Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann Surface
In this paper, the fixed points of involutions on the moduli space of principal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>6</mn></msub></semantics>&l...
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2025-05-01
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| author | Álvaro Antón-Sancho |
| author_facet | Álvaro Antón-Sancho |
| author_sort | Álvaro Antón-Sancho |
| collection | DOAJ |
| description | In this paper, the fixed points of involutions on the moduli space of principal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>6</mn></msub></semantics></math></inline-formula>-bundles over a compact Riemann surface <i>X</i> are investigated. In particular, it is proved that the combined action of a representative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> of the outer involution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>6</mn></msub></semantics></math></inline-formula> with the pull-back action of a surface involution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> admits fixed points if and only if a specific topological obstruction in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mn>2</mn></msup><mfenced separators="" open="(" close=")"><mi>X</mi><mo>/</mo><mi>τ</mi><mo>,</mo><msub><mi>π</mi><mn>0</mn></msub><mfenced separators="" open="(" close=")"><msubsup><mi>E</mi><mn>6</mn><mi>σ</mi></msubsup></mfenced></mfenced></mrow></semantics></math></inline-formula> vanishes. For an involution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>k</mi></mrow></semantics></math></inline-formula> fixed points, it is proved that the fixed point set is isomorphic to the moduli space of principal <i>H</i>-bundles over the quotient curve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, where <i>H</i> is either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mn>4</mn></msub></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">PSp</mo><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> and it consists of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mrow><mi>g</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> components. The complex dimensions of these components are computed, and their singular loci are determined as corresponding to <i>H</i>-bundles admitting non-trivial automorphisms. Furthermore, it is checked that the stability of fixed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>6</mn></msub></semantics></math></inline-formula>-bundles implies the stability of their corresponding <i>H</i>-bundles over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, and the behavior of characteristic classes is discussed under this correspondence. Finally, as an application of the above results, it is proved that the fixed points correspond to octonionic structures on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, and an explicit construction of these octonionic structures is provided. |
| format | Article |
| id | doaj-art-55f3a794388d4b65ac5d0fec8e31528f |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-55f3a794388d4b65ac5d0fec8e31528f2025-08-20T03:27:26ZengMDPI AGAxioms2075-16802025-05-0114642310.3390/axioms14060423Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann SurfaceÁlvaro Antón-Sancho0Department of Mathematics and Experimental Science, Fray Luis de Leon University College, Catholic University of Ávila, C/Tirso de Molina, 44, 47010 Valladolid, SpainIn this paper, the fixed points of involutions on the moduli space of principal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>6</mn></msub></semantics></math></inline-formula>-bundles over a compact Riemann surface <i>X</i> are investigated. In particular, it is proved that the combined action of a representative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> of the outer involution of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>6</mn></msub></semantics></math></inline-formula> with the pull-back action of a surface involution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> admits fixed points if and only if a specific topological obstruction in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mn>2</mn></msup><mfenced separators="" open="(" close=")"><mi>X</mi><mo>/</mo><mi>τ</mi><mo>,</mo><msub><mi>π</mi><mn>0</mn></msub><mfenced separators="" open="(" close=")"><msubsup><mi>E</mi><mn>6</mn><mi>σ</mi></msubsup></mfenced></mfenced></mrow></semantics></math></inline-formula> vanishes. For an involution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mi>k</mi></mrow></semantics></math></inline-formula> fixed points, it is proved that the fixed point set is isomorphic to the moduli space of principal <i>H</i>-bundles over the quotient curve <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, where <i>H</i> is either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>F</mi><mn>4</mn></msub></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">PSp</mo><mo>(</mo><mn>8</mn><mo>,</mo><mi mathvariant="double-struck">C</mi><mo>)</mo></mrow></semantics></math></inline-formula> and it consists of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mrow><mi>g</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> components. The complex dimensions of these components are computed, and their singular loci are determined as corresponding to <i>H</i>-bundles admitting non-trivial automorphisms. Furthermore, it is checked that the stability of fixed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mn>6</mn></msub></semantics></math></inline-formula>-bundles implies the stability of their corresponding <i>H</i>-bundles over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, and the behavior of characteristic classes is discussed under this correspondence. Finally, as an application of the above results, it is proved that the fixed points correspond to octonionic structures on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>/</mo><mi>τ</mi></mrow></semantics></math></inline-formula>, and an explicit construction of these octonionic structures is provided.https://www.mdpi.com/2075-1680/14/6/423principal bundlefixed pointinvolutionRiemann surfaceautomorphismoctonions |
| spellingShingle | Álvaro Antón-Sancho Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann Surface Axioms principal bundle fixed point involution Riemann surface automorphism octonions |
| title | Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann Surface |
| title_full | Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann Surface |
| title_fullStr | Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann Surface |
| title_full_unstemmed | Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann Surface |
| title_short | Involutions of the Moduli Space of Principal <i>E</i><sub>6</sub>-Bundles over a Compact Riemann Surface |
| title_sort | involutions of the moduli space of principal i e i sub 6 sub bundles over a compact riemann surface |
| topic | principal bundle fixed point involution Riemann surface automorphism octonions |
| url | https://www.mdpi.com/2075-1680/14/6/423 |
| work_keys_str_mv | AT alvaroantonsancho involutionsofthemodulispaceofprincipalieisub6subbundlesoveracompactriemannsurface |