标题： 电磁对偶性在$\mathcal{N}=4$超 Yang-Mills 理论中的线缺陷关联函数中的应用 显示英文标题

标题： Electromagnetic Duality for Line Defect Correlators in $\mathcal{N}=4$ Super Yang-Mills Theory

摘要： 我们研究了在存在一个带有电磁荷标记的半BPS线缺陷时，电磁荷为$(p,q)$的应力张量超共形多重态的两个超共形主算符的特定积分关联函数，该理论是具有群$SU(N)$的$\mathcal{N}=4$超对称杨-米尔斯理论（SYM）。$\mathcal{N}=4$SYM 中${\rm SL}(2,\mathbb{Z})$电磁对偶的一个重要结果是，当复耦合常数$\tau=\theta/(2\pi)+4\pi i /g_{_{\rm YM}}^2$适当变换时，具有不同电荷$(p,q)$的线缺陷算符的关联函数必须以非平凡的方式相互关联。 在本工作中，我们引入了一类新的实解析函数，其在${\rm SL}(2,\mathbb{Z})$下的自守性质与$\mathcal{N}=4$SYM 中线缺陷算符在电磁对偶性下的预期变换相匹配。 在大$N$和固定$\tau$的情况下，我们考虑的相关函数与全息对偶的 IIB 超弦理论中扩展$(p,q)$-字符串的两个引力子散射振幅有关。 我们证明了积分两点线缺陷相关函数的大-$N$展开系数由属于此类自守函数的元素的有限线性组合给出，组合系数为有理数。 另一方面，对于任何固定值的$N$，我们猜想线缺陷积分相关函数可以表示为这类自守函数的正式无限级数。 该级数的重求和为积分线缺陷相关函数提供了一个简单的格点求和表达式，体现了其自守性质。 我们明确地对此构造进行了演示，适用于规范群为$SU(2)$和$SU(3)$的情况。 我们的结果可以直接获取存在 't Hooft 线缺陷时的非微扰积分关联函数，这些可观测量通过其他方法很难计算。 显示英文摘要

摘要： We study particular integrated correlation functions of two superconformal primary operators of the stress tensor multiplet in the presence of a half-BPS line defect labelled by electromagnetic charges $(p,q)$ in $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) with gauge group $SU(N)$. An important consequence of ${\rm SL}(2,\mathbb{Z})$ electromagnetic duality in $\mathcal{N}=4$ SYM is that correlators of line defect operators with different charges $(p,q)$ must be related in a non-trivial manner when the complex coupling $\tau=\theta/(2\pi)+4\pi i /g_{_{\rm YM}}^2$ is transformed appropriately. In this work we introduce a novel class of real-analytic functions whose automorphic properties with respect to ${\rm SL}(2,\mathbb{Z})$ match the expected transformations of line defect operators in $\mathcal{N}=4$ SYM under electromagnetic duality. At large $N$ and fixed $\tau$, the correlation functions we consider are related to scattering amplitudes of two gravitons from extended $(p,q)$-strings in the holographic dual type IIB superstring theory. We show that the large-$N$ expansion coefficients of the integrated two-point line defect correlators are given by finite linear combinations with rational coefficients of elements belonging to this class of automorphic functions. On the other hand, for any fixed value of $N$ we conjecture that the line defect integrated correlators can be expressed as formal infinite series over such automorphic functions. The resummation of this series produces a simple lattice sum representation for the integrated line defect correlator that manifests its automorphic properties. We explicitly demonstrate this construction for the cases with gauge group $SU(2)$ and $SU(3)$. Our results give direct access to non-perturbative integrated correlators in the presence of an 't Hooft-line defect, observables otherwise very difficult to compute by other means.