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| 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. |