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

Transveral Dp-D(p+2)-brane intersections in fuzzy funnels

The boundary condition in the nonabelian DBI model of coincident Dp-branes describing their transversal intersection/ending with/on D(p+2)-branes is controled by Nahm's equation and thus exhibits the brane intersection-locus equivalently as:

  1. a fuzzy funnel noncommutative geometry interpolating between the Dp\mathrm{D}p- and the D(p+2)\mathrm{D}(p+2)-brane worldvolumes;

  2. geometric engineering of Yang-Mills monopoles in the worldvolume-theory of the ambient D(p+2)D(p+2)-branes.

(Diaconescu 97, Constable-Myers-Fafjord 99, Hanany-Zaffaroni 99, Gaiotto-Witten 08, Section 2.4, HLPY 08, GZZ 09)

More explicitly, for y∈(0,∞]y \in (0,\infty ] the transversal distance along the stack of NN Dp\mathrm{D}p-branes away from the D(p+2)\mathrm{D}(p+2)-brane, and for

X i∈C ∞((0,∞],𝔲(N))AAAi∈{1,2,3} X^i \in C^\infty\big( (0,\infty], \mathfrak{u}(N) \big) \phantom{AAA} i \in \{1,2,3\}

the three scalar fields on the worldvolume, the boundary condition is:

ddyX 3+[X 1,X 2]=0,ddyX 1+[X 2,X 3]=0,ddyX 2+[X 3,X 1]=0 \frac{d}{d y} X^3 + [X^1, X^2] \;=\; 0 \,, \;\;\; \frac{d}{d y} X^1 + [X^2, X^3] \;=\; 0 \,, \;\;\; \frac{d}{d y} X^2 + [X^3, X^1] \;=\; 0

as y→0y \to 0. These are Nahm's equations, solved by

X i(y)=1yρ i+non-singular X^i(y) = \frac{1}{y} \rho^i + \text{non-singular}

where

ρ:𝔰𝔲(2)βŸΆπ”²(N) \rho \;\colon\; \mathfrak{su}(2) \longrightarrow \mathfrak{u}(N)

is a Lie algebra homomorphism from su(2) to the unitary Lie algebra, and

ρ i≔ρ(Οƒ i) \rho^i \coloneqq \rho(\sigma^i)

is its complex-linear combination of values on the canonical Pauli matrix basis.

Equivalently. ρ\rho is an NN-dimensional complex Lie algebra representation of su(2). Any such is reducible as a direct sum of irreducible representations N (M5)\mathbf{N}^{(M5)}, for which there is exactly one, up to isomorphism, in each dimension N (M5)βˆˆβ„•N^{(M5)} \in \mathbb{N}:

(1)ρ≃⨁i(N i (M2)β‹…N i (M5)). \rho \;\simeq\; \underset{ i }{\bigoplus} \big( N_i^{(M2)} \cdot \mathbf{N}_i^{(M5)} \big) \,.

(Here the notation follows the discussion at M2/M5-brane bound states in the BMN model, which is the M-theory lift of the present situation).

Now each irrep N i (M5)\mathbf{N}_i^{(M5)} may be interpreted as a fuzzy 2-sphere of radius ∝(N i (M5)) 2βˆ’1\propto \sqrt{ \left( N_i^{(M5)}\right)^2 - 1 }, hence as the section of a fuzzy funnel at given y=Ο΅y = \epsilon, whence the totality of (1) represents a system of concentric fuzzy 2-spheres/fuzzy funnels.

graphics from Sati-Schreiber 19c


Moreover, since the complexification of su(2) is the complex special linear Lie algebra 𝔰𝔩(2,β„‚)\mathfrak{sl}(2,\mathbb{C}) (here) the solutions to the boundary conditions are also identified with finite-dimensional 𝔰𝔩(2,β„‚)\mathfrak{sl}(2,\mathbb{C}) Lie algebra representations:

(2)Οβˆˆπ”°π”©(2,β„‚)Rep. \rho \;\in\; \mathfrak{sl}(2,\mathbb{C}) Rep \,.


This is what many authors state, but it is not yet the full picture:

Also the worldvolume Chan-Paton gauge field component AA along yy participates in the brane intersection

A∈C ∞((0,∞],𝔲(N)) A \in C^\infty\big( (0,\infty], \mathfrak{u}(N) \big)

its boundary condition being that

[A,X i]=0AAAAfor alli∈{1,2,3} [A, X^i] \;=\; 0 \phantom{AAAA} \text{for all}\; i \in \{1,2,3\}

as y→0y \to 0 (Constable-Myers 99, Section 3.3, Thomas-Ward 06, p. 16, Gaiotto-Witten 08, Section 3.1.1)

Together with (2) this means that the quadruple of fields (X 1,X 2,X 3,A)(X^1,X^2,X^3,A) constitutes a Lie algebra representation of the general linear Lie algebra

𝔀𝔩(2,β„‚)≃𝔰𝔩(2,β„‚)⏟⟨X 1,X 2,X 3βŸ©βŠ•β„‚βŸβŸ¨A⟩ \mathfrak{gl}(2,\mathbb{C}) \;\simeq\; \underset{ \langle X^1, X^2, X^3 \rangle }{ \underbrace{ \mathfrak{sl}(2,\mathbb{C}) } } \oplus \underset{ \langle A \rangle }{ \underbrace{ \mathbb{C} } }

This makes little difference as far as bare Lie algebra representations are concerned, but it does make a crucial difference when these are regarded as metric Lie representations of metric Lie algebras, since 𝔀𝔩(2,β„‚)\mathfrak{gl}(2,\mathbb{C}) admits further invariant metrics…


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

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:

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 February 1, 2020 at 10:56:51. See the history of this page for a list of all contributions to it.