Hypermultiplets and hypercomplex geometry from 6 to 3 dimensions

TL;DR

This paper develops a unified framework for hypermultiplets across 6, 5, 4, and 3 dimensions, linking hypercomplex geometry with superconformal calculus and extending the c-map to new geometric insights.

Contribution

It provides translation tables for hypermultiplet formulations across dimensions and introduces a superconformal approach to hypercomplex geometry without action.

Findings

Unified description of hypermultiplets in 6 to 3 dimensions

Extension of hypercomplex geometry beyond hyper-Kaehler structures

Superconformal formulation of the c-map and geometric quantities

Abstract

The formulation of hypermultiplets that has been developed for 5-dimensional matter multiplets is by dimensional reductions translated into the appropriate spinor language for 6 and 4 dimensions. We also treat the theories without actions that have the geometrical structure of hypercomplex geometry. The latter is the generalization of hyper-Kaehler geometry that does not require a Hermitian metric and hence corresponds to field equations without action. The translation tables of this paper allow the direct application of superconformal tensor calculus for the hypermultiplets using the available Weyl multiplets in 6 and 4 dimensions. Furthermore, the hypermultiplets in 3 dimensions that result from reduction of vector multiplets in 4 dimensions are considered, leading to a superconformal formulation of the c-map and an expression for the main geometric quantities of the hyper-Kaehler manifolds in the image of this map.