Related papers: Two $2/5$-level mock theta conjecture-like identit…

We apply the new framework for modularity of false theta functions developed by the second author and Nazaroglu to study the asymptotic behavior of Taylor coefficients of false Jacobi forms. The examples we study generate moments of the…

In 1983, Feingold and Frenkel discovered a relation between Siegel modular forms of genus two and a rank-three hyperbolic Kac--Moody algebra extending the affine Lie algebra of type $A_1$. It inspires a problem to explore more general…

In this paper, we construct some maps related to the motivic Galois action on depth-graded motivic multiple zeta values. And from these maps we give some short exact sequences about depth-graded motivic multiple zeta values in depth two and…

For positive integers $m$ we consider the theta functions $f_m(z):=\sum_{mk+1\text{ square }}q^k$. Due to classical identities of Jacobi, it is known that $$f_4\equiv f_6f_{12}\pmod 2.$$ Here we prove that the only triples $(a,b,c)$ for…

The Moore-Tachikawa conjecture is that each connected complex semisimple group $G$ determines a two-dimensional TQFT in a category of Hamiltonian symplectic varieties. While it would be worthwhile to prove this conjecture outright, our…

In this paper we prove a conjectured modular equation of Farkas and Kra, which involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjectured identity…

We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different…

We develop a setting in which one can evaluate certain Hecke-Rogers series in terms of false theta functions. We apply our setting to recent false theta function identities of Chan and Kim as well as Andrews and Warnaar.

In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit…

We study the depth filtration on multiple zeta values, the motivic Galois group of mixed Tate motives over $\mathbb{Z}$ and the Grothendieck-Teichm\"uller group, and its relation to modular forms. Using period polynomials for cusp forms for…

Ramanujan's 1920 last letter to Hardy contains seventeen examples of mock theta functions which he organized into three "orders." The most famous of these is the third-order function $f(q)$ which has received the most attention of any…

We study projective functors (i.e. direct summands of compositions of translations through walls) for parabolic versions of $\cO$ as well as for integral regular blocks outside the critical hyperplanes in the symmetrizable Kac-Moody case.…

We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we…

We propose a new approach to studying hyperbolic Kac-Moody algebras, focussing on the rank-3 algebra $\mathfrak{F}$ first investigated by Feingold and Frenkel. Our approach is based on the concrete realization of this Lie algebra in terms…

Starting from type IIB string theory on an $ADE$ singularity, the $(2,0)$ little string arises when one takes the string coupling $g_s$ to 0. In this setup, we give a unified description of the codimension-two defects of the little string,…

We study the parity of coefficients of classical mock theta functions. Suppose $g$ is a formal power series with integer coefficients, and let $c(g;n)$ be the coefficient of $q^n$ in its series expansion. We say that $g$ is of parity type…

The goal of these lectures is to present an informal but precise introduction to a body of concepts and methods of interest in number theory and string theory revolving around modular forms and their generalizations. Modular invariance lies…

In this article, using the language of jet space, we propose a functional class space for pseudo-local functionals. We test this functional class proposal in a number of examples ranging from string-field-theory to AdS/CFT dualities.…

The modular forms are revisited from a geometric and an algebraic point of view leading to a geometric interpretation of the weak Maass forms connecting them to the Ramanujan Mock Theta functions and to the cusp forms generated from the…

It is well known that Ramanujan conjectured congruences modulo powers of $5$, $7$ and and $11$ for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences…