The Moduli Space of Principal <i>G</i><sub>2</sub>-Bundles and Automorphisms

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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Bibliographic Details
Main Author:	Álvaro Antón-Sancho
Format:	Article
Language:	English
Published:	MDPI AG 2025-03-01
Series:	Mathematics
Subjects:	principal bundle moduli space <i>G</i><sub>2</sub> automorphism fixed point forgetful map
Online Access:	https://www.mdpi.com/2227-7390/13/7/1086
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Summary:	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>.
ISSN:	2227-7390