Hopf-Wess-Zumino term



String theory



Special and general types

Special notions


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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).


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 in

which hence introduced the terminology “Hopf-Wess-Zumino term”. Followup to this terminology includes

  • Jussi Kalkkinen, Kellogg Stelle, Section 3.2 of: Large Gauge Transformations in M-theory, J. Geom. Phys. 48 (2003) 100-132 (arXiv:hep-th/0212081)

  • Shan Hu, Dimitri Nanopoulos, Hopf-Wess-Zumino term in the effective action of the 6d, (2, 0) field theory revisted, JHEP 1110:054, 2011 (arXiv:1110.0861)

  • Alex Arvanitakis, Section 4.1 of Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an L L_\infty-algebroid (arXiv:1804.07303)

More on the relation to the Hopf invariant in

Discussion of the full 6d WZ term is in

Last revised on June 19, 2019 at 03:41:21. See the history of this page for a list of all contributions to it.