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

• Hopfion

• Hopf invariant, Hopf invariant one theorem

• Whitehead integral formula

• functional cup product

• M5-brane in 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. 10 (11 of 18) in String theory dualities from M theory, Nucl. Phys. B 476 (1996) 470-483 [arXiv:hep-th/9604103, doi:10.1016/0550-3213(96)00321-5]

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 $D=11$ 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:

• Kenneth Intriligator, Anomaly Matching and a Hopf-Wess-Zumino Term in 6d, $\mathcal{N}= (2,0)$ Field Theories, Nucl. Phys. B 581 (2000) 257-273 [arXiv:hep-th/0001205, doi:10.1016/S0550-3213(00)00148-6]

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

• Ori Ganor, Lubos Motl, p. 5-6 in: Equations of the (2,0) Theory and Knitted Fivebranes, JHEP 9805:009 (1998) [arXiv:hep-th/9803108, doi:10.1088/1126-6708/1998/05/009]

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, $(2, 0)$ 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 $L_\infty$-algebroid, Advances in Theoretical and Mathematical Physics Volume 23 (2019) Number 5 (arXiv:1804.07303, doi:10.4310/ATMP.2019.v23.n5.a1)

see also:

• David Berman: p. 20 in: M-theory branes and their interactions, Phys. Rept. 456 (2008) 89-126 [arXiv:0710.1707, doi:10.1016/j.physrep.2007.10.002]

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 in view of Hypothesis H:

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5 WZ term level quantization, Comm. Math. Phys. 384 (2021) 403–432 [arXiv:1906.07417, doi:10.1007/s00220-021-03951-0]