Dp-D(p+2)-brane bound state




bound state of Dp- with D(p+2)-branes.



Parallel intersection

Parallel: dissolves (Gava-Narain-Sarmadi 97)

Transversal intersection

The s-rule

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 Dpp-brane ends on any one D(p+2)(p+2)-brane.

For D4-D6 brane intersections:

graphics grabbed from Fazzi 17

For D6-D8 brane intersections:

graphics grabbed from Fazzi 17

graphics grabbed from Gaiotto-Tomasiello 14

As fuzzy funnels/Yang-Mills monopoles

Transversally intersecting Dpp-D(p+2)(p+2)-branes geometrically engineer Yang-Mills monopoles: their moduli space is the moduli space of monopoles/solutions of Nahm's equation

(Diaconescu 97, Hanany-Zaffaroni 99, HLPY 08, GZZ 09)

Specifically for p=6p = 6, i.e. for D6-D8 brane intersections, this fits with the Witten-Sakai-Sugimoto model geometrically engineering quantum chromodynamics, and then gives a geometric engineering of the Yang-Mills monopoles in actual QCD (HLPY 08, p. 16).

Here we are showing

  1. the color D4-branes;

  2. the flavor D8-branes;


    1. the 5d Chern-Simons theory on their worldvolume

    2. the corresponding 4d WZW model on the boundary

    both exhibiting the meson fields

  3. the baryon D4-branes

    (see below at WSS – Baryons)

  4. the Yang-Mills monopole D6-branes

    (see at D6-D8-brane bound state)

  5. the NS5-branes.

Single trace observables as 𝔰𝔲(2)\mathfrak{su}(2)-weight systems on chord diagrams

We discuss how the single trace observables on the fuzzy 2-sphere-sections of Dp-D(p+2) brane intersection fuzzy funnels are given by su(2)-Lie algebra weight systems on chord diagrams (following Ramgoolam-Spence-Thomas 04, McNamara-Papageorgakis 05, see McNamara 06, Section 4 for review).

For more see at weight systems on chord diagrams in physics.

graphics from Sati-Schreiber 19c

While in the commutative large N limit, all powers of the radius function on the fuzzy 2-sphere are equal

limNβ†’βˆžβˆ« S N 2R 2k=4Ο€; \underset{N\to \infty}{\lim} \int_{S^2_N} R^{2 k} \;=\; 4 \pi \,;

for finite NN there is an ordering ambiguity: In fact, the number of functions on the fuzzy 2-sphere at finite NN that all go to the same function R 2kR^{2k} in the large N limit grows rapidly with kk.

At k=1k = 1 there is the single radius observable (?)

∫ S N 2R 2=∫ S N 2βˆ‘iX iβ‹…X i=4Ο€NN 2βˆ’1 \int_{S^2_N} R^2 \;=\; \int_{S^2_N} \underset{i}{\sum} X_i \cdot X_i \;=\; 4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }

At k=2k = 2 there are, under the integral (?), two radius observables:

  1. ∫ S N 2βˆ‘i,jX iX iX jX j \int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j

  2. ∫ S N 2βˆ‘i,jX iX jX jX i\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i

(Here we are using that under the integral/trace, a cyclic permutation of the factors in the integrand does not change the result).

Similarly for higher kk, where the number of possible orderings increases rapidly. The combinatorics that appears here is familiar in knot theory:

Every ordering of operators, up to cyclic permutation, in the single trace observable Tr(R 2) nTr(R^2)^n is encoded in a chord diagram and the value of the corresponding single trace observable is the value of the su(2)-Lie algebra weight system on this chord diagram.

brane intersections/bound states/wrapped branes

S-duality\,bound states:



For parallel intersection

For parallel intersection:

  • E. Gava, K.S. Narain, M.H. Sarmadi, On the Bound States of pp- and (p+2)(p+2)-Branes, Nucl. Phys. B504 (1997) 214-238 (arxiv:hep-th/9704006)

For transversal intersections

As spikes/BIons

On Dp-D(p+2) brane intersections as spikes/BIons?

from the M5-brane:

As fuzzy funnels/Yang-Mills monopoles

On transversal Dp-D(p+2) brane intersections as Yang-Mills monopoles / fuzzy funnel-solutions to Nahm's equation:

For transversal D1-D3 brane intersections:

For transversal D2-D4-brane bound states (with an eye towards AdS/QCD):

  • Alexander Gorsky, Valentin Zakharov, Ariel Zhitnitsky, On Classification of QCD defects via holography, Phys. Rev. D79:106003, 2009 (arxiv:0902.1842)

For transversal D3-D5 brane intersections:

For transversal D6-D8 brane intersections (with an eye towards AdS/QCD):

  • Deog Ki Hong, Ki-Myeong Lee, Cheonsoo Park, Ho-Ung Yee, Section V of: Holographic Monopole Catalysis of Baryon Decay, JHEP 0808:018, 2008 (https:arXiv:0804.1326)

and as transversal D6-D8-brane bound states on a half NS5-brane in type I' string theory:

Relation to Vassiliev braid invariants

Relation of Dp-D(p+2)-brane bound states (hence Yang-Mills monopoles) to Vassiliev braid invariants via chord diagrams computing radii of fuzzy spheres:

Lift to M2-M5-brane bound states

The lift of Dp-D(p+2)-brane bound states in string theory to M2-M5-brane bound states/E-strings in M-theory, under duality between M-theory and type IIA string theory+T-duality, via generalization of Nahm's equation (this eventually motivated the BLG-model/ABJM model):

Single trace observables as weight systems on chord duagrams

Relation of single trace observables in the non-abelian DBI action on Dp-D(p+2)-brane bound states (hence Yang-Mills monopoles) to su(2)-Lie algebra weight systems on chord diagrams computing radii averages of fuzzy spheres:

Last revised on December 24, 2019 at 04:00:30. See the history of this page for a list of all contributions to it.