rank-3 generalized Clifford manifold and its twistor space

TL;DR

This paper introduces rank-3 generalized Clifford manifolds, constructs their twistor spaces, and proves integrability and T-duality compatibility, advancing the understanding of generalized complex geometry.

Contribution

It defines rank-3 generalized Clifford manifolds, constructs their twistor spaces, and demonstrates integrability and T-duality invariance, offering new insights into generalized complex structures.

Findings

Constructed twistor space with integrable generalized complex structure

Established compatibility of Clifford-to-twistor construction with T-duality

Demonstrated Spin(3)-action produces a family of generalized complex structures

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. We further prove that this Clifford-to-twistor construction is compatible with T-duality, in the sense that T-duality preserves the rank-3 Clifford triple, the induced structures, and the associated Spin(3)-rotated family.