nLab (p,q)5-brane

Idea

In type IIB string theory there are bound states of D5-branes with NS5-branes. The bound state of $p \in \mathbb{Z}$ D5-branes with $q \in\mathbb{Z}$ NS5-branes is then called a $(p,q)$-fivebrane or similar.

As $p$ and $q$ varies the species of $(p,q)$5-branes form the lattice $\mathbb{Z}^2$ and are naturally acted on by the S-duality group SL(2,Z).

Properties

$(p,q)$-Brane webs

Label local coordinate functions $x^a$ on 10d Minkowski spacetime $\mathbb{R}^{9,1}$ by $0123455'6789$ and write $v_a \coloneqq \partial_{x^a}$ for the corresponding vector field

Consider a $(1,0)$5-brane (a D5-brane) along the multivector field $v_0 v_1 v_2 v_3 v_4 v_5$ and a $(0,1)$5-brane (an NS-brane) along $v_0 v_1 v_2 v_3 v_4 v_{5'}$

$a =$	$0$	$1$	$2$	$3$	$4$	$5$	$5'$	$6$	$7$	$8$	$9$
D5	—	—	—	—	—	—
NS5	—	—	—	—	—		—

Charge conservation implies that at the brane intersection of the two a $(1,1)$5-brane emerges stretched diagonally along $v_5 + v_{g'}$, i.e. along the multivector field $v_0 v_1 v_2 v_3 v_4 (v_5 + v_{5'})$

Aharony-Hanany 97, Sec. 3

The worldvolume-quantum field theory at the brane intersection point is a geometric engineering of D=5 N=1 SYM .

The T-dual perspective are D4/NS5-brane webs (Witten 97).

The result of connecting several such brane intersections are called $(p,q)$-5brane webs (Aharony-Hanany-Krol 97).

Related entries

• (p,q)-string

• D=5 N=1 SYM

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

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)

References

The original articles are:

• Ofer Aharony, Amihay Hanany, Branes, Superpotentials and Superconformal Fixed Points, Nucl. Phys. B504:239-271, 1997 (arXiv:hep-th/9704170)

• Ofer Aharony, Amihay Hanany, Barak Kol, Webs of $(p,q)$ 5-branes, Five Dimensional Field Theories and Grid Diagrams, JHEP 9801:002,1998 (arXiv:hep-th/9710116)

• Oren Bergman, Gabi Zafrir, Lifting 4d dualities to 5d, JHEP04 (2015) 141 (arXiv:1410.2806)

The T-dual perspective are D4/NS5-brane webs:

• Edward Witten, Solutions Of Four-Dimensional Field Theories Via M Theory, Nucl. Phys. B500:3-42, 1997 (arXiv:hep-th/9703166)

Further intersection with orientifolds:

• Amihay Hanany, Alberto Zaffaroni, Issues on Orientifolds: On the brane construction of gauge theories with $SO(2n)$ global symmetry, JHEP 9907 (1999) 009 (arXiv:hep-th/9903242)

• Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Masato Taki, Futoshi Yagi, More on 5d descriptions of 6d SCFTs, JHEP10 (2016) 126 (arXiv:1512.08239)

• Gabi Zafrir, Brane webs in the presence of an $O5^-$-plane and 4d class S theories of type D, JHEP07 (2016) 035 (arXiv:1602.00130)

• Taro Kimura, Rui-Dong Zhu, Section 2 and 3 of Web Construction of ABCDEFG and Affine Quiver Gauge Theories (arXiv:1907.02382)

On $(p,q)$-5-branes as defect branes:

• Tetsuji Kimura, Defect $(p,q)$ Five-branes, Nucl.Phys. B893 (2015) 1-20 (arXiv:1410.8403, doi:10.1016/j.nuclphysb.2015.01.023)