The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms
Let <i>X</i> be a compact Riemann surface of genus <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>≥</mo><mn>2</mn></mrow></semantics&...
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2025-03-01
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| author | Álvaro Antón-Sancho |
| author_facet | Álvaro Antón-Sancho |
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| description | Let <i>X</i> be a compact Riemann surface of genus <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> be the moduli space of polystable principal bundles over <i>X</i>, the structure group of which is the simple complex Lie group of exceptional type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula>. In this work, it is proved that the only automorphisms that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> admits are those defined as the pull-back action of an automorphism of the base curve <i>X</i>. The strategy followed uses specific techniques that arise from the geometry of the gauge group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula>. In particular, some new results that provide relations between the stability, simplicity, and irreducibility of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula>-bundles over <i>X</i> have been proved in the paper. The inclusion of groups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>2</mn></msub><mo>↪</mo><mo>Spin</mo><mrow><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula> is viewed as the fixed point subgroup of an order of 3 automorphisms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Spin</mo><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> that lifts the triality automorphism is also considered. Specifically, this inclusion induces the forgetful map of moduli spaces of principal bundles <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mrow><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>→</mo><mi>M</mi><mrow><mo>(</mo><mo>Spin</mo><mrow><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In the paper, it is also proved that the forgetful map is an embedding. Finally, some consequences are drawn from the results above on the geometry of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> in relation to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mo>Spin</mo><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-18fbdef733034fdcb20439e0e3c86c3f |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-18fbdef733034fdcb20439e0e3c86c3f2025-08-20T03:06:24ZengMDPI AGMathematics2227-73902025-03-01137108610.3390/math13071086The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and AutomorphismsÁlvaro Antón-Sancho0Department of Mathematics and Experimental Science, Fray Luis de Leon University College of Education, C/Tirso de Molina, 44, 47010 Valladolid, SpainLet <i>X</i> be a compact Riemann surface of genus <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> be the moduli space of polystable principal bundles over <i>X</i>, the structure group of which is the simple complex Lie group of exceptional type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula>. In this work, it is proved that the only automorphisms that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> admits are those defined as the pull-back action of an automorphism of the base curve <i>X</i>. The strategy followed uses specific techniques that arise from the geometry of the gauge group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula>. In particular, some new results that provide relations between the stability, simplicity, and irreducibility of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula>-bundles over <i>X</i> have been proved in the paper. The inclusion of groups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mn>2</mn></msub><mo>↪</mo><mo>Spin</mo><mrow><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mn>2</mn></msub></semantics></math></inline-formula> is viewed as the fixed point subgroup of an order of 3 automorphisms of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Spin</mo><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> that lifts the triality automorphism is also considered. Specifically, this inclusion induces the forgetful map of moduli spaces of principal bundles <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mrow><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>→</mo><mi>M</mi><mrow><mo>(</mo><mo>Spin</mo><mrow><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In the paper, it is also proved that the forgetful map is an embedding. Finally, some consequences are drawn from the results above on the geometry of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><msub><mi>G</mi><mn>2</mn></msub><mo>)</mo></mrow></semantics></math></inline-formula> in relation to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>(</mo><mo>Spin</mo><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/7/1086principal bundlemoduli space<i>G</i><sub>2</sub>automorphismfixed pointforgetful map |
| spellingShingle | Álvaro Antón-Sancho The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms Mathematics principal bundle moduli space <i>G</i><sub>2</sub> automorphism fixed point forgetful map |
| title | The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms |
| title_full | The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms |
| title_fullStr | The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms |
| title_full_unstemmed | The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms |
| title_short | The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms |
| title_sort | moduli space of principal i g i sub 2 sub bundles and automorphisms |
| topic | principal bundle moduli space <i>G</i><sub>2</sub> automorphism fixed point forgetful map |
| url | https://www.mdpi.com/2227-7390/13/7/1086 |
| work_keys_str_mv | AT alvaroantonsancho themodulispaceofprincipaligisub2subbundlesandautomorphisms AT alvaroantonsancho modulispaceofprincipaligisub2subbundlesandautomorphisms |