S-duality, boundary states, and higher-form symmetries on ALE spaces
Abstract
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 S2× S2, and the natural pairing of the two ALE boundary states reproduces the standard Maxwell partition function on S2× S2. 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 AN−1 root lattice, reflecting the mixed electric-magnetic $1$-form anomaly. We also discuss gauging discrete ℤk 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.
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BibTeX
@misc{anber2026sduality,
title = {{$S$-duality, boundary states, and higher-form symmetries on ALE spaces}},
author = {Mohamed M. Anber},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.26224},
eprint = {2605.26224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.26224},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - S-duality, boundary states, and higher-form symmetries on ALE spaces
AU - Mohamed M. Anber
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2605.26224
UR - https://arxiv.org/abs/2605.26224
AB - 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× S^2, and the natural pairing of the two ALE boundary states reproduces the standard Maxwell partition function on S^2× 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 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.
ER -