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The Heterotic String, the Tangent Bundle, and Derived Categories

Paul S. Aspinwall, Ron Y. Donagi

TL;DR

The paper investigates the heterotic string on the tangent bundle of a K3 surface and its duality to Type IIA/F-theory on a Calabi–Yau threefold X, revealing that RR moduli split into components associated with the tangent bundle and with point-like instantons when X is singular. It introduces the relative Fourier–Mukai transform and Mukai vector within the derived category to describe spectral data and RR moduli, and shows how extremal transitions influence gauge symmetry enhancements. The authors demonstrate that the tangent-bundle compactification can exhibit nonperturbative gauge enhancement similar to singular K3 cases, while carefully accounting for RR moduli via spectral data; they argue that these phenomena are best understood in the derived-category framework. Finally, they propose that heterotic T-duality requires specifying data as an object in the derived category, with the Mukai vector and Fourier–Mukai transforms providing a natural, global language for RR moduli and dualities in this setting.

Abstract

We consider the compactification of the E8xE8 heterotic string on a K3 surface with "the spin connection embedded in the gauge group" and the dual picture in the type IIA string (or F-theory) on a Calabi-Yau threefold X. It turns out that the same X arises also as dual to a heterotic compactification on 24 point-like instantons. X is necessarily singular, and we see that this singularity allows the Ramond-Ramond moduli on X to split into distinct components, one containing the (dual of the heterotic) tangent bundle, while another component contains the point-like instantons. As a practical application we derive the result that a heterotic string compactified on the tangent bundle of a K3 with ADE singularities acquires nonperturbatively enhanced gauge symmetry in just the same fashion as a type IIA string on a singular K3 surface. On a more philosophical level we discuss how it appears to be natural to say that the heterotic string is compactified using an object in the derived category of coherent sheaves. This is necessary to properly extend the notion of T-duality to the heterotic string on a K3 surface.

The Heterotic String, the Tangent Bundle, and Derived Categories

TL;DR

The paper investigates the heterotic string on the tangent bundle of a K3 surface and its duality to Type IIA/F-theory on a Calabi–Yau threefold X, revealing that RR moduli split into components associated with the tangent bundle and with point-like instantons when X is singular. It introduces the relative Fourier–Mukai transform and Mukai vector within the derived category to describe spectral data and RR moduli, and shows how extremal transitions influence gauge symmetry enhancements. The authors demonstrate that the tangent-bundle compactification can exhibit nonperturbative gauge enhancement similar to singular K3 cases, while carefully accounting for RR moduli via spectral data; they argue that these phenomena are best understood in the derived-category framework. Finally, they propose that heterotic T-duality requires specifying data as an object in the derived category, with the Mukai vector and Fourier–Mukai transforms providing a natural, global language for RR moduli and dualities in this setting.

Abstract

We consider the compactification of the E8xE8 heterotic string on a K3 surface with "the spin connection embedded in the gauge group" and the dual picture in the type IIA string (or F-theory) on a Calabi-Yau threefold X. It turns out that the same X arises also as dual to a heterotic compactification on 24 point-like instantons. X is necessarily singular, and we see that this singularity allows the Ramond-Ramond moduli on X to split into distinct components, one containing the (dual of the heterotic) tangent bundle, while another component contains the point-like instantons. As a practical application we derive the result that a heterotic string compactified on the tangent bundle of a K3 with ADE singularities acquires nonperturbatively enhanced gauge symmetry in just the same fashion as a type IIA string on a singular K3 surface. On a more philosophical level we discuss how it appears to be natural to say that the heterotic string is compactified using an object in the derived category of coherent sheaves. This is necessary to properly extend the notion of T-duality to the heterotic string on a K3 surface.

Paper Structure

This paper contains 12 sections, 7 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Fourier-Mukai Functor
  3. Some Sheaf Generalities
  4. The Fourier-Mukai Transform of a Sheaf
  5. The Derived Category and the Mukai Vector
  6. The heterotic string on the tangent bundle
  7. Review
  8. Two special cases
  9. Ramond-Ramond moduli and the degenerate Jacobian
  10. Obstructed extremal transitions
  11. The tangent sheaf of an orbifold K3
  12. T-Duality

Key Result

Theorem 1

The functor $\mathbf{S}$ satisfies where $(-1)_E$ is the inversion of each elliptic fibre and $[-1]$ denotes a shift of the complex of one to the right.

Figures (3)

  • Figure 1: The discriminant for an SU(2)-bundle.
  • Figure 2: The discriminant for the tangent bundle and point-like instantons.
  • Figure 3: The discriminant for the tangent sheaf on a ${\mathbb C}^2/{\mathbb Z}_2$ singularity.

Theorems & Definitions (7)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 2
  • Proposition 4
  • Proposition 5