Mock theta functions and characters of N=3 superconformal modules III

Mock theta functions and indefinite modular forms II

In this paper, we compute the Zwegers's modification of the mock theta functions $Φ^{[m,0] \, \ast}$ and study the modular transformation properties of the indefinite modular forms which appear in the explicit formula for the modified functions $\tildeΦ^{[m,0] \, \ast}$.

Mock theta函数与不定模形式

Mock theta functions and indefinite modular forms

本文研究从$D(2,1;a)$余根格得到的带符号mock theta函数的模变换性质，利用Zwegers修正理论证明其线性张成在$SL_2(\mathbf{Z})$下不变。

Representations of affine superalgebras and mock theta functions

We show that the normalized supercharacters of principal admissible modules over the affine Lie superalgebra $\hat{s\ell}_{2|1}$ (resp. $\hat{ps\ell}_{2|2}$) can be modified, using Zwegers' real analytic corrections, to form a modular (resp. $S$-) invariant family of functions. Applying the quantum Hamiltonian reduction, this leads to a new family of positive energy modules over the N=2 (resp. N=4) superconformal algebras with central charge $3(1-\frac{2m+2}{M})$, where $m \in \mathbb{Z}_{\geq 0}$, $M\in \mathbb{Z}_{\geq 2}$, $\gcd(2m+2,M)=1$ if $m>0$ (resp. $6(\frac{m}{M}-1)$, where $m \in \m…

Cone Vertex Algebras, Mock Theta Functions, and Umbral Moonshine Modules

We describe a family of indefinite theta functions of signature $(1,1)$ that can be expressed in terms of trace functions of vertex algebras built from cones in lattices. The family of indefinite theta functions considered has interesting connections with mock theta functions and Appell-Lerch sums. We use these relations to write the McKay-Thompson series of umbral moonshine at lambency $\ell=8,12,16$ in terms of trace functions of vertex algebras modules, and thereby provide the modules for these instances of umbral moonshine.

Representations of affine superalgebras and mock theta functions II

We show that the normalized supercharacters of principal admissible modules, associated to each integrable atypical module over the affine Lie superalgebra $\widehat{sl}_{2|1}$ can be modified, using Zwegers' real analytic corrections, to form an $SL_2(\mathbf{Z})$-invariant family of functions. Using a variation of Zwegers' correction, we obtain a similar result for $\widehat{osp}_{3|2}$. Applying the quantum Hamiltonian reduction, this leads to new families of positive energy modules over the $N=2$ (resp. $N=3$) superconformal algebras with central charge $c=3 (1-\frac{2m+2}{M})$, where $m \…

关于奇自旋A₁^(1)-字符串函数、交叉自旋恒等式以及类似假theta猜想的恒等式

On odd-spin $A_{1}^{(1)}$-string functions, cross-spin identities, and mock theta conjecture-like identities

本文研究了奇自旋A₁^(1)-字符串函数的极有限分解，并发现新的类似假theta猜想的恒等式。

广义仿费米子理论的弦函数、拟θ函数与伪θ函数

On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions

本文研究Kac-Moody代数$A_1^{(1)}$可容许表示弦函数，展示1/2水平弦函数的混合拟模性质及拟θ猜想类恒等式，证明负水平弦函数可由伪θ函数计算。

Demazure Flags, Chebyshev polynomials, Partial and Mock theta functions

We study the level $m$--Demazure flag of a level $\ell$--Demazure module for $\frak{sl}_2[t]$. We define the generating series $A_n^{\ell \rightarrow m}(x,q)$ which encodes the $q$--multiplicity of the level $m$ Demazure module of weight $n$. We establish two recursive formulae for these functions. We show that the specialization to $q=1$ is a rational function involving the Chebyshev polynomials. We give a closed form for $A_n^{\ell \rightarrow \ell+1}(x,q)$ and prove that it is given by a rational function. In the case when $m=\ell+1$ and $\ell=1,2$, we relate the generating series to partia…

关于某系列N=4超共形模的特征标

On the characters of a certain series of N=4 superconformal modules

本文研究由仿射李超代数主容许表示的量子哈密顿约化得到的N=4超共形模，证明其存在一系列特征标为模函数且可由Mumford theta函数显式写出的模。

新的极有限形式的广义欧拉恒等式用于$A_{1}^{(1)}$-弦函数和拟theta猜想-like恒等式

New polar-finite forms of generalized Euler identities for $A_{1}^{(1)}$-string functions and mock theta conjecture-like identities

本文研究了$A_{1}^{(1)}$-弦函数的极有限形式，利用准周期性方法推导出与拟theta猜想类似的恒等式，并引入了mock theta函数进行计算。

Mock Modular Mathieu Moonshine Modules

We construct super vertex operator algebras which lead to modules for moonshine relations connecting the four smaller sporadic simple Mathieu groups with distinguished mock modular forms. Starting with an orbifold of a free fermion theory, any subgroup of Co_0 that fixes a 3-dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain N=4 superconformal algebra. Similarly, any subgroup of Co_0 that fixes a 2-dimensional subspace of the 24-dimensional representation commutes with a certain N=2 superconformal algebra. Through the decomposition of the corre…

关于广义亚费米子理论的弦函数、mock theta函数与false theta函数，II

On string functions of the generalized parafermionic theories, mock theta functions, and false theta functions, II

本文通过发展容许弦函数的拟周期概念并计算Zagier--Zwegers极有限分解，扩展了仿射Kac--Moody代数$A_1^{(1)}$的结果，得到了用Ramanujan的mock theta函数表示的$1/3$、$2/3$和$1/5$层弦函数的新恒等式族。

Higher Depth False Modular Forms

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra $W^0(p)_{A_n}$, $1 \leq n \leq 3$, and from $\hat{Z}$-invari…

Integral Representations of Rank Two False Theta Functions and Their Modularity Properties

False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false t…

The Umbral Moonshine Module for the Unique Unimodular Niemeier Root System

We use canonically-twisted modules for a certain super vertex operator algebra to construct the umbral moonshine module for the unique Niemeier lattice that coincides with its root sublattice. In particular, we give explicit expressions for the vector-valued mock modular forms attached to automorphisms of this lattice by umbral moonshine. We also characterize the vector-valued mock modular forms arising, in which four of Ramanujan's fifth order mock theta functions appear as components.

Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra

We evaluate characters of irreducible representations of the N=2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG-resolution of the admissible N=2 representations and also a new 3,5,7...-resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every $\hat{sl}(2)$…