Contents

# Contents

## Idea

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

## Properties

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

graphics grabbed from Fazzi 17

graphics grabbed from Fazzi 17

graphics grabbed from Gaiotto-Tomasiello 14

#### Relation to Yang-Mills monopoles

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

Specifically for $p = 6$, 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;

with

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

2. the corresponding 4d WZW model on the boundary

both exhibiting the meson fields

3. (see below at WSS – Baryons)

4. (see at D6-D8-brane bound state)

5. the NS5-branes.

#### Single trace observables as $\mathfrak{su}(2)$-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.

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

$\underset{N\to \infty}{\lim} \int_{S^2_N} R^{2 k} \;=\; 4 \pi \,;$

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

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

$\int_{S^2_N} R^2 \;=\; \int_{S^2_N} \underset{i}{\sum} X_i \cdot X_i \;=\; 4 \pi \tfrac{ N }{ \sqrt{N^2 -1} }$

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

1. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j$

2. $\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i$

(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 $k$, 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 $Tr(R^2)^n$ 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:

intersecting$\,$M-branes:

## References

### For parallel intersection

For parallel intersection:

• E. Gava, K.S. Narain, M.H. Sarmadi, On the Bound States of $p$- and $(p+2)$-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:

#### Relation to 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 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 4, 2019 at 13:07:46. See the history of this page for a list of all contributions to it.