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Holomorphic Gauge Fields on $B$-Branes

Andrés Viña

TL;DR

This work extends holomorphic gauge fields from holomorphic vector bundles to B-branes, formulated via the $1$-jet sequence and Atiyah class in the derived category. It constructs a disjoint-support generating family ${S_1,...,S_{n+1}}$ for $D^b({\mathbb P}^n)$ by an iterative hypersurface-with-affine-complement approach, enabling a tight control of morphisms to $\Omega^1$ objects. A key result is that ${\rm Hom}_{D^b({\mathbb P}^n)}(F^{\centerdot}, \Omega^1(F^{\centerdot}))=0$ for any B-brane built from the generators, implying the set of holomorphic gauge fields on any B-brane over ${\mathbb P}^n$ has cardinality $0$ or $1$ (i.e., is strictly less than $2$). The paper also indicates a generalization: if a complex manifold $Y$ admits a tower of divisors with affine complement and vanishing $H^{1,0}$, the same bound holds for B-branes over $Y$. Altogether, the results reveal a strong rigidity of holomorphic gauge structures on B-branes in projective and related settings, connecting derived-category generators to gauge-field moduli.

Abstract

Considering the $B$-branes over a complex manifold as the objects of the bounded derived category of coherent sheaves on that manifold, we extend the definition of holomorphic gauge fields on vector bundles to $B$-branes. We construct a family of coherent sheaves on the complex projective space, which generates the corresponding bounded derived category and such that the supports of the elements of this family are two by two disjoint. Using that family, we prove that the cardinal of the set of holomorphic gauge fields on any $B$-brane over the projective space is less than $2.$

Holomorphic Gauge Fields on $B$-Branes

TL;DR

This work extends holomorphic gauge fields from holomorphic vector bundles to B-branes, formulated via the -jet sequence and Atiyah class in the derived category. It constructs a disjoint-support generating family for by an iterative hypersurface-with-affine-complement approach, enabling a tight control of morphisms to objects. A key result is that for any B-brane built from the generators, implying the set of holomorphic gauge fields on any B-brane over has cardinality or (i.e., is strictly less than ). The paper also indicates a generalization: if a complex manifold admits a tower of divisors with affine complement and vanishing , the same bound holds for B-branes over . Altogether, the results reveal a strong rigidity of holomorphic gauge structures on B-branes in projective and related settings, connecting derived-category generators to gauge-field moduli.

Abstract

Considering the -branes over a complex manifold as the objects of the bounded derived category of coherent sheaves on that manifold, we extend the definition of holomorphic gauge fields on vector bundles to -branes. We construct a family of coherent sheaves on the complex projective space, which generates the corresponding bounded derived category and such that the supports of the elements of this family are two by two disjoint. Using that family, we prove that the cardinal of the set of holomorphic gauge fields on any -brane over the projective space is less than

Paper Structure

This paper contains 14 sections, 11 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Family of generators of $D^b({\mathbb P}^n)$
  3. The functor ${\mathbf A}\to D^b({\mathbf A})$
  4. The complex ${\rm Hom}$
  5. The shifting
  6. The mapping cone
  7. Generators of a derived category
  8. Affine varieties
  9. Hypersufaces with affine complement
  10. Aplication to $D^b({\mathbb P}^n)$
  11. Holomorphic gauge fields
  12. Holomorphic connections on a vector bundle
  13. Holomorphic gauge fields on a $B$-brane.
  14. Holomorphic gauge fields on $B$-branes over ${\mathbb P}^n$

Key Result

Proposition 1

If the sequence $A^{\centerdot}\to B^{\centerdot}\to C^{\centerdot}\to A[1]^{\centerdot}$ is distinguished triangle in $D^b({\mathbf A}),$ then so are the following

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • Definition 9
  • Proposition 10
  • ...and 3 more