nLab self-dual string

Idea

The 6d (2,0)-superconformal QFT admits a soliton/black brane solution which is a string charged under the self-dual B-field.

Externally this is the boundary of the M2-brane ending on the M5-brane. As such this is “the string in M-theory” and hence is 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 10). Discussion of the computation of this elliptic genus includes (Hohenegger-Iqbal 13).

Related concepts

• 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

As a black brane the self-dual string in 6d was maybe first seen in

• Michael Duff, Jian Xin Lu, Black and super $p$-branes in diverse dimensions, Nucl. Phys. B 416 (1994) 301-334 [arXiv:hep-th/9306052]

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 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)

Further discussion includes

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

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

• 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)