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Del Pezzo Surfaces and Affine 7-brane Backgrounds

Tamas Hauer, Amer Iqbal

TL;DR

The paper builds a precise bridge between two realizations of four-dimensional ${\cal N}=2$ theories with $E_N$ global symmetry: IIA geometric engineering on shrinking del Pezzo surfaces and IIB D3-brane probes near affine ${\widehat{E}}_N$ 7-brane backgrounds. Using mirror symmetry, it identifies the lattice of string junctions on affine 7-branes with the K-theory lattice of del Pezzo surfaces, and provides an explicit map between vector bundles on del Pezzo surfaces and junctions, including a detailed matching of charges and intersections. A Fourier–Mukai transform on the del Pezzo side is shown to correspond to the ${\rm SL}(2,\mathbb{Z})$ S-duality of the IIB 7-brane background, establishing a deep duality between geometric data and brane dynamics. The results illuminate the role of affine $E_9$ structures in the dual descriptions, relate moduli spaces of bundles to junction genera, and offer a framework for translating D-brane configurations into sheaf-theoretic data via the Fourier–Mukai transform. This work advances the unification of geometric engineering, mirror symmetry, and brane dualities in the study of ${\cal N}=2$ theories with exceptional global symmetry.

Abstract

A map between string junctions in the affine 7-brane backgrounds and vector bundles on del Pezzo surfaces is constructed using mirror symmetry. It is shown that the lattice of string junctions with support on an affine 7-brane configuration is isomorphic to the K-theory group of the corresponding del Pezzo surface. This isomorphism allows us to construct a map between the states of the N=2, D=4 theories with E_N global symmetry realized in two different ways in Type IIB and Type IIA string theory. A subgroup of the SL(2,Z) symmetry of the \hat{E}_9 7-brane background appears as the Fourier-Mukai transform acting on the D-brane configurations realizing vector bundles on elliptically fibered B_9.

Del Pezzo Surfaces and Affine 7-brane Backgrounds

TL;DR

The paper builds a precise bridge between two realizations of four-dimensional theories with global symmetry: IIA geometric engineering on shrinking del Pezzo surfaces and IIB D3-brane probes near affine 7-brane backgrounds. Using mirror symmetry, it identifies the lattice of string junctions on affine 7-branes with the K-theory lattice of del Pezzo surfaces, and provides an explicit map between vector bundles on del Pezzo surfaces and junctions, including a detailed matching of charges and intersections. A Fourier–Mukai transform on the del Pezzo side is shown to correspond to the S-duality of the IIB 7-brane background, establishing a deep duality between geometric data and brane dynamics. The results illuminate the role of affine structures in the dual descriptions, relate moduli spaces of bundles to junction genera, and offer a framework for translating D-brane configurations into sheaf-theoretic data via the Fourier–Mukai transform. This work advances the unification of geometric engineering, mirror symmetry, and brane dualities in the study of theories with exceptional global symmetry.

Abstract

A map between string junctions in the affine 7-brane backgrounds and vector bundles on del Pezzo surfaces is constructed using mirror symmetry. It is shown that the lattice of string junctions with support on an affine 7-brane configuration is isomorphic to the K-theory group of the corresponding del Pezzo surface. This isomorphism allows us to construct a map between the states of the N=2, D=4 theories with E_N global symmetry realized in two different ways in Type IIB and Type IIA string theory. A subgroup of the SL(2,Z) symmetry of the \hat{E}_9 7-brane background appears as the Fourier-Mukai transform acting on the D-brane configurations realizing vector bundles on elliptically fibered B_9.

Paper Structure

This paper contains 17 sections, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. ${\cal N}=2$ theories with $E_{N}$ global symmetry
  3. Compactification of IIA
  4. The homology of $\widetilde{\cal B}_{N}$ and the $E_N$ root lattice
  5. The K-theory group of del Pezzo surfaces
  6. Probe theory in IIB
  7. ${\bf \widehat{E}_{N< 9}}$
  8. ${\bf \widehat{E}_{9}}$
  9. 3-cycles and string junctions
  10. The mirror of a Calabi-Yau threefold containing ${\cal B}_{9}$
  11. 3-Cycles in $W$
  12. String junctions from 3-cycles
  13. String junctions and Vector bundles
  14. ${\cal B}_9$ and ${\bf \widehat{E}_9}$
  15. ${\cal B}_N$ and ${\bf \widehat{E}_{N}}$
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Two curves (junctions) whose boundary wraps intersecting cycles of the selected elliptic fiber.
  • Figure 2: The non-compact mirror Calabi-Yau $W$, as an $S_{c}^1\times I\!\!R$ fibration over the base of the elliptically fibered ${\cal B}_9$. At the points $z_i$ different cycles of the elliptic fiber of ${\cal B}_9$ are shrinking while at $z_*$ the $S_{c}^1$ shrinks.