Dubrovin $\href{https://doi.org/10.1007/s00023-015-0449-2}{[10]}$ has shown that the spectrum of the quantization (with respect to the first Poisson structure) of the dispersionless Korteweg–de Vries (KdV) hierarchy is given by shifted symmetric functions; the latter are related by the Bloch–Okounkov Theorem $\href{https://doi.org/10.1007/JHEP07(2014)141}{[1]}$ to quasimodular forms on the full modular group. We extend the relation to quasimodular forms to the full quantum KdV hierarchy (and to the more general quantum Intermediate Long Wave hierarchy). These quantum integrable hierarchies have been defined by Buryak and Rossi $\href{https://doi.org/10.1007/s11005-015-0814-6}{[6]}$ in terms of the double ramification cycle in the moduli space of curves. The main tool and conceptual contribution of the paper is a general effective criterion for quasimodularity.

中文翻译：

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量子 KdV 层次结构和拟模形式


Dubrovin $\href{https://doi.org/10.1007/s00023-015-0449-2}{[10]}$ 表明无色散 Korteweg 的量化谱（相对于第一泊松结构） –de Vries (KdV) 层次结构由移位对称函数给出；后者通过 Bloch–Okounkov 定理 $\href{https://doi.org/10.1007/JHEP07(2014)141}{[1]}$ 与全模群上的拟模形式相关。我们将与准模形式的关系扩展到完整的量子 KdV 层次结构（以及更一般的量子中长波层次结构）。这些量子可积层次结构已由 Buryak 和 Rossi $\href{https://doi.org/10.1007/s11005-015-0814-6}{[6]}$ 根据模空间中的双分支循环定义曲线。本文的主要工具和概念贡献是拟模性的通用有效标准。