The moduli space of symplectic bundles over a compact Riemann surface and quaternionic structures

Álvaro Antón-Sancho; Álvaro Antón-Sancho

doi:10.3934/math.2025604

AIMS Mathematics

2025, Volume 10, Issue 6: 13451-13475. doi: 10.3934/math.2025604

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Research article

The moduli space of symplectic bundles over a compact Riemann surface and quaternionic structures

Álvaro Antón-Sancho ^{1,2
,
,}

1.
Department of Mathematics and Experimental Science, Fray Luis de León University College of Education, C/Tirso de Molina, 44, 47010 Valladolid, Spain
2.
Catholic University of Ávila, C/Canteros s/n, 05005 Ávila, Spain

Received: 16 April 2025 Revised: 26 May 2025 Accepted: 06 June 2025 Published: 11 June 2025
MSC : 14H60, 14H10, 57R57

Let $ G = \text{Sp}(4, \mathbb{C}) $, $ K_G $ be a maximal compact subgroup of $ G $, $ H $ be the subgroup of $ G $ generated by one of the non-trivial elements of the quaternion group, viewed as a subgroup of $ G $, and $ X $ be a compact Riemann surface of genus $ g\geq 2 $. The main result of this paper proves that the forgetful map $ F:M(H)\to M(G) $ between moduli spaces of principal bundles over $ X $ induced by the inclusion $ H\hookrightarrow G $ is a closed embedding. From this, $ M(H) $ can be understood as a closed subvariety of $ M(G) $. Moreover, some applications of this result are provided. In particular, it is proved that the bundles in the image of $ F $ admit a quaternionic structure and also a reduction of the structure group to $ \text{Sp}(2, \mathbb{H}) $. From this, some topological constraints are given, including that the image of the forgetful map falls in a single connected component of $ M(G) $. In addition, some applications are provided concerning the representation space $ \mathcal{R}(\pi_1(X), K_G) $, which, by the Narasimhan-Seshadri-Ramanathan correspondence, is isomorphic to $ M(G) $. Specifically, the image of the forgetful map is proved to correspond to the fixed point subset of a certain subvariety of $ \mathcal{R}(\pi_1(X), K_G) $.
- symplectic group,
- injectivity,
- principal bundle,
- moduli space,
- representation
Citation: Álvaro Antón-Sancho. The moduli space of symplectic bundles over a compact Riemann surface and quaternionic structures[J]. AIMS Mathematics, 2025, 10(6): 13451-13475. doi: 10.3934/math.2025604

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Abstract

Let $ G = \text{Sp}(4, \mathbb{C}) $, $ K_G $ be a maximal compact subgroup of $ G $, $ H $ be the subgroup of $ G $ generated by one of the non-trivial elements of the quaternion group, viewed as a subgroup of $ G $, and $ X $ be a compact Riemann surface of genus $ g\geq 2 $. The main result of this paper proves that the forgetful map $ F:M(H)\to M(G) $ between moduli spaces of principal bundles over $ X $ induced by the inclusion $ H\hookrightarrow G $ is a closed embedding. From this, $ M(H) $ can be understood as a closed subvariety of $ M(G) $. Moreover, some applications of this result are provided. In particular, it is proved that the bundles in the image of $ F $ admit a quaternionic structure and also a reduction of the structure group to $ \text{Sp}(2, \mathbb{H}) $. From this, some topological constraints are given, including that the image of the forgetful map falls in a single connected component of $ M(G) $. In addition, some applications are provided concerning the representation space $ \mathcal{R}(\pi_1(X), K_G) $, which, by the Narasimhan-Seshadri-Ramanathan correspondence, is isomorphic to $ M(G) $. Specifically, the image of the forgetful map is proved to correspond to the fixed point subset of a certain subvariety of $ \mathcal{R}(\pi_1(X), K_G) $.

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