algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
(…)
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 - and the -brane worldvolumes;
geometric engineering of Yang-Mills monopoles in the worldvolume-theory of the ambient -branes.
(Diaconescu 97, Constable-Myers-Fafjord 99, Hanany-Zaffaroni 99, Gaiotto-Witten 08, Section 2.4, HLPY 08, GZZ 09)
More explicitly, for the transversal distance along the stack of -branes away from the -brane, and for
the three scalar fields on the worldvolume, the boundary condition is:
as . 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. is an -dimensional complex Lie algebra representation of su(2). Any such is reducible as a direct sum of irreducible representations , for which there is exactly one, up to isomorphism, in each dimension :
(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 may be interpreted as a fuzzy 2-sphere of radius , hence as the section of a fuzzy funnel at given , 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 (here) the solutions to the boundary conditions are also identified with finite-dimensional 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 along participates in the brane intersection
its boundary condition being that
as (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 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 admits further invariant metrics…
The original articles;
Werner Nahm, The construction of all self-dual multi-monopoles by the ADHM method, In: Craigie et al. (eds.), Monopoles in quantum theory, Singapore, World Scientific 1982 (spire:178340)
Werner Nahm, Self-dual monopoles and calorons, In: G. Denardo, G. Ghirardi and T. Weber (eds.), Group theoretical methods in physics, Lecture Notes in Physics 201. Berlin, Springer-Verlag 1984 (doi:10.1007/BFb0016145)
Simon Donaldson, Nahm’s Equations and the Classification of Monopoles, Comm. Math. Phys., Volume 96, Number 3 (1984), 387-407, (euclid:cmp.1103941858)
Further discussion:
Review:
Marcos Jardim, A survey on Nahm transform, J Geom Phys 52 (2004) 313-327 (arxiv:math/0309305)
Reinier Storm, The Yang-Mills Moduli Space and The Nahm Transform (dspace:1874/285043)
See also
In terms of Coulomb branch singularities on super Yang-Mills theories:
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)
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)
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 -representation to a -representation by adjoining the gauge field component to the scalar fields :
Sergey Cherkis, Instantons on Gravitons, around (21) in: Commun. Math. Phys. 306:449-483, 2011 (arXiv:1007.0044)
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 Nahm’s 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)
Last revised on February 1, 2020 at 15:57:22. See the history of this page for a list of all contributions to it.