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Equivalence of Geometric Engineering and Hanany-Witten via Fractional Branes

Andreas Karch, Dieter Lust, Douglas J. Smith

TL;DR

The paper addresses how Geometric Engineering and Hanany-Witten constructions for 4d N=2 gauge theories are related. It builds a unified duality framework by mapping HW setups to fractional branes via T-duality and tying geometric engineering to these pictures through S- and T-dualities and the local mirror map, which reproduces Seiberg-Witten data. The authors provide multi-faceted evidence—5-brane perspectives, deformation/moduli matching, and quantum-corrective consistency—and extend the approach to N=1 theories, suggesting broad applicability. This work offers a cohesive, computationally powerful viewpoint that connects distinct string-theoretic realizations of gauge dynamics and facilitates nonperturbative analyses across frameworks.

Abstract

We present an explicit relation between the Hanany-Witten and Geometric Engineering approaches of realising gauge theories in string theory. The last piece in the puzzle is a T-duality relating arbitrary Hanany-Witten setups and fractional branes.

Equivalence of Geometric Engineering and Hanany-Witten via Fractional Branes

TL;DR

The paper addresses how Geometric Engineering and Hanany-Witten constructions for 4d N=2 gauge theories are related. It builds a unified duality framework by mapping HW setups to fractional branes via T-duality and tying geometric engineering to these pictures through S- and T-dualities and the local mirror map, which reproduces Seiberg-Witten data. The authors provide multi-faceted evidence—5-brane perspectives, deformation/moduli matching, and quantum-corrective consistency—and extend the approach to N=1 theories, suggesting broad applicability. This work offers a cohesive, computationally powerful viewpoint that connects distinct string-theoretic realizations of gauge dynamics and facilitates nonperturbative analyses across frameworks.

Abstract

We present an explicit relation between the Hanany-Witten and Geometric Engineering approaches of realising gauge theories in string theory. The last piece in the puzzle is a T-duality relating arbitrary Hanany-Witten setups and fractional branes.

Paper Structure

This paper contains 23 sections, 21 equations.

Table of Contents

  1. Introduction
  2. The pieces of the puzzle
  3. Geometric Engineering
  4. The setup: IIA on K3 fibered CY
  5. Decoupling Gravity and ALE spaces
  6. The Solution: Local Mirror Symmetry
  7. The Hanany Witten setup
  8. The setup
  9. Corrections: Bending and instantons
  10. The Solution: Lifting to M-theory
  11. Other Gauge Theories
  12. Branes as Probes
  13. D-branes at orbifolds
  14. Moving onto the Coulomb branch: Fractional Branes
  15. Duality between Hanany-Witten and fractional branes
  16. ...and 8 more sections