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Compactification, Geometry and Duality: N=2

Paul S. Aspinwall

TL;DR

This work examines the moduli spaces of four-dimensional N=2 string compactifications, emphasizing a VM×HM split that arises in both Type II on Calabi–Yau threefolds and heterotic on K3×T^2. VM geometry is governed by special Kähler structure and mirror symmetry, enabling exact or highly constrained descriptions, while HM geometry remains more elusive, richly affected by quantum corrections and interwoven with higher-dimensional dynamics, dualities, and extremal transitions. The text develops a detailed map between CY moduli and VM data, explains how dualities constrain global moduli space structure, and uses examples (e.g., mirror pairs, ADE singularities, conifold transitions) to illustrate how VM/HM sectors evolve under quantum corrections and phase transitions. It also discusses the limitations and prospects of HM analysis, including the nonexistence of a universal HM in general and the role of hyperkähler and quaternionic geometries in various limits, highlighting the deep ties between string theory, geometry, and quantum field theory phenomena such as Seiberg–Witten dynamics and F-theory. Overall, the work integrates holonomy, dualities, and geometric transitions to illuminate how N=2 theories encode intricate, stringy geometries beyond classical supergravity.

Abstract

These are notes based on lectures given at TASI99. We review the geometry of the moduli space of N=2 theories in four dimensions from the point of view of superstring compactification. The cases of a type IIA or type IIB string compactified on a Calabi-Yau threefold and the heterotic string compactified on K3xT2 are each considered in detail. We pay specific attention to the differences between N=2 theories and N>2 theories. The moduli spaces of vector multiplets and the moduli spaces of hypermultiplets are reviewed. In the case of hypermultiplets this review is limited by the poor state of our current understanding. Some peculiarities such as ``mixed instantons'' and the non-existence of a universal hypermultiplet are discussed.

Compactification, Geometry and Duality: N=2

TL;DR

This work examines the moduli spaces of four-dimensional N=2 string compactifications, emphasizing a VM×HM split that arises in both Type II on Calabi–Yau threefolds and heterotic on K3×T^2. VM geometry is governed by special Kähler structure and mirror symmetry, enabling exact or highly constrained descriptions, while HM geometry remains more elusive, richly affected by quantum corrections and interwoven with higher-dimensional dynamics, dualities, and extremal transitions. The text develops a detailed map between CY moduli and VM data, explains how dualities constrain global moduli space structure, and uses examples (e.g., mirror pairs, ADE singularities, conifold transitions) to illustrate how VM/HM sectors evolve under quantum corrections and phase transitions. It also discusses the limitations and prospects of HM analysis, including the nonexistence of a universal HM in general and the role of hyperkähler and quaternionic geometries in various limits, highlighting the deep ties between string theory, geometry, and quantum field theory phenomena such as Seiberg–Witten dynamics and F-theory. Overall, the work integrates holonomy, dualities, and geometric transitions to illuminate how N=2 theories encode intricate, stringy geometries beyond classical supergravity.

Abstract

These are notes based on lectures given at TASI99. We review the geometry of the moduli space of N=2 theories in four dimensions from the point of view of superstring compactification. The cases of a type IIA or type IIB string compactified on a Calabi-Yau threefold and the heterotic string compactified on K3xT2 are each considered in detail. We pay specific attention to the differences between N=2 theories and N>2 theories. The moduli spaces of vector multiplets and the moduli spaces of hypermultiplets are reviewed. In the case of hypermultiplets this review is limited by the poor state of our current understanding. Some peculiarities such as ``mixed instantons'' and the non-existence of a universal hypermultiplet are discussed.

Paper Structure

This paper contains 38 sections, 17 theorems, 78 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. General Structure
  3. Holonomy
  4. U-Duality
  5. Eight supercharges
  6. Type II compactification
  7. Heterotic compactification
  8. $E_H$ and its bundle
  9. $S_H$ and its bundle
  10. Who gets corrected?
  11. The Moduli Space of Vector Multiplets
  12. The special Kähler geometry of ${\Scr M}_V$
  13. ${\Scr M}_V$ in the type IIA string
  14. Before corrections and five dimensions
  15. Mirror Pairs
  16. ...and 23 more sections

Key Result

Theorem 1

If a Riemannian manifold is complete, simply connected and if the holonomy of its tangent bundle with respect to the Levi-Civita connection is reducible, then this manifold is a product metrically.

Figures (8)

  • Figure 1: A K3 fibration.
  • Figure 2: The rigid limit of an $\operatorname{SU}(2)$ theory.
  • Figure 3: Moduli Space of a Torus.
  • Figure 4: Moduli Space of the Quintic.
  • Figure 5: Classification of elliptic fibres.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 2
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Proposition 8
  • ...and 7 more