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Holomorphic Yang-Mills fields on $B$-branes

Andrés Viña

TL;DR

The work generalizes holomorphic gauge fields and Yang-Mills theory from vector bundles to B-branes, treating branes as objects in $D^b(X)$ and introducing a holomorphic gauge field as a lift to the first jet complex. The Atiyah class emerges as the obstruction to existence, and, when present, the gauge fields form a finite-dimensional affine space; for stratified manifolds and in particular flag varieties, uniqueness (or at most one) gauge field results are proved. The Yang-Mills functional is defined via curvature data on cohomology sheaves and extended to branes, leading to an algebraic description: Yang-Mills fields on a reflexive brane correspond to the zeros of $m$ cubic-degree equations in $m$ variables, where $m={\rm dim}\,{\rm Ext}^0({\mathscr F}^{\bullet},\Omega^1({\mathscr F}^{\bullet}))$. The paper thus bridges derived-category brane theory with holomorphic gauge theory, providing concrete computational frameworks (Euler-Poincaré mapping, cones, and generator sets) and yielding concrete bounds in the flag-variety case. These insights advance the mathematical understanding of holomorphic gauge theories in complex geometry and their categorical underpinnings.

Abstract

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.

Holomorphic Yang-Mills fields on $B$-branes

TL;DR

The work generalizes holomorphic gauge fields and Yang-Mills theory from vector bundles to B-branes, treating branes as objects in and introducing a holomorphic gauge field as a lift to the first jet complex. The Atiyah class emerges as the obstruction to existence, and, when present, the gauge fields form a finite-dimensional affine space; for stratified manifolds and in particular flag varieties, uniqueness (or at most one) gauge field results are proved. The Yang-Mills functional is defined via curvature data on cohomology sheaves and extended to branes, leading to an algebraic description: Yang-Mills fields on a reflexive brane correspond to the zeros of cubic-degree equations in variables, where . The paper thus bridges derived-category brane theory with holomorphic gauge theory, providing concrete computational frameworks (Euler-Poincaré mapping, cones, and generator sets) and yielding concrete bounds in the flag-variety case. These insights advance the mathematical understanding of holomorphic gauge theories in complex geometry and their categorical underpinnings.

Abstract

Considering -branes over a complex manifold as objects of the bounded derived category of coherent sheaves over , we define holomorphic gauge fields on -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 -branes. For a given -brane, we show that its Atiyah class is the obstruction to the existence of gauge fields. When is the variety of complete flags in a -dimensional complex vector space, we prove that any -brane over admits at most one holomorphic gauge field. Furthermore, we establish that the set of Yang-Mills fields on a given -brane, if nonempty, is in bijective correspondence with the points of an algebraic set defined by complex polynomials of degree less than four in indeterminates, where is the dimension of the space of morphisms from the brane to its tensor product with the sheaf of holomorphic one-forms.
Paper Structure (18 sections, 22 theorems, 105 equations)

This paper contains 18 sections, 22 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Holomorphic gauge fields on $B$-branes
  3. Holomorphic connections on a sheaf.
  4. The first jet sheaf
  5. Gauge fields on a $B$-brane
  6. $B$-branes on stratified manifolds
  7. Generators of a derived category
  8. Generators of the derived category $D^b(X)$.
  9. Flag varieties
  10. Holomorphic Yang-Mills fields
  11. Hermitian sheaves
  12. Yang-Mills functional
  13. Yang-Mills fields on reflexive sheaves
  14. The case ${\rm rank}\,{\mathscr F}=1$
  15. Yang-Mills fields on $B$-branes
  16. ...and 3 more sections

Key Result

Proposition 1

The vanishing of $a({\mathscr F}^{\bullet})$ is a necessary and sufficient condition for the existence of gauge fields on the brane ${\mathscr F}^{\bullet}.$ Furthermore, the set of gauge fields on ${\mathscr F}^{\bullet},$ if is nonempty, is an affine space over the finite dimensional vector space

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 1
  • Lemma 1
  • Corollary 1
  • Proposition 2
  • Proposition 3
  • ...and 27 more