nLab self-dual string

Idea

The equations of motion on the M5-brane (the 6d (2,0)-superconformal QFT in a suitable limit) admits a soliton/black brane solution which is a string charged under the self-dual B-field [Howe, Lambert & West 1997, §3].

Externally this is the boundary of the M2-brane ending on the M5-brane. As such this is “the string in M-theory” and as such also called the M-string.

Properties

Elliptic genus and M5-brane charge

The elliptic genus of the self-dual string at the boundary of the M2-brane on the M5-brane has been suggested to be directly analogous to the Todd genus of the boundary of the open superstring ending on D-branes and hence has been suggested to be the M5-brane charge in Sati 2010. Discussion of the computation of this elliptic genus includes Hohenegger & Iqbal 2013.

Related concepts

• M-string

• open membrane theory

• 3-brane in 6d

Table of branes appearing in supergravity/string theory (for classification see at brane scan).

brane	in supergravity	charged under gauge field	has worldvolume theory
black brane	supergravity	higher gauge field	SCFT
D-brane	type II	RR-field	super Yang-Mills theory
$(D = 2n)$	type IIA	$\,$	$\,$
D(-2)-brane	$\,$	$\,$
D0-brane	$\,$	$\,$	BFSS matrix model
D2-brane	$\,$	$\,$	$\,$
D4-brane	$\,$	$\,$	D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane	$\,$	$\,$	D=7 super Yang-Mills theory
D8-brane	$\,$	$\,$
$(D = 2n+1)$	type IIB	$\,$	$\,$
D(-1)-brane	$\,$	$\,$	$\,$
D1-brane	$\,$	$\,$	2d CFT with BH entropy
D3-brane	$\,$	$\,$	N=4 D=4 super Yang-Mills theory
D5-brane	$\,$	$\,$	$\,$
D7-brane	$\,$	$\,$	$\,$
D9-brane	$\,$	$\,$	$\,$
(p,q)-string	$\,$	$\,$	$\,$
(D25-brane)	(bosonic string theory)
NS-brane	type I, II, heterotic	circle n-connection	$\,$
string	$\,$	B2-field	2d SCFT
NS5-brane	$\,$	B6-field	little string theory
D-brane for topological string			$\,$
A-brane			$\,$
B-brane			$\,$
M-brane	11D SuGra/M-theory	circle n-connection	$\,$
M2-brane	$\,$	C3-field	ABJM theory, BLG model
M5-brane	$\,$	C6-field	6d (2,0)-superconformal QFT
M9-brane/O9-plane			heterotic string theory
M-wave
topological M2-brane	topological M-theory	C3-field on G₂-manifold
topological M5-brane	$\,$	C6-field on G₂-manifold
S-brane
SM2-brane, membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane	6d (2,0)-superconformal QFT
self-dual string		self-dual B-field
3-brane in 6d

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

The equations of motion of the self-dual string as a solution of D=6 supergravity was maybe first seen in

• Michael Duff, Jian Xin Lu, §11 in: Black and super $p$-branes in diverse dimensions, Nucl. Phys. B 416 (1994) 301-334 [arXiv:hep-th/9306052, doi:10.1016/0550-3213(94)90586-X]

The appearance of self-dual string in six dimensions with an ADE classification was first observed in

• Edward Witten, Some Comments On String Dynamics [arXiv:hep-th/9507121]

As a black brane (“soliton”) solution to the equations of motion on the M5-brane:

• Paul S. Howe, Neil Lambert, Peter West, §3 in: The Self-Dual String Soliton, Nucl. Phys. B 515 (1998) 203-216 [arXiv:hep-th/9709014, doi:10.1016/S0550-3213(97)00750-5]

As a Green-Schwarz sigma-model the brane scan-cocycle for the superstring in 6d is given in

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, (III.7.15) of Supergravity and Superstrings - A Geometric Perspective, World Scientific, 1991

  (in D'Auria-Fré formulation of supergravity)

As the brane intersection of M2-branes onto an M5-brane (cf. M-string):

• Yoji Michishita: The M2-brane Soliton on the M5-brane with Constant 3-Form, Journal of High Energy Physics 2000 JHEP09 (2000) [doi:10.1088/1126-6708/2000/09/036, arXiv:hep-th/0008247]

• Donam Youm: BPS Solitons in M5-Brane Worldvolume Theory with Constant Three-Form Field, Phys. Rev. D 63 (2001) 045004 [doi:10.1103/PhysRevD.63.045004, arXiv:hep-th/0009082]

• Anirban Basu, Jeffrey A. Harvey, §3.2 in: The M2-M5 Brane System and a Generalized Nahm’s Equation, Nucl. Phys. B 713 (2005) 136-150 [arXiv:hep-th/0412310, doi:10.1016/j.nuclphysb.2005.02.007]

Further discussion:

• John Schwarz, Self-Dual Superstring in Six Dimensions [arXiv:hep-th/9604171]

• Vasilis Niarchos, Konstadinos Siampos, Entropy of the self-dual string soliton, J. High Energ. Phys. 2012 134 (2012). [arXiv:1206.2935, doi:10.1007/JHEP07(2012)134]

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

In view of M-strings and the M-string elliptic genus:

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