AI 中文总结
研究为具有负接触列表的$(\PP^1,0,\infty)$平稳理论构造主2 - 托达tau序列，用无限楔李代数元素等编码，经特化恢复相关不变量，主tau序列更大使广田方程成耦合系统，借助多种定义和构造完成。
AI 中文摘要
我们为具有任意固定有限负接触列表的$(\PP^1,0,\infty)$的平稳相对理论构造了一个主2 - 托达tau序列。负接触块由无限楔李代数元素的指数的有限乘积编码，在$\infty$处的块取为整个相应左块的伴随。在加入独立的正轮廓托达时间后，得到的带电真空矩阵元素形成一个真正的2 - 托达tau序列。按照标准的$Q,u$特化，其正符号度系数恢复了具有规定正负接触的平稳相对格罗莫夫 - 威滕不变量，这些不变量出现在诱导的系数形式的广田方程中并受其约束。主tau序列比枚举扇区更大：它通常包含非零的、非真空的符号度为零的系数以及辅助非零格分量，对此未给出枚举解释。因此，广田方程形成一个耦合的主系统，并且在没有额外输入的情况下，仅靠正度不变量是不封闭的。该构造使用了负接触的大根定义、管的约翰逊 - 库马兰 - 吴算子公式以及2 - 托达层级的京都/普吕克构造。
英文摘要
We construct a master $2$-Toda tau sequence for the stationary relative theory of $(\PP^1,0,\infty)$ with any fixed finite lists of negative contacts. The negative-contact blocks are encoded by finite products of exponentials of infinite-wedge Lie algebra elements, with the block at $\infty$ taken as the adjoint of the entire corresponding left block. After adjoining independent positive-profile Toda times, the resulting charged-vacuum matrix elements form a genuine $2$-Toda tau sequence. Following the standard $Q,u$ specialization, its positive signed-degree coefficients recover the stationary relative Gromov--Witten invariants with the prescribed positive and negative contacts, which occur in and are constrained by the induced coefficientwise Hirota equations. The master tau sequence is larger than the enumerative sector: it generally contains nonzero, non-vacuum signed-degree-zero coefficients, as well as auxiliary nonzero lattice components, for which no enumerative interpretation is asserted. Consequently the Hirota equations form a coupled master system and do not, without additional input, close on the positive-degree invariants alone. The construction uses the large-root definition of negative contact, the Johnson--Kumaran--Wu operator formula for the tube, and the Kyoto/Plücker construction of the $2$-Toda hierarchy.
Commentsv2: 17 pages, small corrections. v1: 17 pages, AI generated, human reviewed, comments very welcome
