nLab Hopf-Wess-Zumino term

Contents

Context

String theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Green-Schwarz sigma-model of the M5-brane contains a 6d higher WZW term. The first summand of this term was called the Hopf-Wess-Zumino term in Intriligator 00, due to its resemblance of the Whitehead integral formula for the Hopf invariant if the C-field is assumed to be classified by a smooth function to the 4-sphere. But in fact the full term is a homotopy Whitehead integral formula (“functional cup product” or “homotopy period”) still computing the Hopf invariant if the C-field is assumed to be a cocycle in twisted cohomotopy in degree 4 (FSS 19).

Similar Hopf terms can be considered in all dimensions 4k+24k+2 (TN 89).

References

The general concept of Hopf-Wess-Zumino terms was considered in

The higher WZW term of the M5-brane was first proposed in

and had been settled by the time of

The resemblence of the first summand of the term to the Whitehead integral formula for the Hopf invariant was noticed and hence the terminology “Hopf-Wess-Zumino term” was introduced in:

by observing that the “scalar” fields on the M5-brane worldvolume parameterize a map to the 4-sphere, as previously highlighted in

Articles following this terminology “Hopf-Wess-Zumino term”:

see also:

More on the relation to the Hopf invariant in

Discussion of the full 6d WZ term in view of Hypothesis H:

Last revised on July 2, 2024 at 12:22:59. See the history of this page for a list of all contributions to it.