BPS/CFT对应中的爆破，以及Painlevé VI
Blowups in BPS/CFT correspondence, and Painlevé VI
AI总结：
本文研究了带缺陷的四维超对称规范理论，提出关联不同形变参数配分函数的恒等式，借此推导出Painlevé VI的tau函数与Liouville理论共形块的关系公式，并探讨了其在Garnier-Schlesinger系统中的推广及相关物理应用。
摘要：
我们研究了存在面缺陷和点状缺陷（爆破）的四维超对称规范理论，并提出一个恒等式，关联不同$Ω$-形变参数$({\varepsilon}_{1}, {\varepsilon}_{2})$值下的配分函数。作为推论，我们得到了O. Gamayun、N. Iorgov和O. Lysovyy于2012年猜想的公式，将tau函数$τ_{PVI}$与Liouville理论的$c=1$共形块联系起来，并提出了其在Garnier-Schlesinger系统情形下的推广。为此，我们阐明了Painlevé VI的半经典tau函数$τ_{PVI}$及其推广的概念。我们还对球面配分函数、与环面流形上四维${\mathcal{N}}=2$理论相关的${\mathcal{N}}=(4,4)$ sigma模型中的边界算子乘积展开进行了评述，讨论了conifold上的交叉瞬子，阐明了BPZ/KZ对应的某些方面，以及其在量子化中的应用。
英文摘要：
We study four dimensional supersymmetric gauge theory in the presence of surface and point-like defects (blowups) and propose an identity relating partition functions at different values of $Ω$-deformation parameters $({\varepsilon}_{1}, {\varepsilon}_{2})$. As a consequence, we obtain the formula conjectured in 2012 by O$.$Gamayun, N$.$Iorgov, and O$.$Lysovyy, relating the tau-function $τ_{PVI}$ to $c=1$ conformal blocks of Liouville theory and propose its generalization for the case of Garnier-Schlesinger system. To this end we clarify the notion of the quasiclassical tau-function $τ_{PVI}$ of Painlevé VI and its generalizations. We also make some remarks about the sphere partition functions, the boundary operator product expansion in the ${\mathcal{N}}=(4,4)$ sigma models related to four dimensional ${\mathcal{N}}=2$ theories on toric manifolds, discuss crossed instantons on conifolds, elucidate some aspects of the BPZ/KZ correspondence, and applications to quantization.
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