高级Appell函数、模变换和字符
Higher-Level Appell Functions, Modular Transformations, and Characters
摘要：
研究了无穷维李超代数的特征标所涉及的一类不定θ级数的模变换性质。该\textit{level-$\ell$ Appell functions} $K_\ell$满足开放的准周期性关系与附加的θ函数项出现在翻译的“周期”。将θ函数的公知解释推广为线丛的部分，$K_\ell$函数进入秩- $(\ell+1)$束$V(\ell,τ)$的部分的构造。我们评估的$K_\ell$功能的模块化的转换，并构造SL（2，Z）子群的作用，使$V(\ell,τ)$的部分构造从$K_\ell$不变。将$K_\ell$的模变换性质应用于仿射李超代数^sl（2| 1）在有理水平k>-1和N=2的超Virasoro代数上，导出了具有“可容许”特征标的模变换，这些特征标在谱流下不是周期的，因而不能用θ函数有理地表示.这给出了一个例子，其中构造一个模群作用涉及在非有理共形模型中的表示之间的扩展。
英文摘要：
We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The \textit{level-$\ell$ Appell functions} $K_\ell$ satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the ``period.'' Generalizing the well-known interpretation of theta functions as sections of line bundles, the $K_\ell$ function enters the construction of a section of a rank-$(\ell+1)$ bundle $V(\ell,τ)$. We evaluate modular transformations of the $K_\ell$ functions and construct the action of an SL(2,Z) subgroup that leaves the section of $V(\ell,τ)$ constructed from $K_\ell$ invariant. Modular transformation properties of $K_\ell$ are applied to the affine Lie superalgebra ^sl(2|1) at rational level k>-1 and to the N=2 super-Virasoro algebra, to derive modular transformations of ``admissible'' characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.
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