We study the analytic structure of semiclassical conformal blocks, namely of the 1-point conformal block on the torus and of the 4-point conformal block on the sphere, as functions of the intermediate dimension. We interpret their discontinuities, which can be revealed with the use of a particular resummation procedure, holographically in terms of configurations of geodesics in AdS 3 . In other words, we study the behavior of the Ω-deformed N $$ \mathcal{N} $$ = 2 SYM theory in the Nekrasov-Shatashvili limit near those singular points, where naively the W-boson with non-vanishing angular momentum becomes massless. Upon a proper resummation of instanton contributions, these singularities disappear, which is similar to the Seiberg-Witten solution in the undeformed case where there is no massless W-boson. It is shown that the order parameter undergoes a non-analytic behavior near positions of the poles. The jump of the order parameter is interpreted holographically in terms of geodesic networks.

中文翻译：

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走向极点

我们研究半经典共形块的解析结构，即圆环上的 1 点共形块和球体上的 4 点共形块的解析结构，作为中间维度的函数。我们解释了它们的不连续性，这可以通过使用特定的恢复程序，全息地根据 AdS 3 中的测地线配置来揭示。换句话说，我们研究了 Ω 变形 N $$ \mathcal{N} $$ = 2 SYM 理论在这些奇异点附近的 Nekrasov-Shatashvili 极限中的行为，其中天真地具有非消失角动量的 W 玻色子变得没有质量。在对瞬时子贡献进行适当的总结后，这些奇点消失了，这类似于没有无质量 W 玻色子的未变形情况下的 Seiberg-Witten 解。结果表明，阶次参数在极点位置附近发生非解析行为。阶参数的跳跃是根据测地线网络全息解释的。