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rank-3 generalized Clifford manifold and its twistor space

Guangzhen Ren, Kai Tang, Qingyan Wu

Abstract

We introduce the notion of a rank-3 generalized Clifford manifold, defined by a triple of generalized complex structures satisfying Clifford-type relations. We show that every such structure canonically induces a generalized hypercomplex structure. We further describe a natural Spin(3)-action by Clifford rotations, which produces an $S^2 \times S^2$-family of generalized complex structures. The corresponding twistor space is then constructed, and we prove that the induced almost generalized complex structure is integrable. In contrast to the standard pure-spinor approach, the integrability of the twistor-space structure is established entirely in terms of the generalized Nijenhuis tensor.

rank-3 generalized Clifford manifold and its twistor space

Abstract

We introduce the notion of a rank-3 generalized Clifford manifold, defined by a triple of generalized complex structures satisfying Clifford-type relations. We show that every such structure canonically induces a generalized hypercomplex structure. We further describe a natural Spin(3)-action by Clifford rotations, which produces an -family of generalized complex structures. The corresponding twistor space is then constructed, and we prove that the induced almost generalized complex structure is integrable. In contrast to the standard pure-spinor approach, the integrability of the twistor-space structure is established entirely in terms of the generalized Nijenhuis tensor.
Paper Structure (6 sections, 19 theorems, 140 equations)

This paper contains 6 sections, 19 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. generalized complex structure
  4. generalized real structure
  5. Generalized Clifford manifold
  6. the twistor space of generalized Clifford manifold

Key Result

Theorem 1.1

Let $\left(\mathcal{I}_1, \mathcal{I}_2, \mathcal{I}_3\right)$ be an alomost generalized Clifford structure over $\mathbb{T} M$, then is equivalent to where $\mathcal{J}_i$ is the induced almost generalized complex structure defined by coincide00000,

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 30 more