A tt⁎tt⁎-bundle associated with a harmonic map from a Riemann surface into a sphere

A tt⁎tt⁎-bundle is constructed by a harmonic map from a Riemann surface into an n  -dimensional sphere. This tt⁎tt⁎-bundle is a high-dimensional analogue of a quaternionic line bundle with a Willmore connection. For the construction, a flat connection is decomposed into four parts by a fiberwise complex structure.