nLab s-rule

Idea

In string theory, specifically for intersecting D-brane models, what has come to be known as the s-rule is the conjecture that the configuration of Dp-D(p+2)-brane bound states with the Dp-branes stretching from the D(p+2)-branes to NS5-branes, can be supersymmetric only if at most one D$p$-brane ends on any one D$(p+2)$-brane (Hanany-Witten 97).

Examples

D3-D5 brane intersections

For D3-D5 brane intersections:

Hanay-Witten 97

D4-D6 brane intersections

For D4-D6 brane intersections:

graphics grabbed from Fazzi 17

D6-D8 brane intersections

For D6-D8 brane intersections:

graphics grabbed from Gaiotto-Tomasiello 14

graphics grabbed from Fazzi 17

graphics grabbed from GKSTY 02

Related concepts

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

The s-rule is attributed to

• Amihay Hanany, Edward Witten, Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics, Nucl. Phys. B 492:152-190, 1997 (arXiv:hep-th/9611230)

which considers D3-D5 brane intersections, speaks of “s-configurations” as those with more than one D3-brane ending on any D5-brane and gives plausibility arguments that these s-configurations are not supersymmetric.

Review:

• Amit Giveon, David Kutasov, p. 83 onwards in: Brane Dynamics and Gauge Theory, Rev. Mod. Phys. 71:983-1084, 1999 (arXiv:hep-th/9802067)

Relation to the Pauli exclusion principle:

• Constantin Bachas, Michael Green, A Classical Manifestation of the Pauli Exclusion Principle, JHEP 9801 (1998) 015 (arXiv:hep-th/9712187)

See also

• Wikipedia, Hanany-Witten transition – The S-rule

For D2-D4 brane intersections:

• Minoru Eto, Youichi Isozumi, Muneto Nitta, Keisuke Ohashi, Kazutoshi Ohta, Norisuke Sakai, Section 3 of: D-brane Construction for Non-Abelian Walls, Phys. Rev. D71 (2005) 125006 (arXiv:hep-th/0412024)

For D3-D5 brane intersections:

• Constantin Bachas, Michael Green, Adam Schwimmer, Section 2.3 of: $(8,0)$ Quantum mechanics and symmetry enhancement in type I’ superstrings, JHEP 9801 :006, 1998 (arXiv:hep-th/9712086)

• Akikazu Hashimoto, Peter Ouyang, Masahito Yamazaki, Section 3.6 of: Boundaries and Defects of N=4 SYM with 4 Supercharges, Part II: Brane Constructions and 3d N=2 Field Theories, JHEP 1410 : 108, 2014 (arXiv:1406.5501)

• Federico Carta, Hirotaka Hayashi, Hilbert series and mixed branches of $T[SU(N)]$ theory, JHEP 02 (2017) 037 (arXiv:1609.08034)

For D4-D6 brane intersections:

• Hirosi Ooguri, Cumrun Vafa, Section 6.2 of: Geometry of $N = 1$ Dualities in Four Dimensions, Nucl. Phys. B 500:62-74, 1997 (arXiv:hep-th/9702180)

• Kentaro Hori, Hirosi Ooguri, Yaron Oz, Section 3 of: Strong Coupling Dynamics of Four-Dimensional N=1 Gauge Theories from M Theory Fivebrane, Adv. Theor. Math. Phys.1:1-52, 1998 (arXiv:hep-th/9706082)

For D6-D8 brane intersections:

• Davide Gaiotto, Alessandro Tomasiello, p. 24-25 in: Holography for $(1,0)$ theories in six dimensions, JHEP 12 (2014) 003 (arXiv:1404.0711)

Review includes:

• Marco Fazzi, around Figure 3.5 and Figure 3.13 in: Higher-dimensional field theories from type II supergravity (arxiv:1712.04447)