Twisted Cohomotopy Implies Level Quantization of the Full 6d Wess-Zumino Term of the M5-Brane

Abstract

The full 6d Hopf–Wess–Zumino term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess–Zumino term. We prove that the anomaly cancellation condition is implied by the hypothesis that the C-field is charge-quantized in twisted Cohomotopy theory. The proof follows by a twisted/parametrized generalization of the Hopf invariant, after identifying the full 6d Hopf–Wess–Zumino term with a twisted homotopy Whitehead integral formula, which we establish.