nLab self-dual string




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.


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.

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

branein supergravitycharged under gauge fieldhas worldvolume theory
black branesupergravityhigher gauge fieldSCFT
D-branetype IIRR-fieldsuper Yang-Mills theory
(D=2n)(D = 2n)type IIA\,\,
D0-brane\,\,BFSS matrix model
D4-brane\,\,D=5 super Yang-Mills theory with Khovanov homology observables
D6-brane\,\,D=7 super Yang-Mills theory
(D=2n+1)(D = 2n+1)type IIB\,\,
D1-brane\,\,2d CFT with BH entropy
D3-brane\,\,N=4 D=4 super Yang-Mills theory
(D25-brane)(bosonic string theory)
NS-branetype I, II, heteroticcircle n-connection\,
string\,B2-field2d SCFT
NS5-brane\,B6-fieldlittle string theory
D-brane for topological string\,
M-brane11D SuGra/M-theorycircle n-connection\,
M2-brane\,C3-fieldABJM theory, BLG model
M5-brane\,C6-field6d (2,0)-superconformal QFT
M9-brane/O9-planeheterotic string theory
topological M2-branetopological M-theoryC3-field on G2-manifold
topological M5-brane\,C6-field on G2-manifold
membrane instanton
M5-brane instanton
D3-brane instanton
solitons on M5-brane6d (2,0)-superconformal QFT
self-dual stringself-dual B-field
3-brane in 6d

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

ddpartition function in dd-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpinKOM Spin \to KO
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory


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

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

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

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

As the brane intersection of M2-branes onto an M5-brane:

Further discussion:

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

Last revised on June 12, 2024 at 13:35:03. See the history of this page for a list of all contributions to it.