标题： 最高权向量，移位的拓扑递归和量子曲线 显示英文标题

标题： Highest weight vectors, shifted topological recursion and quantum curves

摘要： 我们通过考虑其划分函数为特定$\mathcal{W}$-代数表示的最高权向量的Airy结构，扩展了拓扑递归理论。 这样的最高权向量仅在表示满足某些条件时才作为Airy结构的划分函数出现。 在拓扑递归的谱曲线表述中，我们证明这种推广相当于向相关函数$ \omega_{g,1}$添加特定项，从而得到一个“偏移的拓扑递归”公式。 然后我们证明，由此偏移版本的拓扑递归构造出的波函数是带有$ \hbar$-依赖项的谱曲线的量化族的WKB解。 从反方向来看，从一个$\hbar$-联络出发，如果我们发现对于Airy结构所找到的相同条件也成立，则该联络为拓扑类型。 当这种情况发生时，由此产生的偏移环方程可以通过之前获得的偏移拓扑递归来求解。 显示英文摘要

摘要： We extend the theory of topological recursion by considering Airy structures whose partition functions are highest weight vectors of particular $\mathcal{W}$-algebra representations. Such highest weight vectors arise as partition functions of Airy structures only under certain conditions on the representations. In the spectral curve formulation of topological recursion, we show that this generalization amounts to adding specific terms to the correlators $ \omega_{g,1}$, which leads to a ``shifted topological recursion'' formula. We then prove that the wave-functions constructed from this shifted version of topological recursion are WKB solutions of families of quantizations of the spectral curve with $ \hbar$-dependent terms. In the reverse direction, starting from an $\hbar$-connection, we find that it is of topological type if the exact same conditions that we found for the Airy structures are satisfied. When this happens, the resulting shifted loop equations can be solved by the shifted topological recursion obtained earlier.