F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
Let XX be a compact Riemann surface of genus g≥2g\ge 2 and ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E6{E}_{6}-Higgs bundles over XX. We consider the automorphisms σ+{\sigma }_{+} of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ+(E,φ)=(E∗,−φt){\sigma }_{+}\lef...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-12-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2022-0543 |
| _version_ | 1811171924032618496 |
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| author | Antón-Sancho Álvaro |
| author_facet | Antón-Sancho Álvaro |
| author_sort | Antón-Sancho Álvaro |
| collection | DOAJ |
| description | Let XX be a compact Riemann surface of genus g≥2g\ge 2 and ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E6{E}_{6}-Higgs bundles over XX. We consider the automorphisms σ+{\sigma }_{+} of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ+(E,φ)=(E∗,−φt){\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}), induced by the action of the outer involution of E6{E}_{6} in ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}), and σ−{\sigma }_{-} defined by σ−(E,φ)=(E∗,φt){\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}), which results from the combination of σ+{\sigma }_{+} with the involution of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}), which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ+{\sigma }_{+} and σ−{\sigma }_{-}, as F4{F}_{4}-Higgs bundles, F4{F}_{4}-Higgs pairs associated with the fundamental irreducible representation of F4{F}_{4}, and PSp(8,C){\rm{PSp}}\left(8,{\mathbb{C}})-Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp(8,C){\rm{Sp}}\left(8,{\mathbb{C}}). Finally, we describe the reduced notions of semistability and polystability for these objects. |
| first_indexed | 2024-04-10T17:22:17Z |
| format | Article |
| id | doaj.art-0b85e6ecceb34fbf8835976cfd11dda5 |
| institution | Directory Open Access Journal |
| issn | 2391-5455 |
| language | English |
| last_indexed | 2024-04-10T17:22:17Z |
| publishDate | 2022-12-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj.art-0b85e6ecceb34fbf8835976cfd11dda52023-02-05T08:27:17ZengDe GruyterOpen Mathematics2391-54552022-12-012011723173310.1515/math-2022-0543F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surfaceAntón-Sancho Álvaro0Department of Mathematics and Experimental Science, School of Education Fray Luis de Leon, Catholic University of Ávila C/ Tirso de Molina, 44, 47010 Valladolid, SpainLet XX be a compact Riemann surface of genus g≥2g\ge 2 and ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E6{E}_{6}-Higgs bundles over XX. We consider the automorphisms σ+{\sigma }_{+} of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ+(E,φ)=(E∗,−φt){\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}), induced by the action of the outer involution of E6{E}_{6} in ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}), and σ−{\sigma }_{-} defined by σ−(E,φ)=(E∗,φt){\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}), which results from the combination of σ+{\sigma }_{+} with the involution of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}), which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ+{\sigma }_{+} and σ−{\sigma }_{-}, as F4{F}_{4}-Higgs bundles, F4{F}_{4}-Higgs pairs associated with the fundamental irreducible representation of F4{F}_{4}, and PSp(8,C){\rm{PSp}}\left(8,{\mathbb{C}})-Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp(8,C){\rm{Sp}}\left(8,{\mathbb{C}}). Finally, we describe the reduced notions of semistability and polystability for these objects.https://doi.org/10.1515/math-2022-0543higgs pairslie group e6 automorphismstabilityfixed points14d2014h1014h60 |
| spellingShingle | Antón-Sancho Álvaro F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface Open Mathematics higgs pairs lie group e6 automorphism stability fixed points 14d20 14h10 14h60 |
| title | F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface |
| title_full | F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface |
| title_fullStr | F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface |
| title_full_unstemmed | F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface |
| title_short | F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface |
| title_sort | f4 and psp 8 c higgs pairs understood as fixed points of the moduli space of e6 higgs bundles over a compact riemann surface |
| topic | higgs pairs lie group e6 automorphism stability fixed points 14d20 14h10 14h60 |
| url | https://doi.org/10.1515/math-2022-0543 |
| work_keys_str_mv | AT antonsanchoalvaro f4andpsp8chiggspairsunderstoodasfixedpointsofthemodulispaceofe6higgsbundlesoveracompactriemannsurface |