nLab (p,q)-string

under construction

Idea

In type II string theory: a bound state of $p$-F1-branes and $q$ D1-branes.

Under S-duality the modular group $SL_2(\mathbb{Z})$ acts on these $(p,q)$-tuples and mixes the $F1$- and the $D1$-brane states.

from M-branes to F-branes: superstrings, D-branes and NS5-branes

M-theory on $S^1_A \times S^1_B$-elliptic fibration	KK-compactification on $S^1_A$	type IIA string theory	T-dual KK-compactification on $S^1_B$	type IIB string theory	geometrize the axio-dilaton	F-theory on elliptically fibered-K3 fibration	duality between F-theory and heterotic string theory	heterotic string theory on elliptic fibration
M2-brane wrapping $S_A^1$	double dimensional reduction $\mapsto$	type IIA superstring	$\mapsto$	type IIB superstring	$\mapsto$	“	$\mapsto$	heterotic superstring
M2-brane wrapping $S_B^1$	$\mapsto$	D2-brane	$\mapsto$	D1-brane	$\mapsto$	“
M2-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$	$\mapsto$	$p$ strings and $q$ D2-branes	$\mapsto$	(p,q)-string	$\mapsto$	“
M5-brane wrapping $S_A^1$	double dimensional reduction $\mapsto$	D4-brane	$\mapsto$	D5-brane	$\mapsto$	“
M5-brane wrapping $S_B^1$	$\mapsto$	NS5-brane	$\mapsto$	NS5-brane	$\mapsto$	“	$\mapsto$	NS5-brane
M5-brane wrapping $p$ times around $S_A^1$ and $q$ times around $S_B^1$	$\mapsto$	$p$ D4-brane and $q$ NS5-branes	$\mapsto$	(p,q)5-brane	$\mapsto$	“
M5-brane wrapping $S_A^1 \times S_B^1$	$\mapsto$		$\mapsto$	D3-brane	$\mapsto$	“
KK-monopole/A-type ADE singularity (degeneration locus of $S^1_A$-circle fibration, Sen limit of $S^1_A \times S^1_B$ elliptic fibration)	$\mapsto$	D6-brane	$\mapsto$	D7-branes	$\mapsto$	A-type nodal curve cycle degeneration locus of elliptic fibration (Sen 97, section 2)		SU-gauge enhancement
KK-monopole orientifold/D-type ADE singularity	$\mapsto$	D6-brane with O6-planes	$\mapsto$	D7-branes with O7-planes	$\mapsto$	D-type nodal curve cycle degeneration locus of elliptic fibration (Sen 97, section 3)		SO-gauge enhancement
exceptional ADE-singularity	$\mapsto$		$\mapsto$		$\mapsto$	exceptional ADE-singularity of elliptic fibration	$\mapsto$	E6-, E7-, E8-gauge enhancement

(e.g. Johnson 97, Blumenhagen 10)

Related concepts

• S-duality

• F-theory

brane intersections/bound states/wrapped branes/polarized branes

• D-branes and anti D-branes form bound states by tachyon condensation, thought to imply the classification of D-brane charge by K-theory

• intersecting D-branes/fuzzy funnels:

  • Dp-D(p+2)-brane bound state and Yang-Mills monopoles

    • D0-D2 brane bound state

    • D1-D3 brane bound state

    • D2-D4 brane bound state

    • D3-D5 brane bound state

    • D4-D6 brane bound state

    • D6-D8-brane bound state

  • Dp-D(p+4)-brane bound state and Yang-Mills instantons

    • D0-D4 brane bound state

    • D1-D5 brane bound state

    • D2-D6 brane bound state

    • D3-D7 brane bound state

    • D4-D8 brane bound state

    • D5-D9 brane bound state

      small instanton

• Dp-D(p+6) brane bound state

  • D0-D6 brane bound state

  • D1-D7 brane bound state

  • D2-D8 brane bound state

• NS5-D4-D2 bound states

• Hanany-Witten effect

S-duality$\,$bound states:

• (p,q)-string

• (p,q)5-branes

• string network

• 5-brane web

intersecting$\,$M-branes:

• M5-MO9 brane bound state

• M2-M5 brane bound state

  • M-string, E-string

• membrane triple junction

References

• Igor Bandos, Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane, Nucl.Phys.B599:197-227,2001 (arXiv:hep-th/0008249)