Mock theta functions and characters of N=3 superconformal modules
Minoru Wakimoto
本文利用 Zwegers 关于 mock theta 函数修正的理论，研究了 N=3 超共形模的特征标问题。
本文利用 Zwegers 关于 mock theta 函数修正的理论，研究 N=3 超共形模的特征标。
In this paper we study the characters of N=3 superconformal modules by using the Zwegers' theory on modification of mock theta functions.
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In this paper we study the branching functions of tensor products of N=3 superconformal modules.
We give a characterization of modified (in the sense of Zwegers) mock theta functions, parallel to that of ordinary theta functions. Namely, modified mock theta functions are characterized by their analyticity properties, elliptic transformation properties, and by being annihilated by certain second order differential operators.
We obtain an explicit formula for the mock theta function $Φ^{[m,s]}$ in the case when either $m$ or $s$ is a half of an odd integer by using the coroot lattice of $D(2,1;a)$. This enables us, together with the recurrence formula for $Φ^{[m,s]}$, to know $Φ^{[m,s]}$ for all $m$ and $s$. As its application, we deduce explicit formulas for the character and the supercharacter of N=3 modules obtained from the quantum Hamiltonian reduction of $\hat{osp}(3|2)$-modules.
Mock theta functions and characters of N=3 superconformal modules IV
本文利用李超代数 $D(2,1,a)$ 的余根格和 Kac-Peterson 恒等式，得到 mock theta 函数的显式公式，并用于证明 N=3 模张量积分支函数的猜想。
We study modular invariance of normalized supercharacters of tame integrable modules over an affine Lie superalgebra, associated to an arbitrary basic Lie superalgebra $ \mathfrak{g}. $ For this we develop a several step modification process of multivariable mock theta functions, where at each step a Zwegers' type "modifier" is used. We show that the span of the resulting modified normalized supercharacters is $ SL_2(\mathbb{Z}) $-invariant, with the transformation matrix equal, in the case the Killing form on $\mathfrak{g}$ is non-degenerate, to that for the subalgebra $ \mathfrak{g}^! $ of $…
It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an $SL_2(\mathbf{Z})$-invariant space. This result extends to admissible $\hat{\frak{g}}$-modules, where $\frak{g}$ is a simple Lie algebra or $osp_{1|n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $\hat{\frak{g}}$-modules when $\frak{g} =sl_2$ (resp. $=osp_{1|2}$) one obtains minimal series modules over the Virasoro (resp. $N=1$ superconformal algebras), which form modular invariant families. Another instance of modular invari…
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 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。
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…
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.
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 \…
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。
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。
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…
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。
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。
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…
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。
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…
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…
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.
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 $\hat{sl}(2)$…
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