Nahm Equations Research Articles

Aspects of q-discretized Nahm equations

There is a conjecture by Ward that almost all of integrable equations are derived from (anti-)self-dual (ASD) Yang–Mills equations. This conjecture is supported by many concrete examples, e.g., the Nahm equations. In this work, we consider a situation that if the ASD conditions are slightly loosened, as to how it affects the integrability of the equations. For this purpose, we consider a q-analog of the Nahm equations, as a non-ASD system. The analysis is performed on the reduced system which is a q-analog of the Euler–Arnold top, by the singularity confinement test and the estimation of the algebraic entropy.

Towards the quantum geometry of the M5-brane in a constantC-field from multiple membranes

We show that the Nahm equation which describes a fuzzy D3-brane in the presence of a B-field can be derived as a boundary condition of the F1-strings ending on the D3-brane, and that the modifications of the original Nahm equation by a B-field can be understood in terms of the noncommutative geometry of the D3-brane. Naturally this is consistent with the alternative derivation by quantising the open strings in the B-field background. We then consider a configuration of multiple M2-branes ending on an M5-brane with a constant 3-form C-field. By analogy with the case of strings ending on a D3-brane with a constant B-field, one can expect that this system can be described in terms of the boundary of the M2-branes moving within a certain kind of quantum geometry on the M5-brane worldvolume. By repeating our analysis, we show that the analogue of the B-field modified Nahm equation, the C-field modified Basu-Harvey equation can also be understood as a boundary condition of the M2-branes. We then compare this to the M5-brane BIon description and show that the two descriptions match provided we postulate a new type of quantum geometry on the M5-brane worldvolume. Unlike the D-brane case, this is naturally expressed in terms of a relation between a 3-bracket of the M5-brane worldvolume coordinates and the C-field.

FINITE-GAP INTEGRATION OF THE SU(2) BOGOMOLNY EQUATIONS

AbstractThe Atiyah–Drinfeld–Hitchin–Manin–Nahm (ADHMN) construction of magnetic monopoles is given in terms of the (normalizable) solutions of an associated Weyl equation. We focus here on solving this equation directly by algebro-geometric means. The (adjoint) Weyl equation is solved using an ansatz of Nahm in terms of Baker–Akhiezer functions. The solution of Nahm's equation is not directly used in our development.

Topological branes, p-algebras and generalized Nahm equations

Inspired by the recent advances in multiple M2-brane theory, we consider the generalizations of Nahm equations for arbitrary p-algebras. We construct the topological p-algebra quantum mechanics associated to them and we show that this can be obtained as a truncation of the topological p-brane theory previously studied by the authors. The resulting topological p-algebra quantum mechanics is discussed in detail and the relation with the M2–M5 system is pointed out in the p=3 case, providing a geometrical argument for the emergence of the 3-algebra structure in the Bagger–Lambert–Gustavsson theory.

Yang–Mills instantons and dyons on group manifolds

We consider Euclidean SU(N) Yang–Mills theory on the space G×R, where G is a compact semisimple Lie group, and introduce first-order BPS-type equations which imply the full Yang–Mills equations. For gauge fields invariant under the adjoint G-action these BPS equations reduce to first-order matrix equations, to which we give instanton solutions. In the case of G=SU(2)≅S3, our matrix equations are recast as Nahm equations, and a further algebraic reduction to the Toda chain equations is presented and solved for the SU(3) example. Finally, we change the metric on G×R to Minkowski and construct finite-energy dyon-type Yang–Mills solutions. The special case of G=SU(2)×SU(2) may be used in heterotic flux compactifications.

M2-M5 systems in 𝒩 = 6 Chern-Simons theory

We study two aspects of M5-branes in N=6 U(N)xU(N) Chern-Simons gauge theory. We first examine multiple M2-branes ending on a M5-brane. We study Basu-Harvey type cubic equations, fuzzy funnel configurations, and derive the M5-brane tension from the N=6 theory. We also find a limit in which the above M2-M5 system reduces to a D2-D4 system and we recover the Nahm equation from the N=6 theory. We then examine domain wall configurations in mass-deformed N=6 theory with a manifest SU(2)xSU(2)xU(1) global symmetry. We derive tensions of domain walls connecting between arbitrary M5-brane vacua of the deformed theory and observe their consistency with gravity dual expectations.

Classification of the BPS states in Bagger-Lambert theory

We classify, in a group theoretical manner, the BPS configurations in the multiple M2-brane theory recently proposed by Bagger and Lambert. We present three types of BPS equations preserving various fractions of supersymmetries: in the first type we have constant fields and the interactions are purely algebraic in nature; in the second type the equations are invariant under spatial rotation SO(2), and the fields can be time-dependent; in the third class the equations are invariant under boost SO(1,1) and provide the eleven-dimensional generalizations of the Nahm equations. The BPS equations for different number of supersymmetries exhibit the division algebra structures: octonion, quarternion or complex.

Selfdual strings and loop space Nahm equations

We give two independent arguments why the classical membrane fields should be take values in a loop algebra. The first argument comes from how we may construct selfdual strings in the M5 brane from a loop space version of the Nahm equations. The second argument is that there appears to be no infinite set of finite-dimensional Lie algebras (such as su(N) for any N) that satisfies the algebraic structure of the membrane theory.

Affine A3(1) N = 2 Monopole as the D Module and Affine ADHMN Sheaf

A Higgs–Yang–Mills monopole scattering spherical symmetrically along light cones is given. The left incoming anti-self-dual a plane fields are holomorphic, but the right outgoing SD β plane fields are antiholomorphic, meanwhile the diffeomorphism symmetry is preserved with mutual inverse affine rapidity parameters μ and μ−1. The Dirac wave function scattering in this background also factorized respectively into the (anti)holomorphic amplitudes. The holomorphic anomaly is realized by the center term of a quasi Hopf algebra corresponding to an integrable conformal affine massive field. We find explicit Nahm transformation matrix (Fourier-Mukai transformation) between the Higgs YM BPS (flat) bundles (D modules) and the affinized blow up ADHMN twistors (perverse sheafs). Thus we establish the algebra for the 't Hooft–Hecke operators in the Hecke correspondence of the geometric Langlands program.

Nahm transform for integrable connections on the Riemann sphere

Dans ce texte, nous definissons la transformee de Nahm pour les connexions integrables ayant des singularites regulieres et une singularite irreguliere sur la sphere de Riemann. Apres une definition en terme de cohomologie L^2, nous donnons une description algebrique en terme d'hypercohomologie. En se servant de cette double interpretation, nous decrivons l'objet transformee a la fois par des formules analytiques explicites et geometriquement en utilisant la courbe spectrale du probleme. Finalement, nous demontrons que la correspondance definie est (a un signe pres) une involution.

On Monopoles and Domain Walls

The purpose of this paper is to describe a relationship between maximally supersymmetric domain walls and magnetic monopoles. We show that the moduli space of domain walls in non-abelian gauge theories with N flavors is isomorphic to a complex, middle dimensional, submanifold of the moduli space of U(N) magnetic monopoles. This submanifold is defined by the fixed point set of a circle action rotating the monopoles in the plane. To derive this result we present a D-brane construction of domain walls, yielding a description of their dynamics in terms of truncated Nahm equations. The physical explanation for the relationship lies in the fact that domain walls, in the guise of kinks on a vortex string, correspond to magnetic monopoles confined by the Meissner effect.

A note on the M2–M5 brane system and fuzzy spheres

This note covers various aspects of recent attempts to describe membranes ending on fivebranes using fuzzy geometry. In particular, we examine the Basu-Harvey equation and its relation to the Nahm equation as well as the consequences of using a non-associative algebra for the fuzzy three-sphere. This produces the tantalising result that the fuzzy funnel solution corresponding to Q coincident membranes ending on a five-brane has $Q^{3/2}$ degrees of freedom.

Moduli Space of IIB Superstring and SUYMin AdS5×S5

This paper consists of two parts.In part I, we interpret the hidden symmetry of the moduli space ofIIB superstring on AdS5×S5 in terms of the chiralembedding in AdS5, which turns out to be the ℂℙ3conformal affine Toda model. We review how the position μ ofpoles in the Riemann–Hilbert formulation of dressingtransformation and the value of loop parameter μ in thevertex operator of affine algebra determine the moduli space ofthe soliton solutions, which describes the moduli space of theGreen–Schwarz superstring. We show also how this affine SU(4)symmetry affinizes the conformal symmetry in the twistor space, andhow a soliton string corresponds to a Robinson congruence withtwist and dilation spin coefficients μ of twistor.In part II, by extending the dressing symmetric action of IIB stringin AdS5×S5 to the D3 brane, we find a gaugedWZW action of Higgs Yang–Mills field including the 2-cocycle ofaxially anomaly. The left and right twistor structures of left andright α-planes glue into an ambitwistor. The symmetrygroup of Nahm equations is centrally extended to an affine group,thus we explain why the spectral curve is given by affine Toda.

M2-branes stretching between M5-branes

A generalization of Nahm's equation has been recently conjectured by Basu and Harvey to be the BPS condition describing the bound state of a stack of M2-branes ending on an M5-brane. In this note exact solutions are presented for the proposed BPS equation - which is from the point of view of the M2-brane world-volume dynamics - with boundary conditions appropriate for M2-branes stretching between two M5-branes. Unfortunately, since the action for multiple M5-branes or for multiple coincident M2-branes is not known, one can only resort to consistency checks of the proposal instead of a direct comparison of the M2 and M5 world-volume point of views. The existence of our solutions should be seen as such a consistency check of the conjecture, and also as a source of new insight into the dynamics of multiple M2 and M5-branes.

Brackets, forms and invariant functionals

In the context of generalized geometry we first show how the Courant bracket helps to define connections with skew torsion and then investigate a five-dimensional invariant functional and its associated geometry, which involves three Courant-commuting sections of $T \bigoplus T\sp *$ . A Hamiltonian flow arising from this corresponds to a version of the Nahm equations, and we investigate the sixdimensional geometrical structure this describes.

Hamiltonian formulation and integrability of a complex symmetric nonlinear system

The integrability of a complex generalization of the ‘elegant’ system, proposed by Fairlie [D. Fairlie, Phys. Lett. A 119 (1987) 438] and its relation to the Nahm equation and the Manakov top is discussed.

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.

The M2–M5 brane system and a generalized Nahm's equation

We propose an equation that describes M2-branes ending on M5-branes, and which generalizes the description of the D1–D3 system via Nahm's equation. The simplest solution to this equation constructs the transverse geometry in terms of a fuzzy three sphere. We show that the solution passes a number of consistency checks including a calculation of the energy of the system, matching to the self-dual string solution in the M5-brane worldvolume, and a study of simple fluctuations about the ground state configuration. We write down certain terms in the effective action of multiple membranes, which includes a sextic scalar coupling.

A survey on Nahm transform

We review the construction known as the Nahm transform in a generalised context, which includes all the examples of this construction already described in the literature. The Nahm transform for translation invariant instantons on R 4 is presented in an uniform manner. We also briefly analyse two new examples, the second of which being the first example involving a four-manifold that is not hyperkähler.

Complexification and Hypercomplexification of Manifolds with a Linear Connection

We give a simple interpretation of the adapted complex structure of Lempert–Szöke and Guillemin–Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperkähler metric.