bound state of Dp- with D(p+2)-branes.
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:
a fuzzy funnel noncommutative geometry interpolating between the $\mathrm{D}p$- and the $\mathrm{D}(p+2)$-brane worldvolumes;
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
the three scalar fields on the worldvolume, the boundary condition is:
as $y \to 0$. These are Nahm's equations, solved by
where
is a Lie algebra homomorphism from su(2) to the unitary Lie algebra, and
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 $N^{(M5)} \in \mathbb{N}$:
(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:
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
its boundary condition being that
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
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 metricsβ¦
Specifically for $p = 6$, 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
with
the 5d Chern-Simons theory on their worldvolume
the corresponding 4d WZW model on the boundary
both exhibiting the meson fields
(see below at WSS β Baryons)
the Yang-Mills monopole D6-branes
(see at D6-D8-brane bound state)
the NS5-branes.
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 D$p$-brane ends on any one D$(p+2)$-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
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
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 (?)
At $k = 2$ there are, under the integral (?), two radius observables:
$\int_{S^2_N} \underset{i,j}{\sum} X_i X_i X_j X_j$
$\int_{S^2_N} \underset{i,j}{\sum} X_i X_j X_j X_i$
(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.
Parallel: dissolves (Gava-Narain-Sarmadi 97)
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+6) brane bound state
intersecting$\,$M-branes:
For parallel intersection:
On Dp-D(p+2) brane intersections as spikes/BIons
Curtis Callan, Juan Maldacena, Brane Dynamics From the Born-Infeld Action, Nucl. Phys. B513 (1998) 198-212 (arXiv:hep-th/9708147)
Paul Howe, Neil Lambert, Peter West, The Self-Dual String Soliton, Nucl. Phys. B515 (1998) 203-216 (arXiv:hep-th/9709014)
Gary Gibbons, Born-Infeld particles and Dirichlet p-branes, Nucl. Phys. B514: 603-639, 1998 (arXiv:hep-th/9709027)
from the M5-brane:
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:
Duiliu-Emanuel Diaconescu, D-branes, Monopoles and Nahm Equations, Nucl. Phys. B503 (1997) 220-238 (arxiv:hep-th/9608163)
Amihay Hanany, Edward Witten, Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics, Nucl. Phys. B492:152-190, 1997 (arxiv:hep-th/9611230)
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)
Jessica K. Barrett, Peter Bowcock, Using D-Strings to Describe Monopole Scattering (arxiv:hep-th/0402163)
Jessica K. Barrett, Peter Bowcock, Using D-Strings to Describe Monopole Scattering - Numerical Calculations (arxiv:hep-th/0512211)
Steven Thomas, John Ward, Electrified Fuzzy Spheres and Funnels in Curved Backgrounds, JHEP 0611:019, 2006 (arXiv:hep-th/0602071)
For transversal D2-D4-brane bound states (with an eye towards AdS/QCD):
For transversal D3-D5 brane intersections:
For transversal D6-D8 brane intersections (with an eye towards AdS/QCD):
and as transversal D6-D8-brane bound states on a half NS5-brane in type I' string theory:
Making explicit the completion of the $\mathfrak{su}(2)_{\mathbb{C}} \simeq \mathfrak{sl}(2,\mathbb{C})$-representation to a $\mathfrak{gl}(2,\mathbb{C})$-representation by adjoining the gauge field component $A_y$ to the scalar fields $\vec X$:
Sergey Cherkis, Instantons on Gravitons, around (21) in: Commun. Math. Phys. 306:449-483, 2011 (arXiv:1007.0044)
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:
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 Field Theory and Fuzzy Funnels, 2006 (spire:1351861, pdf, pdf)
Constantinos Papageorgakis, p. 161-162 of: On matrix D-brane dynamics and fuzzy spheres, 2006 (pdf)
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):
Anirban Basu, Jeffrey Harvey, The M2-M5 Brane System and a Generalized Nahmβs Equation, Nucl.Phys. B713 (2005) 136-150 (arXiv:hep-th/0412310)
Jonathan Bagger, Neil Lambert, Sunil Mukhi, Constantinos Papageorgakis, Section 2.2.1 of Multiple Membranes in M-theory, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 (arXiv:1203.3546, doi:10.1016/j.physrep.2013.01.006)
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:
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)
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.