AI 中文总结
研究四维\(\mathcal{N}=2\)\(\mathrm{SU}(2)\)规范理论NS极限中休恩方程共振解，推导爆破方程重新求和奇异展开，揭示其解析结构，确定控制能隙闭合等的质量位点，开发并演示\(N_f=(1,1)\)理论的重新求和程序。
AI 中文摘要
我们研究了在具有基本超多重态的四维\(\mathcal{N}=2\)\(\mathrm{SU}(2)\)规范理论的涅克拉索夫 - 沙塔什维利（NS）极限中出现的休恩方程及其合流极限的共振解。在库仑分支参数\(a\)的共振位点\(2a / \hbar \in \mathbb{Z}\)处，弗洛凯乘数合并，体和表面缺陷NS函数的瞬子展开产生阶数不断增加的极点。我们推导了仅涉及NS函数的爆破方程，并用它们来重新求和这些奇异展开。由此得到的重新求和的体和表面缺陷NS函数揭示了规范理论解在共振位点附近的解析结构，包括被逐项瞬子展开所掩盖的辅助参数和弗洛凯解的分支结构。在共振时，重新求和的辅助参数和适当归一化的缺陷波函数允许有限极限，这些极限描述了谱隙边缘的周期或反周期解，并使我们能够构建它们的对数伴随。然后，我们确定了控制能隙闭合和半单共振单值性的不同嵌套质量位点。在较大的位点上，带边辅助参数合并，而在较小的位点上，两个独立的共振（反）周期弗洛凯解幸存。我们为\(n_i \leq 2\)（\(i = 0,1\)）的\(N_f = (n_0,n_1)\)理论开发了一般的重新求和程序，并针对\(N_f = (1,1)\)理论进行了明确演示。
英文摘要
We study resonant solutions of the Heun equation and its confluent limits that arise in the Nekrasov-Shatashvili (NS) limit of four-dimensional $\mathcal{N}=2$ $\mathrm{SU}(2)$ gauge theories with fundamental hypermultiplets. At the resonant loci $2a/\hbar\in\mathbb{Z}$ in the Coulomb branch parameter $a$, the Floquet multipliers coalesce and the instanton expansions of the bulk and surface defect NS functions develop poles of increasing order. We derive blow-up equations involving exclusively NS functions and use them to resum these singular expansions. The resulting resummed bulk and surface defect NS functions reveal the analytic structure of the gauge-theoretic solutions near the resonant loci, including the branch structure of the accessory parameter and of the Floquet solutions that is obscured by the term-by-term instanton expansion. At resonance, the resummed accessory parameters and suitably normalized defect wavefunctions admit finite limits that describe periodic or antiperiodic solutions at the edges of spectral gaps and allow us to construct their logarithmic companions. We then identify distinct nested mass loci governing gap closure and semisimple resonant monodromy. On the larger locus the band-edge accessory parameters coalesce, while on the smaller locus two independent resonant (anti)periodic Floquet solutions survive. We develop the general resummation procedure for $N_f=(n_0,n_1)$ theories with $n_i\leq 2$ $(i=0,1)$, and demonstrate it explicitly for the $N_f=(1,1)$ theory.
Comments47 pages, 1 figure
