coordinates. Using the theory of the tetragonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit Riemann theta function representations of the Baker–Akhiezer function and the meromorphic function, and in particular, that of solutions for the entire Satsuma–Hirota hierarchy.

Binary theta functions and Borcherds products

We obtain infinite product expansions in the sense of Borcherds for theta functions associated with certain positive-definite binary quadratic and binary hermitian forms. Among other things, we show that every weight 1 binary theta function is a Borcherds product. In particular, binary theta functions have zeros only at quadratic irrationals.

Bounds for theta sums in higher rank. I

Theta sums are finite exponential sums with a quadratic form in the oscillatory phase. This paper establishes new upper bounds for theta sums in the case of smooth and box truncations. This generalises a classic 1977 result of Fiedler, Jurkat and Körner for one-variable theta sums and, in the multi-variable case, improves previous estimates obtained by Cosentino and Flaminio in 2015. Key steps in our approach are the automorphic representation of theta functions and their growth in the cusps of the underlying homogeneous space.

Nonterminating transformations and summations associated with some q-Mellin–Barnes integrals

In many cases one may encounter an integral which is of q-Mellin–Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting q-Mellin–Barnes integrals and using them we derive transformation and summation formulas for nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey–Wilson moments, nonterminating well-poised ϕ23, nonterminating very-well-poised W45, W78, products of two nonterminating ϕ12's, square of a nonterminating well-poised ϕ12, a nonterminating W910, two nonterminating W1112's and several nonterminating summations which arise from the Askey–Roy and Gasper integrals.

On string functions and double-sum formulas

String functions are important building blocks of characters of integrable highest modules over affine Kac--Moody algebras. Kac and Peterson computed string functions for affine Lie algebras of type $A_{1}^{(1)}$ in terms of Dedekind eta functions. We produce new relations between string functions by writing them as double-sums and then using certain symmetry relations. We evaluate the series using special double-sum formulas that express Hecke-type double-sums in terms of Appell--Lerch functions and theta functions, where we point out that Appell--Lerch functions are the building blocks of Ramanujan's classical mock theta functions.

The mixed mock modularity of certain duals of generalized quantum modular forms of Hikami and Lovejoy

We resolve a question of Hikami and Lovejoy that sits at the intersection of quantum modular forms, mock modular forms, and knot theory. We express a family of Hecke–Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. They obtained their family of Hecke–Appell-type sums by considering certain duals of generalized quantum modular forms, where the quantum modular forms are motivated by the colored Jones polynomial for special torus knots.

Heat coefficients for magnetic Laplacians on the complex projective space P n(ℂ)

We denote by Δ ν the Fubini-Study Laplacian perturbed by a uniform magnetic field whose strength is proportional to ν. When acting on bounded functions on the complex projective n-space, this operator has a discrete spectrum consisting on eigenvalues β m , m ∈ Z + . For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of Δ ν . Using a suitable polynomial decomposition of the multiplicity of each β m , we write down a trace formula for the heat operator associated with Δ ν in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as t ↘ 0 + by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with Δ ν .

Graph schemes, graph series, and modularity

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert–Poincaré series of a certain infinite jet scheme. We study new q-representations and examine modular properties of several examples including Dynkin diagrams of finite and affine type. Notably, we obtain new formulas for graph series of type A7 and A8 in terms of “sum of tails” series, and of type D4 and D5 in the form of indefinite theta functions of signature (1,1). We also study examples related to sums of powers of divisors corresponding to 5-cycles. For several examples of graphs, we prove that graph series are mixed quantum modular forms and also mixed mock modular forms.

Convolution and Square in Abelian Groups I

We prove that the functional equation f ⋆ f ( 2 t ) = λ f ( t ) 2 , for t in Z / d Z with d odd, admits a nonzero solution f if λ = a ​ + ​ i b with a, b positive integers such that a ​ + ​ b ​ = ​ d and a ​ ≡ ( d + 1 ) 2 4 mod 4. The proof involves theta functions on elliptic curves with complex multiplication.

The three missing terms in Ramanujan’s septic theta function identity

On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function varphi (q): φ(e-7π7)=7-3/4φ(e-π7){1+()2/7+()2/7+()2/7}.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\varphi (e^{-7\\pi \\sqrt{7}}) = 7^{-3/4}\\varphi (e^{-\\pi \\sqrt{7}})\\big \\{1 + (\\quad )^{2/7} + (\\quad )^{2/7} + (\\quad )^{2/7}\\big \\}. \\end{aligned}$$\\end{document}We give the three missing terms. In addition, we calculate the class invariant G_{343} and further special values of varphi (e^{-npi }), for n = 7, 21, 35, and 49.

Symmetries of Calabi-Yau prepotentials with isomorphic flops

Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kähler cone made up from an infinite number of individual Kähler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kähler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold.

The lambda extensions of the Ising correlation functions

We revisit, with a pedagogical heuristic motivation, the lambda extension of the low-temperature row correlation functions of the two-dimensional Ising model. In particular, using these one-parameter series to understand the deformation theory around selected values of , namely with m and n integers, we show that these series yield perturbation coefficients, generalizing form factors, that are D-finite functions. As a by-product these exact results provide an infinite number of highly non-trivial identities on the complete elliptic integrals of the first and second kind. These results underline the fundamental role of Jacobi theta functions and Jacobi forms, the previous D-finite functions being (relatively simple) rational functions of Jacobi theta functions, when rewritten in terms of the nome of elliptic functions.

The mixed mock modularity of a new $U$-type function related to the Andrews-Gordon identities

In this paper we resolve a question by Bringmann, Lovejoy, and Rolen on a new vector-valued $U$-type function. We obtain an expression for a corresponding family of Hecke--Appell-type sums in terms of mixed mock modular forms; that is, we express the sum in terms of Appell functions and theta functions. This $U$-type function appears from considering the special polynomials related to generating functions for the partitions occurring in Gordon’s generalization of the Rogers--Ramanujan identities.