(p,q)5-brane in nLab

Contents

Contents

Idea

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

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

Properties

(p,q)(p,q)-Brane webs

Label local coordinate functions x ax^a on 10d Minkowski spacetime ℝ 9,1\mathbb{R}^{9,1} by 0123455′67890123455'6789 and write v a≔∂ x av_a \coloneqq \partial_{x^a} for the corresponding vector field

Consider a (1,0)(1,0)5-brane (a D5-brane) along the multivector field v 0v 1v 2v 3v 4v 5v_0 v_1 v_2 v_3 v_4 v_5 and a (0,1)(0,1)5-brane (an NS-brane) along v 0v 1v 2v 3v 4v 5′v_0 v_1 v_2 v_3 v_4 v_{5'}

a=a =	00	11	22	33	44	55	5′5'	66	77	88	99
D5	—	—	—	—	—	—
NS5	—	—	—	—	—		—

Charge conservation implies that at the brane intersection of the two a (1,1)(1,1)5-brane emerges stretched diagonally along v 5+v g′v_5 + v_{g'}, i.e. along the multivector field v 0v 1v 2v 3v 4(v 5+v 5′)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)(p,q)-5brane webs (Aharony-Hanany-Krol 97).

• (p,q)-string

• D=5 N=1 SYM

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)(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)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 −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)(p,q)-5-branes as defect branes:

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