We study the relation between class \(\mathcal {S}\) theories on punctured tori and isomonodromic deformations of flat SL(N) connections on the two-dimensional torus with punctures. Turning on the self-dual \(\Omega \)-background corresponds to a deautonomization of the Seiberg–Witten integrable system which implies a specific time dependence in its Hamiltonians. We show that the corresponding \(\tau \)-function is proportional to the dual gauge theory partition function, the proportionality factor being a nontrivial function of the solution of the deautonomized Seiberg–Witten integrable system. This is obtained by mapping the isomonodromic deformation problem to \(W_N\) free fermion correlators on the torus.

我们研究了关于穿孔环面的类\(\mathcal {S}\)理论与带穿孔的二维环面上的平面SL ( N ) 连接的等单向变形之间的关系。打开自对偶\(\Omega \)背景对应于 Seiberg-Witten 可积系统的非自治化，这意味着其哈密顿量具有特定的时间依赖性。我们证明了相应的\(\tau \)函数与对偶规范论的配分函数成正比，比例因子是非自治化 Seiberg-Witten 可积系统解的非平凡函数。这是通过将等单向变形问题映射到\(W_N\) 获得的 环面上的自由费米子相关器。