Topological Torsion

Topological Torsion in 4 dimensions implies that the 3-form A^dA is not zero. Consider a map from 4 dimensions to a 3 dimensional sphere (R4 to S3). Three functions (on R4) lead to 4D gradients which form the tangent vectors to three dimensional spherical surface. The adjoint vector (in R4) to this tangent space is a non-integrable 1-form, A. Not only is A^dA not equal to zero, but also dA^dA is not zero. It is possible, however, to renormalize the adjoint vector such that it has a zero divergence. Closed integrals of A^dA when dA^dA=0 form topological period integrals of torsion.

Image from the Hopf Archives, Univ of Indiana.

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Last update 01/23/2009
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