Nahm Equations Research Articles

Constituent monopoles through the eyes of fermion zero-modes

We use the fermion zero-modes in the background of multi-caloron solutions with non-trivial holonomy as a probe for constituent monopoles. We find in general indication for an extended structure. However, for well separated constituents these become point-like. We analyze this in detail for the SU(2) charge 2 case, where one is able to solve the relevant Nahm equation exactly, beyond the piecewize constant solutions studied previously. Remarkably the zero-mode density can be expressed in the high temperature limit as a function of the conserved quantities that classify the solutions of the Nahm equation.

Darboux covariant equations of von Neumann type and their generalizations

Lax pairs with operator valued coefficients, which are explicitly connected by means of an additional condition, are considered. This condition is proved to be covariant with respect to the Darboux transformation of a general form. Nonlinear equations arising from the compatibility condition of the Lax pairs in the matrix case include, in particular, Nahm equations, and Volterra, Bogoyavlenskii and Toda lattices. The examples of other one-, two- and multi-field lattice equations are also presented.

On noncommutative merons and instantons

The Yang-Mills (YM) and self-dual Yang-Mills (SDYM) equations on the noncommutative Euclidean four-dimensional space are considered. We introduce an ansatz for a gauge potential reducing the noncommutative SDYM equations to a difference form of the Nahm equations. By constructing solutions to the difference Nahm equations, we obtain solutions of the noncommutative SDYM equations. They are noncommutative generalizations of the known solutions to the SDYM equations such as the Minkowski solution, the one-instanton solution and others. Using the noncommutative deformation of the Corrigan-Fairlie-'t Hooft-Wilzek ansatz, we reduce the noncommutative YM equations to equations on a scalar field which have meron solutions in the commutative limit and show that they have no such solutions in the noncommutative case. To overcome this difficulty, another ansatz reducing the noncommutative YM equations to a system of difference equations on matrix-valued functions is used. For self-dual configurations this system is reduced to the difference Nahm equations.

Calibrations, monopoles, and fuzzy funnels

We present new non-Abelian solitonic configurations in the low energy effective theory describing a collection of N parallel D1-branes. These configurations preserve $1/4,1/8,1/16$ and $1/32$ of the spacetime supersymmetry. They are solutions to a set of generalized Nahm equations which are related to self-duality equations in eight dimensions. Our solutions represent D1-branes which expand into fuzzy funnel configurations ending on collections of intersecting D3-branes. Supersymmetry dictates that such intersecting D3-branes must lie on a calibrated three-surface of spacetime and we argue that the generalized Nahm equations encode the data for the construction of magnetic monopoles on the relevant three-surfaces.

Einstein–Weyl spaces and dispersionless Kadomtsev–Petviashvili equation from Painlevé I and II

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlev\'e transcendents. The first construction is a hodograph transformation based on Einstein--Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are constant on a central quadric. We show how the solutions to the dKP equations can be used to construct some three-dimensional Einstein--Weyl structures, and four--dimensional anti-self-dual null-K\"ahler metrics.

Non-collapsing membrane instantons in higher dimensions

We introduce a particular embedding of seven-dimensional self-duality membrane equations in C 3× R which breaks G 2 invariance down to SU(3). The world-volume membrane instantons define SU(3) special lagrangian submanifolds of C 3. We discuss in detail solutions for spherical and toroidal topologies assuming factorization of time. We show that the extra dimensions manifest themselves in the solutions through the appearance of a non-zero conserved charge which prevents the collapse of the membrane. We find non-collapsing rotating membrane instantons which contract from infinite size to a finite one and then they bounce to infinity in finite time. Their motion is periodic. These generalized complex Nahm equations, in the axially symmetric case, lead to extensions of the continuous Toda equation to complex space.

ON FAIRLIE'S MOYAL FORMULATION OF M(ATRIX)-THEORY

Starting from the Moyal formulation of M theory in the large N-limit, we propose to reexamine the associated membrane equations of motion in ten dimensions formulated in terms of Poisson bracket. Among the results obtained, we rewrite the coupled first order Nahm equations into a simple form leading in turn to their systematic relation with SU (∞) Yang–Mills equations of motion. The former are interpreted as the vanishing condition of some conserved currents which we propose. We develop also an algebraic analysis in which an ansatz is considered and find an explicit form for the membrane solution of our problem. Typical solutions known in literature can also emerge as special cases of the proposed solution.

Nahm's equations, N = 1* domain walls, and D-strings in AdS5 × S5

We consider the following two problems: classical domain walls in the $N=1^*$ mass deformation of the maximally supersymmetric Yang Mills theory, and D-strings as external magnetic sources in the context of the AdS/CFT correspondence. We show that they are both described by Nahm's equations with unconventional boundary conditions, and analyze the relevant moduli space of solutions. We argue that general `fuzzy sphere' configurations of D-strings in AdS$_5$ correspond to Wilson-'t Hooft lines in higher representations of the dual SU(n) gauge theory.

ADHM construction of instantons on the torus

We apply the ADHM instanton construction to SU(2) gauge theory on T^n x R^(4-n)for n=1,2,3,4. To do this we regard instantons on T^n x R^(4-n) as periodic (modulo gauge transformations) instantons on R^4. Since the R^4 topological charge of such instantons is infinite the ADHM algebra takes place on an infinite dimensional linear space. The ADHM matrix M is related to a Weyl operator (with a self-dual background) on the dual torus tilde T^n. We construct the Weyl operator corresponding to the one-instantons on T^n x R^(4-n). In order to derive the self-dual potential on T^n x R^(4-n) it is necessary to solve a specific Weyl equation. This is a variant of the Nahm transformation. In the case n=2 (i.e. T^2 x R^2) we essentially have an Aharonov Bohm problem on tilde T^2. In the one-instanton sector we find that the scale parameter, lambda, is bounded above, (lambda)^2 tv<4 pi, tv being the volume of the dual torus tilde T^2.

The non-abelian self dual string on the light cone

We construct the scalar profile for the non-abelian self dual string connecting two M5-branes compactified on a light-like circle. The construction is based on a conjectured modified version of Nahm's equations describing a D2-brane, with a magnetic field on it, suspended between two D4-branes. Turning on a constant magnetic field on the D2-brane corresponds to a boost in the eleventh direction. In the limit of infinite boost the D4-branes correspond to light-like compactified M5-branes. The solution for the scalar profile of the brane remains finite in this limit and displays all the correct expected features such as smooth interpolation between the unbroken and broken phase with the correct value for the Higgs field at infinity.

From SYM perturbation theory to closed strings in matrix theory

For the purpose of better understanding the AdS/CFT correspondence it is useful to have a description of the theory for all values of the 't Hooft coupling, and for all $N$. We discuss such a description in the framework of Matrix theory for SYM on D4-branes, which is given in terms of quantum mechanics on the moduli space of solutions of the Nahm equations. This description reduces to both SYM perturbation theory and to closed string perturbation theory, each in its appropriate regime of validity, suggesting a way of directly relating the variables in the two descriptions. For example, it shows explicitly how holes in the world-sheets of the 't Hooft expansion close to give closed surfaces.

On symmetry reductions of topological/cohomological Yang-Mills theories

General aspects of symmetry reductions of topological/cohomological Yang-Mills theories are discussed. It is shown that cohomological quantum field theories associated with the Nahm equations can be derived from the Donaldson-Witten theory by a symmetry reduction with respect to a three-dimensional Abelian group.

Nahm transformation on the lattice

The Nahm transformation is a duality mapping between self-dual Yang-Mills configurations on the torus, which exchanges the number of colours with the topological charge. We show how lattice gauge theory techniques can be used to implement it numerically. The method is presented and its precision illustrated with some applications.

Nahm dualities on the torus – a synthesis

We give a unified description of self-dual SU(2) gauge fields on tori of size l t × l s 3, based on a mixture of analytical and numerical methods using the Nahm transformation, extended to the case of twisted boundary conditions. We show how torus calorons ( l t / l s small) are Nahm dual to the torus instantons ( l t / l s large). Holonomies are dual to the locations of constituents, this duality becoming exact in the limiting cases l s or l t →∞. Implications for the moduli spaces are discussed.

Deformed Nahm equation and a noncommutative BPS monopole

A deformed Nahm equation for the BPS equation in the noncommutative N=4 supersymmetric U(2) Yang-Mills theory is obtained. Using this, we constructed explicitly a monopole solution of the noncommutative BPS equation to the linear order of the noncommutativity scale. We found that the leading order correction to the ordinary SU(2) monopole lies solely in the overall U(1) sector and that the overall U(1) magnetic field has an expected long range component of magnetic dipole moment.

The generalized Moyal–Nahm and continuous Moyal–Toda equations

We present in detail a class of solutions to the 4D SU(∞) Moyal–anti-self-dual Yang–Mills (ASDYM) equations (an effective 6D theory) that are related to reductions of the generalized Moyal–Nahm equations using the Ivanova–Popov ansatz. The former yields solutions to the ASDYM/SDYM equations for arbitrary gauge groups in four dimensions. A further dimensional reduction of the above effective 6D equations yields solutions to the Moyal–anti-self-dual gravitational equations in four dimensions. The self-dual Yang–Mills/self-dual gravity case requires a separate study. The SU(2) Toda lattice and SU(∞) (continuous) Moyal–Toda lattice equations are derived from the Moyal–Nahm equations. An explicit map taking the Moyal heavenly form (after a rotational Killing symmetry reduction of the Moyal heavenly equations) into the SU(2) Toda lattice field is found. Finally, the generalized Moyal–Nahm equations are conjectured that contain the (continuous) SU(∞) Moyal–Toda lattice equations, after a suitable reduction process. Embeddings of the different types of Moyal–Toda lattice equations into the Moyal–Nahm equations are described.

String Junction from Worldsheet Gauge Theory

We study the three-pronged strings (three string junctions) from the point of view of D-string worldsheet gauge theory. We justify interpreting kink solutions in the 2-dimensional gauge theory as pronged strings by examining BPS energy bounds obtained from this theory. When the ends of the pronged string are on D3-branes, the configuration preserves 1/4 supersymmetry with BPS conditions including the Nahm equation. Using solutions of the Nahm equation explicitly, we treat string junction configuration ending on the D3-branes from the viewpoint of the D-string. In particular, when two separated pronged strings end on the same D3-branes, they interact with each other through the D3-branes and the resultant trajectories of three-pronged strings are found to be curved non-trivially.

Moyal–Nahm equations

Various aspects of the Nahm equations in three and seven dimensions are investigated. The residues of the variables at simple poles in the seven-dimensional case form an algebra. A large class of matrix representations of this algebra is constructed. The large N limit of these equations is taken by replacing the commutators by Moyal brackets, and a set of nontrivial solutions in a generalized form of Wigner distribution functions is obtained.

Nahm's Equations and Generalizations of the Neumann System

In this paper a completely integrable Hamiltonian system on T*M is constructed for every Riemannian symmetric space M. We show that the solutions of this system correspond to the solutions of Nahm's equations for suitably chosen maps. Nahm's equations were introduced by Nahm as a rewriting of Bogomolny equations for magnetic monopoles. We represent our system as a degeneration of a certain case of Hitchin's algebraically integrable system. We prove the complete integrability of our system by means of this representation. Some concrete examples of our Hamiltonian system on T*M are described. When M = Sn, we obtain the classical system of C. Neumann. If the configuration space M of our system is the n-dimensional hyperbolic space, we get the Minkowskian analogue of the C. Neumann system. Other examples that we describe are a many-body C. Neumann system, a spherical pendulum, and a spherical pendulum with an additional magnetic force. 1991 Mathematics Subject Classification: 11D25, 11G05, 14G05.

On Nahm's transformation with twisted boundary conditions

Following two different tracks, we arrive at a definition of Nahm's transformation valid for self-dual fields on the four-dimensional torus T 4 with non-zero twist tensor. The transform is again a self-dual gauge field defined on a new torus T ^ 4 and with non-zero twist tensor. It preserves the property of being an involution.