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Nahm's equation

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Chern-Weil theory

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

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(…)

Properties

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 iC ((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 y0y \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)𝔲(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 iρ(σ 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)N^{(M5)} \in \mathbb{N}:

(1)ρi(N i (M2)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)) 21\propto \sqrt{ \left( N_i^{(M5)}\right)^2 - 1 }, hence as the section of a fuzzy funnel at given y=ϵy = \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,)\mathfrak{sl}(2,\mathbb{C}) (here) the solutions to the boundary conditions are also identified with finite-dimensional 𝔰𝔩(2,)\mathfrak{sl}(2,\mathbb{C}) Lie algebra representations:

(2)ρ𝔰𝔩(2,)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

AC ((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 y0y \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

𝔤𝔩(2,)𝔰𝔩(2,)X 1,X 2,X 3A \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 𝔤𝔩(2,)\mathfrak{gl}(2,\mathbb{C}) admits further invariant metrics…


References

General

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 in SYM

In terms of Coulomb branch singularities on super Yang-Mills theories:

In term of Dp-D(p+2) brane intersections

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) 𝔰𝔩(2,)\mathfrak{su}(2)_{\mathbb{C}} \simeq \mathfrak{sl}(2,\mathbb{C})-representation to a 𝔤𝔩(2,)\mathfrak{gl}(2,\mathbb{C})-representation by adjoining the gauge field component A yA_y to the scalar fields X\vec X:

Lift to M-theory

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):

Last revised on February 1, 2020 at 10:57:22. See the history of this page for a list of all contributions to it.