nLab
fuzzy funnel

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 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.


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

limN→∞∫ S N 2R 2k=4π; \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 2∑iX i⋅X i=4πNN 2−1 \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 2∑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 2∑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:

intersecting\,M-branes:

References

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:

Last revised on December 9, 2019 at 03:02:14. See the history of this page for a list of all contributions to it.