nLab Dp-D(p+2)-brane bound state




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



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}


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…

D6βŠ₯D8\mathrm{D}6 \perp \mathrm{D}8-brane intersections

Specifically for p=6p = 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;


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

    2. the corresponding 4d WZW model on the boundary

    both exhibiting the meson fields

  3. the baryon D4-branes

    (see below at WSS – Baryons)

  4. the Yang-Mills monopole D6-branes

    (see at D6-D8-brane bound state)

  5. the NS5-branes.

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

For D4-D6 brane intersections:

graphics grabbed from Fazzi 17

For D6-D8 brane intersections:

graphics grabbed from Fazzi 17

graphics grabbed from Gaiotto-Tomasiello 14

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.

graphics from Sati-Schreiber 19c

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.

Parallel intersection

Parallel: dissolves (Gava-Narain-Sarmadi 97)

brane intersections/bound states/wrapped branes/polarized branes

S-duality\,bound states:



For parallel intersection

For parallel intersection:

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

As fuzzy funnels/Yang-Mills 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:

Making explicit the completion of the 𝔰𝔲(2) ℂ≃𝔰𝔩(2,β„‚)\mathfrak{su}(2)_{\mathbb{C}} \simeq \mathfrak{sl}(2,\mathbb{C})-representation to a 𝔀𝔩(2,β„‚)\mathfrak{gl}(2,\mathbb{C})-representation by adjoining the gauge field component A yA_y to the scalar fields Xβ†’\vec X:

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 in the non-abelian DBI action 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:

In solid state physics

Relation of Yang-Mills monopoles as Dp/D(p+2)-brane intersections and Yang-Mills instantons as Dp/D(p+4)-brane intersections to the K-theory classification of topological phases of matter via AdS/CFT duality in solid state physics:

Last revised on February 12, 2020 at 12:17:07. See the history of this page for a list of all contributions to it.