nLab fuzzy funnel

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 $\mathrm{D}p$- and the $\mathrm{D}(p+2)$-brane worldvolumes;

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

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

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

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

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

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

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

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

where

$$\rho \;\colon\; \mathfrak{su}(2) \longrightarrow \mathfrak{u}(N)$$

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

$$\rho^i \coloneqq \rho(\sigma^i)$$

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

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

(1)$$\rho \;\simeq\; \underset{ i }{\bigoplus} \big( N_i^{(M2)} \cdot \mathbf{N}_i^{(M5)} \big) \,.$$

(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 $\mathbf{N}_i^{(M5)}$ may be interpreted as a fuzzy 2-sphere of radius $\propto \sqrt{ \left( N_i^{(M5)}\right)^2 - 1 }$, hence as the section of a fuzzy funnel at given $y = \epsilon$, 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 $\mathfrak{sl}(2,\mathbb{C})$ (here) the solutions to the boundary conditions are also identified with finite-dimensional $\mathfrak{sl}(2,\mathbb{C})$ Lie algebra representations:

(2)$$\rho \;\in\; \mathfrak{sl}(2,\mathbb{C}) Rep \,.$$

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

Also the worldvolume Chan-Paton gauge field component $A$ along $y$ participates in the brane intersection

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

its boundary condition being that

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

as $y \to 0$ (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 $(X^1,X^2,X^3,A)$ constitutes a Lie algebra representation of the general linear Lie algebra

$$\mathfrak{gl}(2,\mathbb{C}) \;\simeq\; \underset{ \langle X^1, X^2, X^3 \rangle }{ \underbrace{ \mathfrak{sl}(2,\mathbb{C}) } } \oplus \underset{ \langle A \rangle }{ \underbrace{ \mathbb{C} } }$$

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 $\mathfrak{gl}(2,\mathbb{C})$ admits further invariant metrics…

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.

Related concepts

• Nahm's equation

• fuzzy sphere

• noncommutative torus

brane intersections/bound states/wrapped branes/polarized branes

• D-branes and anti D-branes form bound states by tachyon condensation, thought to imply the classification of D-brane charge by K-theory

• intersecting D-branes/fuzzy funnels:

  • Dp-D(p+2)-brane bound state and Yang-Mills monopoles

    • D0-D2 brane bound state

    • D1-D3 brane bound state

    • D2-D4 brane bound state

    • D3-D5 brane bound state

    • D4-D6 brane bound state

    • D6-D8-brane bound state

  • Dp-D(p+4)-brane bound state and Yang-Mills instantons

    • D0-D4 brane bound state

    • D1-D5 brane bound state

    • D2-D6 brane bound state

    • D3-D7 brane bound state

    • D4-D8 brane bound state

    • D5-D9 brane bound state

      small instanton

• Dp-D(p+6) brane bound state

  • D0-D6 brane bound state

  • D1-D7 brane bound state

  • D2-D8 brane bound state

• NS5-D4-D2 bound states

• Hanany-Witten effect

S-duality$\,$bound states:

• (p,q)-string

• (p,q)5-branes

• string network

• 5-brane web

intersecting$\,$M-branes:

• M5-MO9 brane bound state

• M2-M5 brane bound state

  • M-string, E-string

• membrane triple junction

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:

• Neil Constable, Robert Myers, Oyvind Tafjord, The Noncommutative Bion Core, Phys. Rev. D61 (2000) 106009 (arXiv:hep-th/9911136)

• Robert Myers, Section 4 of: Nonabelian D-branes and Noncommutative Geometry, J. Math. Phys. 42: 2781-2797, 2001 (arXiv:hep-th/0106178)

• Neil Constable, Neil Lambert, Calibrations, Monopoles and Fuzzy Funnels, Phys. Rev. D66 (2002) 065016 (arXiv:hep-th/0206243)

For D3-D5 brane intersections

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

• Davide Gaiotto, Edward Witten, Section 3.4.3 of: Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory, J Stat Phys (2009) 135: 789 (arXiv:0804.2902)

• Marco Fazzi, Section 3.2.3 of: Higher-dimensional field theories from type II supergravity (arxiv:1712.04447)

For D6-D8 brane intersections

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

• Davide Gaiotto, Alessandro Tomasiello, Holography for $(1,0)$ theories in six dimensions, JHEP 12 (2014) 003 (arXiv:1404.0711)

• Fazzi 17, p. 41-46

For D1-D5-brane intersections

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

• Neil Constable, Robert Myers, Oyvind Tafjord, Non-abelian Brane Intersections, JHEP 0106:023, 2001 (arXiv:hep-th/0102080)

For D1-D7-brane intersections

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

• Paul Cook, Robert de Mello Koch, Jeff Murugan, Non-Abelian BIonic Brane Intersections, Phys. Rev. D68:126007, 2003 (arXiv:hep-th/0306250)

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:

• Sanyaje Ramgoolam, Bill Spence, S. Thomas, Section 3.2 of: Resolving brane collapse with $1/N$ corrections in non-Abelian DBI, Nucl. Phys. B703 (2004) 236-276 (arxiv:hep-th/0405256)

• Simon McNamara, Constantinos Papageorgakis, Sanyaje Ramgoolam, Bill Spence, Appendix A of: Finite $N$ effects on the collapse of fuzzy spheres, JHEP 0605:060, 2006 (arxiv:hep-th/0512145)

• Simon McNamara, Section 4 of: Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels, 2006 (spire:1351861, pdf, pdf)

• Constantinos Papageorgakis, p. 161-162 of: On matrix D-brane dynamics and fuzzy spheres, 2006 (pdf)