Contents

# Contents

## Idea

The microscopic geometry of transversal Dp-D(p+2)-brane intersections and Dp-D(p+4)-brane intersections look like warped non-commutative metric cones on fuzzy spheres (namely on the spheres around the lower dimensional D-branes inside the higher dimensional D-branes). These have hence been called fuzzy funnels. graphics grabbed from Fazzi 17, Fig. 3.14, taken in turn from Gaiotto-Tomassiello 14, Figure 5 graphics grabbed from Fazzi 17

## Properties

### 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:

### General

• Rajsekhar Bhattacharyya, Robert de Mello Koch, Fluctuating Fuzzy Funnels, JHEP 0510 (2005) 036 (arXiv:hep-th/0508131)

### For D1-D3-brane intersections

On D1-D3 brane intersections as fuzzy funnels on fuzzy 2-spheres:

### For D3-D5 brane intersections

On D3-D5 brane intersections as fuzzy funnels on fuzzy 2-spheres:

### For D6-D8 brane intersections

On D6-D8 brane intersections as fuzzy funnels on fuzzy 2-spheres:

### For D1-D5-brane intersections

On D1-D5 brane intersections as fuzzy funnels on fuzzy 4-spheres:

### For D1-D7-brane intersections

On D1-D7 brane intersections as fuzzy funnels on fuzzy 6-spheres:

### 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: