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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2407.06193 |
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| _version_ | 1866913772673695744 |
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| author | Viña, Andrés |
| author_facet | Viña, Andrés |
| contents | Considering $B$-branes over a complex manifold $X$ as objects of the bounded derived category of coherent sheaves over $X$, we define holomorphic gauge fields on $B$-branes and introduce the Yang-Mills functional for these fields. These definitions extend well-known concepts in the context of vector bundles to the setting of $B$-branes.
For a given $B$-brane, we show that its Atiyah class is the obstruction to the existence of gauge fields. When $X$ is the variety of complete flags in a $3$-dimensional complex vector space, we prove that any $B$-brane over $X$ admits at most one holomorphic gauge field.
Furthermore, we establish that the set of Yang-Mills fields on a given $B$-brane, if nonempty, is in bijective correspondence with the points of an algebraic set defined by $m$ complex polynomials of degree less than four in $m$ indeterminates, where $m$ is the dimension of the space of morphisms from the brane to its tensor product with the sheaf of holomorphic one-forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06193 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Holomorphic Yang-Mills fields on $B$-branes Viña, Andrés Algebraic Geometry Mathematical Physics 2020: 53C05, 58E15, 18G10 Considering $B$-branes over a complex manifold $X$ as objects of the bounded derived category of coherent sheaves over $X$, we define holomorphic gauge fields on $B$-branes and introduce the Yang-Mills functional for these fields. These definitions extend well-known concepts in the context of vector bundles to the setting of $B$-branes. For a given $B$-brane, we show that its Atiyah class is the obstruction to the existence of gauge fields. When $X$ is the variety of complete flags in a $3$-dimensional complex vector space, we prove that any $B$-brane over $X$ admits at most one holomorphic gauge field. Furthermore, we establish that the set of Yang-Mills fields on a given $B$-brane, if nonempty, is in bijective correspondence with the points of an algebraic set defined by $m$ complex polynomials of degree less than four in $m$ indeterminates, where $m$ is the dimension of the space of morphisms from the brane to its tensor product with the sheaf of holomorphic one-forms. |
| title | Holomorphic Yang-Mills fields on $B$-branes |
| topic | Algebraic Geometry Mathematical Physics 2020: 53C05, 58E15, 18G10 |
| url | https://arxiv.org/abs/2407.06193 |