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Geometric Model for Complex Non-Kaehler Manifolds with SU(3) Structure

Edward Goldstein, Sergey Prokushkin

Abstract

For a given complex n-fold M we present an explicit construction of all complex (n+1)-folds which are principal holomorphic T2-fibrations over M. For physical applications we consider the case of M being a Calabi-Yau 2-fold. We show that for such M, there is a subclass of the 3-folds that we construct, which has natural families of non-Kaehler SU(3)-structures satisfying the conditions for N = 1 supersymmetry in the heterotic string theory compactified on the 3-folds. We present examples in the aforementioned subclass with M being a K3-surface and a 4-torus.

Geometric Model for Complex Non-Kaehler Manifolds with SU(3) Structure

Abstract

For a given complex n-fold M we present an explicit construction of all complex (n+1)-folds which are principal holomorphic T2-fibrations over M. For physical applications we consider the case of M being a Calabi-Yau 2-fold. We show that for such M, there is a subclass of the 3-folds that we construct, which has natural families of non-Kaehler SU(3)-structures satisfying the conditions for N = 1 supersymmetry in the heterotic string theory compactified on the 3-folds. We present examples in the aforementioned subclass with M being a K3-surface and a 4-torus.

Paper Structure

This paper contains 12 sections, 12 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Global model -- the construction
  3. Holomorphic principal T2 fiber bundles
  4. Intrinsic torsion of $SU(3)$-structures
  5. Metric scaling and non-constant dilaton
  6. Examples
  7. $SU(3)$-fibrations over K3 surfaces
  8. $SU(3)$-fibrations over $T^4$
  9. Cohomology and Hodge numbers of $M^{P,Q}$
  10. Hodge numbers $h^{1,0}$ and $h^{0,1}$ of $M^{P,Q}$
  11. Cohomology of $M^{P,Q}$
  12. Pulling up Special Lagrangians

Key Result

Theorem 1

Let $\omega_P$ and $\omega_Q$ be closed 2-forms on a complex Hermitian $n$-fold $(M,g)$ s.t. the following two conditions hold: 1) $\omega_P+i\omega_Q$ has no component in $\Lambda^{0,2}T^{\ast}M$. 2) $\frac{\omega_P}{2\pi}$ and $\frac{\omega_Q}{2\pi}$ represent integral cohomology classes. Then the

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Definition 1.0.1
  • Theorem 3
  • Theorem 4
  • Theorem 1
  • Lemma 2.0.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • ...and 3 more