Searcharxiv⌕ Search

arXiv subjects

Kathrin Bringmann

Publications and source records attributed to Kathrin Bringmann.

At least 109 records · Page 6Linked to original sources
arXiv2018

On characters of $L_{\frak{sl}_n}(-Λ_0)$-modules

We use recent results of Rolen, Zwegers, and the first author to study characters of irreducible (highest weight) modules for the vertex operator algebra $L_{\frak{sl}_\ell}(-Λ_0)$. We establish asymptotic behaviors of characters for the (ordinary) irreducible $L_{\frak{sl}_\ell}(-Λ_0)$-modules. As a consequence we prove that their quantum dimensions are one, as predicted by representation theory. We also establish a full asymptotic expansion of irreducible characters for $\frak{sl}_3$. Finally, we determine a decomposition formula for the full characters in terms of unary theta and false theta functions which allows us to study their modular properties.

math.NT↗
arXiv2018

Higher depth quantum modular forms, multiple Eichler integrals, and $\frak{sl}_3$ false theta functions

We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose "companions" in the lower half-plane can be also realized both as double Eichler integrals and as non-holomorphic theta series having values of "double error" functions as coefficients. In particular, we prove that the false theta of $\frak{sl}_3$, appearing in the character of the vertex algebra $W^0(p)_{A_2}$, can be written as the sum of two depth two quantum modular forms of positive integral weight.

math.NT↗
arXiv2018

Indefinite theta functions arising in Gromov-Witten Theory of elliptic orbifolds

In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in Zwegers' thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim, and Lau's functions, inspired by recent work of Alexandrov, Banerjee, Manschot, and Pioline on similar functions. Combined with the work of Lau and Zhou, as well as the aforementioned work of Zwegers and two of the authors, this affords a complete understanding of the modularity transformation properties of the open Gromov-Witten potentials of elliptic orbifolds of the form $\mathbb P^1_{a,b,c}$ computed by Cho, Hong, Kim, and Lau. It is hoped that this will provide a fuller picture of the mirror-symmetric properties of these orbifolds in subsequent works.

math.NT↗
arXiv2017

Number-theoretic generalization of the Monster denominator formula

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(τ)$ in terms of the Hecke system of $\operatorname{SL}_2(\mathbb{Z})$-modular functions $j_n(τ)$. This formula can be reformulated entirely number-theoretically. Namely, it is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form in $τ$ with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. In this survey we discuss this generalization, and we offer an introduction to the theory of polar harmonic Maass forms. We conclude with applications to formulas of Ramanujan and Green's functions.

math.NT↗
arXiv2017

On a modularity conjecture of Andrews, Dixit, Schultz, and Yee for a variation of Ramanujan's $ω(q)$

We analyze the mock modular behavior of $\bar{P}_ω(q)$, a partition function introduced by Andrews, Dixit, Schultz, and Yee. This function arose in a study of smallest parts functions related to classical third order mock theta functions, one of which is $ω(q)$. We find that the modular completion of $\bar{P}_ω(q)$ is not simply a harmonic Maass form, but is instead the derivative of a linear combination of products of various harmonic Maass forms and theta functions. We precisely describe its behavior under modular transformations and find that the image under the Maass lowering operator lies in a relatively simpler space.

math.NT↗
arXiv2017

Taylor coefficients of non-holomorphic Jacobi forms and applications

In this paper, we prove modularity results of Taylor coefficients of certain non-holomorphic Jacobi forms. It is well-known that Taylor coefficients of holomorphic Jacobi forms are quasimoular forms. However recently there has been a wide interest for Taylor coefficients of non-holomorphic Jacobi forms for example arising in combinatorics. In this paper, we show that such coefficients still inherit modular properties. We then work out the precise spaces in which these coefficients lie for two examples.

math.NT↗
arXiv2017

Zhu reduction for Jacobi $n$-point functions and applications

We establish precise Zhu reduction formulas for Jacobi $n$-point functions which show the absence of any possible poles arising in these formulas. We then exploit this to produce results concerning the structure of strongly regular vertex operator algebras, and also to motivate new differential operators acting on Jacobi forms. Finally, we apply the reduction formulas to the Fermion model in order to create polynomials of quasi-Jacobi forms which are Jacobi forms.

math.QA↗
arXiv2017

Differential operators on polar harmonic Maass forms and elliptic duality

In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

math.NT↗
arXiv2017

On Divisors of Modular Forms

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(τ)$ given in terms of the Hecke system of $\operatorname{SL}_2(\mathbb Z)$-modular functions $j_n(τ)$. It is prominent in Zagier's seminal paper on traces of singular moduli, and in the Duncan-Frenkel work on Moonshine. The formula is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. We use these functions to study divisors of modular forms.

math.NT↗
arXiv2017

On some special families of $q$-hypergeometric Maass forms

Using special polynomials introduced by Hikami and the second author in their study of torus knots, we construct classes of $q$-hypergeometric series lying in the Habiro ring. These give rise to new families of quantum modular forms, and their Fourier coefficients encode distinguished Maass cusp forms. The cuspidality of these Maass waveforms is proven by making use of the Habiro ring representations of the associated quantum modular forms. Thus, we provide an example of using the $q$-hypergeometric structure of associated series to establish modularity properties which are otherwise non-obvious. We conclude the paper with a number of motivating questions and possible connections with Hecke characters, combinatorics, and still mysterious relations between $q$-hypergeometric series and {the passage} from positive to negative coefficients of Maass waveforms.

math.NT↗
arXiv2016

Images of Maass-Poincaré series in the lower half-plane

In this note we extend integral weight harmonic Maass forms to functions defined on the upper and lower half-planes using the method of Poincaré series. This relates to Rademacher's "expansion of zero" principle, which was recently employed by Rhoades to link mock theta functions and partial theta functions.

math.NT↗
arXiv2016

Polar harmonic Maass forms and their applications

In this survey, we present recent results of the authors about non-meromorphic modular objects known as polar harmonic Maass forms. These include the computation of Fourier coefficients of meromorphic modular forms and relations between inner products of meromorphic modular forms and higher Green's functions evaluated at CM-points.

math.NT↗
arXiv2016

Formulas for Jacobi forms and generalized Frobenius partitions

Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting combinatorial structure and led to proofs of a number of congruences. In this paper, we show how Andrews' generating functions can be cast in the framework of Eichler and Zagier's Jacobi forms. This reformulation allows us to compute explicit formulas for the generalized Frobenius partition generating functions (and in fact provides formulas for further functions of potential combinatorial interest), and it leads to a recursion formula to calculate them in terms of infinite $q$-products.

math.NT↗
arXiv2016

A classification of harmonic Maass forms

We give a classification of the Harish-Chandra modules generated by the pullback to $\text{SL}_2(\mathbb R)$ of harmonic Maass forms for congruence subgroups of $\text{SL}_2(\mathbb Z)$ with exponential growth allowed at the cusps. We assume that the weight is integral but include vector-valued forms. Due to the weak growth condition, these modules need not be irreducible. Elementary Lie algebra considerations imply that there are 9 possibilities, and we show, by giving explicit examples, that all of them arise from harmonic Maass forms. Finally, we briefly discuss the case of forms that are not harmonic but rather are annihilated by a power of the Laplacian, where much more complicated Harish-Chandra modules can arise.

math.NT↗
arXiv2016

Regularized inner products and errors of modularity

We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized inner products of weakly holomorphic modular forms of weights $0$ and $3/2$. These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic cohomology group introduced by Bruggeman, Choie, and the second author. This provides explicit representatives of the cohomology classes constructed abstractly and in a very general setting in their work.

math.NT↗
arXiv2016

Ramanujan-like formulas for Fourier coefficients of all meromorphic cusp forms

In this paper, we investigate Fourier expansions of meromorphic modular forms. Over the years, a number of special cases of meromorphic modular forms were shown to have Fourier expansions closely resembling the expansion of the reciprocal of the weight $6$ Eisenstein series which was computed by Hardy and Ramanujan. By investigating meromorphic modular forms within a larger space of so-called polar harmonic Maass forms, we prove in this paper that all negative-weight meromorphic modular forms (and furthermore all quasi-meromorpic modular forms) have Fourier expansions of this type, granted that they are bounded towards $i\infty$.

math.NT↗
arXiv2016

On a completed generating function of locally harmonic Maass forms

While investigating the Doi-Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itelf modular, it can be naturally completed to obtain a half-integral weight modular object.

math.NT↗