论文标题
周围旋转Hurwitz数字
Around spin Hurwitz numbers
论文作者
论文摘要
我们对旋转Hurwitz数字进行了评论，该数字用自旋结构计算了受损坏的覆盖物。它们与特殊的$ q $ schur函数有关，这些功能实际上与Sergeev组的字符有关。这样一来，人们就可以将整个故事纳入矩阵模型和可集成层次结构的现代背景。 Hurwitz分区功能实际上比常规$τ$函数更广泛，但在特定情况下会减少。我们解释一下，KDV和Veselov-Novikov层次结构的特殊$ d $ -soliton $τ$ - 生成旋转hurwitz数字$ h^\ pm \ left（γ^r_d \ right）$和$ h^\ pm pm \ weft（γ^r_d，δ\ right）$。旋转Hurwitz数字的生成功能是BKP可集成层次结构的超几何$τ$ - 我们提出了他们的费米金实现。我们还解释了如何完全根据$ q $ schur函数构建此类型的$τ$插入。在这种方法中，在特殊基因座的$ q $ schur功能的分解公式中发挥了重要作用。
We present a review of the spin Hurwitz numbers, which count the ramified coverings with spin structures. They are related to peculiar $Q$ Schur functions, which are actually related to characters of the Sergeev group. This allows one to put the whole story into the modern context of matrix models and integrable hierarchies. Hurwitz partition functions are actually broader than conventional $τ$-functions, but reduce to them in particular circumstances. We explain, how a special $d$-soliton $τ$-functions of KdV and Veselov-Novikov hierarchies generate the spin Hurwitz numbers $H^\pm\left( Γ^r_d \right)$ and $H^\pm\left( Γ^r_d,Δ\right)$. The generating functions of the spin Hurwitz numbers are hypergeometric $τ$-functions of the BKP integrable hierarchy, and we present their fermionic realization. We also explain how one can construct $τ$-functions of this type entirely in terms of the $Q$ Schur functions. An important role in this approach is played by factorization formulas for the $Q$ Schur functions on special loci.
