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 functions of \(\mathfrak {sl}_3\), 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.

中文翻译：

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更高深度的量子模块形式，多个Eichler积分和$$ \ mathfrak {sl} _3 $$ sl 3个假theta函数

我们介绍和研究更高深度的量子模块化形式。我们构造了两个来自第2级假theta函数的例子，它们在下半平面的“同伴”也可以实现为双重Eichler积分和具有“ double error”函数值作为系数的非亚纯正theta级数。特别地，我们证明\（\ mathfrak {sl} _3 \）的错误theta函数出现在顶点代数\（W ^ 0（p）_ {A_2} \）的情况下，可以写为正整数权重的两个深度的两个量子模块形式的和。