It is a well-known result by Hanany and Witten that, when two five-branes move across each other, D3-branes stretching between them are generated. Later the same brane configurations played a crucial role in understanding the worldvolume theory of multiple M2-branes. Recently the partition function of multiple M2-branes was transformed to the Fredholm determinant for quantum algebraic curves, where the characteristic 3/2 power law of degrees of freedom is reproduced and the determinant enjoys large symmetries given by exceptional Weyl groups. The large exceptional Weyl group reproduces the Hanany-Witten brane transitions and, besides, contains brane transitions unknown previously. Aiming at understanding the new brane transitions better, we generalize our previous study on the $D_5$ quantum curve to the $E_7$ case, which requires delicate handling of degeneracies. By combining the results of these two cases, we propose a “local” rule for the brane transitions.

Hanany 和 Witten 的一个众所周知的结果是，当两个五膜相互移动时，会产生在它们之间伸展的 D3 膜。后来相同的膜配置在理解多个 M2 膜的世界体积理论方面发挥了关键作用。最近，多个 M2-膜的配分函数被转换为量子代数曲线的 Fredholm 行列式，其中重现了自由度的特征 3/2 幂律，并且行列式享有由例外 Weyl 群给出的大对称性。大的特殊 Weyl 群再现了 Hanany-Witten 膜跃迁，此外，还包含以前未知的膜跃迁。为了更好地理解新的膜转变，我们概括了我们之前对$D_5$ 量子曲线 ${乙}_7$情况，这需要对退化进行精细处理。通过结合这两种情况的结果，我们提出了膜转换的“局部”规则。