group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 (TN 89).
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 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
and had been settled by the time of
Igor Bandos, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, Dmitri Sorokin, Mario Tonin, (1) in: Covariant Action for the Super-Five-Brane of M-Theory, Phys. Rev. Lett. 78 (1997) 4332-4334 [arXiv:hep-th/9701149, doi:10.1103/PhysRevLett.78.4332]
Paolo Pasti, Dmitri Sorokin, Mario Tonin, (17) in: Covariant Action for a Five-Brane with the Chiral Field, Phys. Lett. B 398 (1997) 41 [arXiv:hep-th/9701037, doi:10.1016/S0370-2693(97)00188-3]
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”:
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, doi:10.1016/S0393-0440(03)00027-5]
Shan Hu, Dimitri Nanopoulos, Hopf-Wess-Zumino term in the effective action of the 6d, field theory revisted, JHEP 1110:054 (2011) [arXiv:1110.0861, doi:10.1007/JHEP10(2011)054]
Alex Arvanitakis, Section 4.1 of Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an -algebroid, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 5 (arXiv:1804.07303, doi:10.4310/ATMP.2019.v23.n5.a1)
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.