$S$-duality, boundary states, and higher-form symmetries on ALE spaces
$S$-对偶、边界态以及ALE空间上的高阶形式对称性
Mohamed M. Anber
AI总结 研究ALE空间上Maxwell理论的阿贝尔$S$-对偶,通过将路径积分分解为theta函数块并解释为边界态,揭示了向量值模协变性,并探讨了1-形式对称性背景下的混合反常。
Comments 40 pages+appendices, 4 figures
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我们研究$A$型渐近局部欧几里得(ALE)空间上Maxwell理论的阿贝尔$S$-对偶。与闭四流形不同,ALE空间上的Maxwell路径积分自然不是一个标量配分函数。相反,它分解为theta函数块,这些块由渐近透镜空间边界上的平坦$U(1)$和乐扇区标记。我们将这些块解释为由ALE路径积分准备的希尔伯特空间边界态的分量。根据这种解释,普通模性的明显失效被模群作用下的向量值模协变性所取代。我们通过将Eguchi-Hanson空间与其定向反转粘合来明确检验这一图像。得到的闭四流形微分同胚于$S^2\times S^2$,两个ALE边界态的自然配对再现了$S^2\times S^2$上的标准Maxwell配分函数。然后,我们通过打开电和磁$1$-形式对称性背景来精炼构造。在它们存在的情况下,ALE theta块不是普通函数,而是与$A_{N-1}$根格相关联的Cartan环面上的线丛的截面,反映了电-磁混合$1$-形式反常。我们还讨论了规范$1$-形式对称性的离散$\mathbb Z_k$子群,并表明向量值边界态结构在规范后仍然是自然的协变框架。在这个意义上,ALE空间表现为四维Maxwell理论的手征构建块:单个ALE块携带扇区分辨的边界数据,而粘合将这些扇区配对以产生一个普通的闭流形配分函数,类似于二维CFT中左、右移动共形块的配对。
We study Abelian $S$-duality of Maxwell theory on $A$-type asymptotically locally Euclidean (ALE) spaces. Unlike on closed four-manifolds, the Maxwell path integral on an ALE space is not naturally a scalar partition function. Rather, it decomposes into theta-function blocks labeled by flat $U(1)$ holonomy sectors on the asymptotic lens-space boundary. We interpret these blocks as components of the Hilbert-space boundary state prepared by the ALE path integral. With this interpretation, the apparent failure of ordinary modularity is replaced by vector-valued modular covariance under the action of the modular group. We test this picture explicitly for Eguchi-Hanson space by gluing it to its orientation reversal. The resulting closed four-manifold is diffeomorphic to $S^2\times S^2$, and the natural pairing of the two ALE boundary states reproduces the standard Maxwell partition function on $S^2\times S^2$. We then refine the construction by turning on electric and magnetic $1$-form symmetry backgrounds. In their presence, the ALE theta blocks are not ordinary functions, but sections of a line bundle over the Cartan torus associated with the $A_{N-1}$ root lattice, reflecting the mixed electric-magnetic $1$-form anomaly. We also discuss gauging discrete $\mathbb Z_k$ subgroups of the $1$-form symmetries and show that the vector-valued boundary-state structure remains the natural covariant framework after gauging. In this sense, ALE spaces behave as chiral building blocks for four-dimensional Maxwell theory: individual ALE blocks carry sector-resolved boundary data, while gluing pairs these sectors to produce an ordinary closed-manifold partition function, much like the pairing of left- and right-moving conformal blocks in two-dimensional CFT.