# Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino term of the M5-brane

## TL;DR

This paper demonstrates that the anomaly cancellation condition for the M5-brane's 6d Wess-Zumino term is implied by charge quantization in twisted Cohomotopy theory, linking higher topology to physical quantization.

## Contribution

It establishes a novel connection between anomaly cancellation in M5-branes and charge quantization in twisted Cohomotopy, using a twisted homotopy Whitehead integral formula.

## Key findings

- Charge quantization in twisted Cohomotopy implies level quantization of the Wess-Zumino term.
- Identification of the 6d Hopf-Wess-Zumino term with a twisted homotopy Whitehead integral.
- Proof of anomaly cancellation condition via higher topological methods.

## 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.