nLab M5-brane charge

Idea

Where D-brane charge is the charge in K-theory carried by D-branes and computed as the partition function/index of the “superparticles” on the D-brane being the boundary field theory of the open superstring ending on them, so open M2-branes ending on M5-branes should induced a charge given by the partition function of their bounding self-dual strings (“M-strings”), which is now an elliptic genus. Hence M5-brane charge should be in elliptic cohomology (Sati 10).

Related concepts

• Cohomotopy charge

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

That the elliptic genus of the self-dual string on the M5-brane should be the M5-brane charge in elliptic cohomology in higher analogy to D-brane charge in K-theory is due to

• Hisham Sati, Geometric and topological structures related to M-branes, part I, Proc. Symp. Pure Math. 81 (2010), 181-236 arXiv:1001.5020,

Discussion of computation of the elliptic genus/Witten genus of the self-dual string on the M5-brane includes

• Stefan Hohenegger, Amer Iqbal, M-strings, Elliptic Genera and N=4 String Amplitudes (arXiv:1310.1325)