nLab
Hopf-Wess-Zumino term

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 $4k+2$ (TN 89).

Related concepts

• Hopf invariant, Hopf invariant one theorem

• Whitehead integral formula

• functional cup product

• M-theory

References

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

• Frank Wilczek and Anthony Zee, Linking Numbers, Spin, and Statistics of Solitons, Phys. Rev. Lett. 51, 2250, 1983 (doi:10.1103/PhysRevLett.51.2250)

• Hsiung Chia Tze, Soonkeon Nam, Topological Phase Entanglements of Membrane Solitons in Division Algebra $\sigma$ Models With a Hopf Term, Annals Phys. 193 (1989) 419-471 (spire:25058, doi:10.1016/0003-4916(89)90005-5)

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

• Ofer Aharony, p. 11 of String theory dualities from M theory, Nucl. Phys. B476:470-483, 1996 (arXiv:hep-th/9604103)

and had been settled by the time of

• Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, Covariant Action for the Super-Five-Brane of M-Theory, Phys. Rev. Lett. 78 (1997) 4332-4334 (arXiv:hep-th/9701149)

  Paolo Pasti, Dmitri Sorokin and M. Tonin, Covariant Action for a D=11 Five-Brane with the Chiral Field, Phys. Lett. B398 (1997) 41.

The resemblence of the first summand of the term to the Whitehead integral formula for the Hopf invariant was noticed in

• Kenneth Intriligator, Anomaly Matching and a Hopf-Wess-Zumino Term in 6d, $\mathcal{N}= (2,0)$ Field Theories, Nucl. Phys. B581 (2000) 257-273 (arXiv:hep-th/0001205)

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 $L_\infty$-algebroid, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 5 (arXiv:1804.07303, doi:10.4310/ATMP.2019.v23.n5.a1)

More on the relation to the Hopf invariant in

• Hisham Sati, Framed M-branes, corners, and topological invariants, J. Math. Phys. 59 (2018), 062304 (arXiv:1310.1060)

Discussion of the full 6d WZ term is in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5 WZ term level quantization (arXiv:1906.07417)