nLab Nahm's equation

Contents

Context

Chern-Weil theory

Algebraic Quantum Field Theory

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

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

(…)

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 15:57:22. See the history of this page for a list of all contributions to it.