S-duality, boundary states, and higher-form symmetries on ALE spaces
- 1Durham University
Abstract
A bstract : 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.
Related
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BibTeX
@article{anber2026sduality,
title = {{S-duality, boundary states, and higher-form symmetries on ALE spaces}},
author = {Mohamed M. Anber},
journal = {Journal of High Energy Physics},
year = {2026},
month = oct,
doi = {10.1007/jhep10(2026)183},
url = {https://link.springer.com/article/10.1007/JHEP10(2026)183},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - S-duality, boundary states, and higher-form symmetries on ALE spaces
AU - Mohamed M. Anber
PY - 2026
DA - 2026/10/16
JO - Journal of High Energy Physics
DO - 10.1007/jhep10(2026)183
UR - https://link.springer.com/article/10.1007/JHEP10(2026)183
AB - A bstract : 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.
SN - 1029-8479
ER -