classical solutions of the chiral model unitons and holomorphic vector bundles this paper deals with classical solutions of thesu 2 chiral model on 2 and of a generalized chiral model on 2 1 such solutions are shown to correspond to certain holomorphic vector bundles over minitwistor space with an appropriate boundary condition the solutions called 1 unitons in 9 correspond to bundles over a compact 2 dimensional complex manifold and the problem becomes one of algebraic geometry

a note on the index bundle over the moduli space of monopoles donaldson has shown that the moduli space of monopolesmk is diffeomorphic to the spaceratk of based rational maps from the two sphere to itself we use this diffeomorphism to give an explicit description of the bundle onratk obtained by pushing out the index bundle frommk this gives an alternative and more explicit proof of some earlier results of cohen and jones

monopoles and the gibbons manton metric we show that in the region where monopoles are well separated the l 2 metric on the moduli space of n monopoles is exponentially close to the l 2 invariant hyperkahler metric proposed by gibbons and manton the proof is based on a description of the gibbons manton metric as a metric on a certain moduli space of solutions to nahm s equations and on twistor methods in particular we show how the twistor description of monopole metrics determines the asymptotic metric the construction of the gibbons manton metric in terms of nahm s equations yields a class of interesting pseudo hyperkahler metrics for example we show for each semisimple lie group g and a maximal torus t g the existence of a g t invariant pseudo hyperkahler manifold whose hyperkahler quotients by t are precisely kronheimer s hyperkahler metrics on g t a similar result holds for kronheimer s ale spaces

a note on monopole moduli spaces we discuss the structure of the framed moduli space of bogomolny monopoles for arbitrary symmetry breaking and extend the definition of its stratification to the case of arbitrary compact lie groups we show that each stratum is a union of submanifolds for which we conjecture that the natural l2 metric is hyper kahler the dimensions of the strata and of these submanifolds are calculated and it is found that for the latter the dimension is always a multiple of four

on the bilinear structure associated to bezoutians abstract this paper is partly a survey of known results on quadratic forms that are hard to find in the literature our main focus is a twisted form of a construction due to bezout this skew bezoutian is a symplectic resp quadratic space associated to a pair of reciprocal or skew reciprocal coprime polynomials of same degree the isometry group of this space turns out to contain a certain associated hypergeometric group using the skew bezoutian we construct explicit isometries of bilinear spaces with given invariants such as the characteristic polynomial or jordan form and in the quadratic case the spinor norm

monopoles and rational maps a note on a theorem of donaldson in a recent paper donaldson gave a description of the moduli space of su 2 monopoles in terms of rational maps this was done indirectly via the associated solution of nahm s equations we give here an interpretation of these rational maps in terms of the monopole s spectral curve and then as scattering data for the monopole itself

calorons nahm s equations on s1 and bundles over mathbb p 1 times mathbb p 1 the moduli space of solutions to nahm s equations of rank k k j on the circle and hence of su 2 calorons of charge k j is shown to be equivalent to the moduli of holomorphic rank 2 bundles on mathbb p 1 times mathbb p 1 trivialized at infinity infty times mathbb p 1 cup mathbb p 1 times infty with c 2 k and equipped with a flag of degree j along mathbb p 1 times 0 an explicit matrix description of these spaces is given by a monad construction

rounding effects of quenched randomness on first order phase transitions frozen in disorder in an otherwise homogeneous system is modeled by interaction terms with random coefficients given by independent random variables with a translation invariant distribution for such systems it is proven that ind 2 dimensions there can be no first order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter discontinuities which would amount to a continuous symmetry breaking in systems which are stochastically invariant under the action of a continuous subgroup ofo n are suppressed by the randomness in dimensionsd 4 specific implications are found in the random field ising model for which we conclude that ind 2 dimensions at all h the gibbs state is unique for almost all field configurations and in the random bond potts model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point the results are explained by the argument of imry and ma 1 the proofs involve the analysis of fluctuations of free energy differences which are shown using martingale techniques to be gaussian on the suitable scale

why instantons are monopoles it is shown that instantons are hyperbolic monopoles for the loop group with non maximal symmetry breaking at infinity

the order parameter in a spin glass various possible precise definitions of an edwards anderson type of order parameter for an ising model spin glass are considered using boundary conditions for a finite system states of an infinite system and a duplicate system approach several of these definitions are shown to yield identical results

combined effect of pores concavity and aspect ratio on the elastic properties of a porous material the present paper focuses on materials containing superellipsoidal pores described by equation x 1 2 p x 2 2 p x 3 2 p 1 this shape is concave when 0 p p 0 5 they are oblate when 1 combined effect of two shape factors parameter p describing the concavity and aspect ratio describing extend of oblateness prolateness on compliance contribution tensor is analyzed numerically and approximated analytically for most important case of p 1 it is shown that in this case the combined effect is equal to the product of two effects considered separately the obtained approximate expressions are used to calculate the effective elastic properties of a heterogeneous material containing pores of superellipsoidal shape

algebraic cycles and infinite loop spaces in this paper we use recent results about the topology of chow varieties to answer an open question in infinite loop space theory that is we construct an infinite loop space structure on a certain product of eilenberg maclane spaces so that the total chern map is an infinite loop map an analogous result for the total stiefel whitney map is also proved further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined

nahm s transformation for instantons we describe in mathematical detail the nahm transformation which maps anti self dual connections on the four torus s1 4 onto anti self dual connections on the dual torus this transformation induces a map between the relevant instanton moduli spaces and we show that this map is a hyperkahler isometry

the topology of instanton moduli spaces i the atiyah jones conjecture in this paper we study the global geometry and topology of the moduli spaces of based su 2 instantons over the 4 sphere s4 these instanton moduli spaces have a rich history and have been analyzed from many points of view originally these spaces which we denote by mk were defined as solution spaces modulo gauge equivalence to certain partial differential equations namely the self duality equations associated to the yang mills functional in su 2 gauge theory they have been successfully studied from this point of view by taubes t1 t2 uhlenbeck u and others an important alternative approach was initiated by ward w who related instantons to certain holomorphic bundles on cp3 and was continued by atiyah and ward in aw this allowed the classification of instantons on 4 in terms of quaternionic linear algebra by atiyah drinfeld hitchin and manin adhm this holomorphic approach was further extended by donaldson d who showed that these bundles were determined by their restriction to a cp2 and that the restricted bundles only had to satisfy the constraint of being trivial on a fixed line in c2r2 hurtubise hul then exploited this fact to study the moduli spaces mk as did atiyah a to show that mk arise naturally in the theory of holomorphic maps into loop groups this latter approach was continued by gravesen g atiyah and jones aj obtained the first results and formulated the foundational questions on the global topology of these moduli spaces recall that an element of mk is a based gauge equivalence class of a connection on the principal su 2 bundle over 4 denoted by pk with second chern class k satisfying the self duality equations there is a natural forgetful map to the based equivalence classes of all connections in pk atiyah and jones aj showed that this target space which we denote by bk is homotopy equivalent

an intermediate phase with slow decay of correlations in one dimensional 1 x y 2 percolation ising and potts models we rigorously establish the existence of an intermediate ordered phase in one dimensional 1 x y 2 percolation ising and potts models the ising model truncated two point function has a power law decay exponent which ranges from its low and high temperature value of two down to zero as the inverse temperature and nearest neighbor coupling vary similar results are obtained for percolation and potts models

stratifying monopoles and rational maps a stratification of the moduli space of monopoles and of the space of rational maps into a flag variety is presented it is shown that the map associating a rational map to a monopole preserves these strata these strata explain some problems in the intepretation of the parameters of the moduli space in terms of superpositions of fundamental monopoles this interpretation is not valid on the individual strata the space of fundamental monopoles is described and shown to be the same as the corresponding space of rational maps

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

monopoles on asymptotically euclidean 3 manifolds on considere une generalisation de la theorie de yang mills higgs sur r 3 euclidien a des sommes convexes de r 3 avec des 3 varietes closes compactes

min max theory for the yang mills higgs equations in each monopole sector there exist an infinite number of finite energy solutions to the prasad sommerfield limit of the su 2 yang mills higgs equations on 3 whose energy is greater than any finite number

on the critical percolation probabilities we prove that the critical probabilities of site percolation on the square lattice satisfy the relation p c p 1 furthermore we prove the continuity of the function percolation probability

infinite clusters in dependent automorphism invariant percolation on trees we study dependent bond percolation on the homogeneous tree t n of order n 2 under the assumption of automorphism invariance excluding a trivial case we find that the number of infinite clusters a s is either 0 or furthermore each infinite cluster a s has either 1 2 or infinitely many topological ends and infinite clusters with infinitely many topological ends have a s a branching number greater than 1 we also show that if the marginal probability that a single edge is open is at least 2 n 1 then the existence of infinite clusters has to have positive probability several concrete examples are considered

effective properties of composite materials containing voids recent results of theoretical and practical importance prove that the two dimensional in plane effective average young s modulus for an isotropic elastic material containing voids is independent of the poisson s ratio of the matrix material this result is true regardless of the shape and morphology of the voids so long as isotropy is maintained the present work uses this proof to obtain explicit analytical forms for the effective young s modulus property forms which simplify greatly because of this characteristic in some cases the optimal morphology for the voids can be identified giving the shapes of the voids at fixed volume that maximize the effective young s modulus in the two dimensional situation recognizing that two dimensional isotropy is a subset of three dimensional transversely isotropic media it is shown in this more general case that three of the five properties are independent of poisson s ratio leaving only two that depend upon it for three dimensionally isotropic composite media containing voids it is shown that a somewhat comparable situation exists whereby the three dimensional young s modulus is insensitive to variations in poisson s ratio v m over the range 0 v m although the same is not true for negative values of v m this further extends the practical usefulness of the two dimensional result to three dimensional conditions for realistic values of v m

invaded cluster algorithm for potts models the invaded cluster algorithm a method for simulating phase transitions is described in detail theoretical albeit nonrigorous justification of the method is presented and the algorithm is applied to potts models in two and three dimensions the algorithm is shown to be useful for both first order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities the dynamic properties of the invaded cluster algorithm are studied numerical evidence suggests that the algorithm has no critical slowing for ising models s1063 651x 96 09208 2

cavities vis a vis rigid inclusions elastic moduli of materials with polygonal inclusions in this paper we explore the correspondence between rigid inclusions and cavities dundurs 1989 j appl mech 56 786 790 as applied to the effective elastic moduli of materials with polygonal rigid inclusions and cavities in the analysis we use a complex variable method of elasticity and a conformal transformation to solve for the stress field due to a single rigid inclusion then we use a far field approach to obtain the effective elastic constants of composites with a dilute concentration of rigid polygonal inclusions by employing the dundurs correspondence we can automatically obtain the result for the effective elastic moduli of materials with cavities finally we use effective medium theories to predict the elastic moduli of materials containing a finite concentration of inclusions

the classification of monopoles for the classical groups by studying a construction of nahm we compute the moduli spaces of monopoles with maximal symmetry breaking at infinity forsu n so n andsp n these are found to be equivalent to spaces of holomorphic maps from 1 into flag manifolds

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

large q asymptotics of the random bond potts model we numerically examine the large q asymptotics of the q state random bond potts model special attention is paid to the parametrisation of the critical line which is determined by combining the loop representation of the transfer matrix with zamolodchikov s c theorem asymptotically the central charge seems to behave like c q 1 2 log 2 q o 1 very accurate values of the bulk magnetic exponent x 1 are then extracted by performing monte carlo simulations directly at the critical point as q infinity these seem to tend to a non trivial limit x 1 0 192 0 002

instantons and geometric invariant theory we show that the yang mills instantons can be described in terms of certain holomorphic bundles on the projective plane the proof uses explicit matrix descriptions arising from monads and an analysis of the corresponding groups of symmetries

instantons in two and four dimensions it is shown that yang mills instantons in four dimensions can naturally be identified with the instantons of a two dimensional theory with values in the loop group

effective elastic properties of solids with defects of irregular shapes pores and defects in real materials often have very irregular shapes thus micromechanical modeling based on the analytical solutions of elasticity becomes inapplicable the objective of this paper is to present a computational procedure to calculate the contribution of the irregularly shaped defects into the effective moduli of two dimensional elastic solids in this procedure the cavity compliance tensor is constructed numerically for an individual defect and then used in the elastic potential based approach to predict the effective moduli of porous solids two computational methods are used in this paper to calculate the components of a cavity compliance tensor finite element analysis and numerical conformal mapping application of this procedure to the regular hole shapes produces results that are in good correspondence with analytical predictions

on the unicity of discontinuous transitions in the two dimensional potts and ashkin teller models we investigate the uniqueness of the discontinuous phase transition of the q state potts model and the symmetric q s s cubic model generalized ashkin teller model on the integer square lattice with q 1 for the ferromagnetic cases when there is a discontinuous phase transition we show that the discontinuity can only occur at a point of self duality

monopoles and clusters we define and study certain hyperkahler manifolds which capture the asymptotic behaviour of the su 2 monopole metric in regions where monopoles break down into monopoles of lower charges the rate at which these new metrics approximate the monopole metric is exponential as for the gibbons manton metric

on the construction of monopoles for the classical groups forg a classical group an equivalence is exhibited between a g monopoles over 3 with maximal symmetry breaking at infinity b families of rank g algebraic curves int 1 along with divisors on those curves satisfying certain constraints c solutions of nahm s equations over rank g intervals satisfying the appropriate boundary conditions a and b are linked by twistor techniques b and c via the krichever method for solving non linear differential equations and a and c via the adhmn construction providing a unified picture of techniques for solution amongst other things an asymptotic formula for the higgs field of the monopole is computed

on the complete integrability of the discrete nahm equations the discrete nahm equations a system of matrix valued difference equations arose in the work of braam and austin on half integral mass hyperbolic monopoles

monopoles and spectral curves for arbitrary lie groups the definition of the spectral curve of a monopole is extended to any connected compact simple lie groupk it is found there are rankk curves whose degrees are related to the topological weights of the monopole

on the construction of monopoles we show that any self dual su 2 monopole may be constructed either by ward s twistor method or nahm s use of the adhm construction the common factor in both approaches is an algebraic curve whose jacobian is used to linearize the non linear ordinary differential equations which arise in nahm s method we derive the non singularity condition for the monopole in terms of this curve and apply the result to prove the regularity of axially symmetric solutions

periodic instantons and the loop group we construct a large class of periodic instantons conjecturally we produce all periodic instantons this confirms a conjecture of garland and murray that relates periodic instantons to orbits of the loop group acting on an extension of its lie algebra

symmetric calorons and the rotation map we study su 2 calorons also known as periodic instantons and consider invariance under isometries of s1 r3 coupled with a non spatial isometry called the rotation map in particular we investigate the fixed points under various cyclic symmetry groups our approach utilises a construction akin to the adhm construction of instantons what we call the monad matrix data for calorons derived from the work of charbonneau and hurtubise to conclude we present an example of how investigating these symmetry groups can help to construct new calorons by deriving nahm data in the case of charge 2

the phase boundary in dilute and random ising and potts ferromagnets the authors present an approach to dilute ising and potts models based on the fortuin kasteleyn random cluster representation which is simultaneously rigorous intuitive and surprisingly simple their analysis yields with no dimensional restrictions or other caveats the following asymptotic form of the phase boundary for the regular dilute model in which bonds have constant ferromagnetic coupling j with probability p and are vacant with probability 1 p the critical temperature scales as exp j ktc p approximately mod p pc mod implying that the crossover exponent is phi 1 if the constant couplings are replaced by a distribution f j with mass near j 0 quite different crossover behaviour is observed for example if f j approximately jalpha then for p near pc tc p approximately mod p pc mod 1 alpha

some solutions of the yang mills equations in dimension greater than four the yang mills equations for gauge fields of an arbitrary semisimple lie group g in euclidean spaces rn with dimension n 4 is considered for gauge fields aa we introduce the ansatze which reduce the yang mills equations in rn to a set of nonlinear differential matrix equations rouhani ward equations this permits us to write out the explicit form of cosmological plane wave and other types of solutions

elastic fields generated by inhomogeneities far field asymptotics its shape dependence and relation to the effective elastic properties we discuss connections between the effective elastic properties of a solid with inhomogeneities and the far field asymptotics of the elastic fields generated by them we focus on the dependence of the far field asymptotics on the inhomogeneity shape this shape dependence in the inhomogeneity problem is in contrast with shape independence of the far field in the eigenstrain problem for the latter the far field applies to inclusions of any shape we show that the external fields in the eigenstrain and the inhomogeneity problems are interrelated by the same tensor that characterizes the compliance contribution of an inhomogeneity

effective elastic properties of solids with two dimensional inclusions of irregular shapes abstract a computational procedure to calculate the contribution of irregularly shaped inclusions into the effective moduli of two dimensional elastic solids is proposed this procedure is based on the analysis of a representative volume element subjected to a prescribed macrostress for each type of inclusion the compliance contribution tensor is constructed using a combination of kolosov muskhelishvili complex variable technique and numerical conformal mapping application of this procedure to regularly shaped inhomogeneities produces results that are in good correspondence with analytical predictions the effective properties are first derived in the approximation of non interacting inclusions then several first order micromechanical models for interacting inclusions are considered

percolation phenomena in low and high density systems we consider the 2d quenched disordered q state potts ferromagnets and show that in the translation invariant measure averaged over the disorder at self dual points any amalgamation of q 1 species will fail to percolate despite an overall high density of 1 q 1 further in the dilute bond version of these systems if the system is just above threshold then throughout the low temperature phase there is percolation of a single species despite a correspondingly small density finally we demonstrate both phenomena in a single model by considering a perturbation of the dilute model that has a self dual point we also demonstrate that these phenomena occur by a similar mechanism in a simple coloring model invented by o haggstrom

the atiyah jones conjecture the purpose of this note is to announce our proof of the atiyah jones conjecture concerning the topology of the moduli spaces of based su 2 instantons over s 4 full details and proofs appear in our paper bhmm1

quenched randomness at first order transitions abstract a rigorous theorem due to aizenman and wehr asserts that there can be no latent heat in a two dimensional system with quenched random impurities we examine this result and its possible extensions to higher dimensions in the context of several models for systems whose pure versions undergo a strong first order transition we show that there is an asymptotically exact mapping to the random field ising model at the level of the interface between the ordered and disordered phases this provides a physical explanation for the above result and also implies a correspondence between the problems in higher dimensions including scaling relations between their exponents the particular example of the q state potts model in two dimensions has been considered in detail by various authors and we review the numerical results obtained for this case turning to weak fluctuation driven first order transitions we describe analytic renormalization group calculations which show how the continuous nature of the transition is restored by randomness in two dimensions

monopoles and harmonic maps recently jarvis has proved a correspondence between su n monopoles and rational maps of the riemann sphere into flag manifolds furthermore he has outlined a construction to obtain the monopole fields from the rational map in this paper we examine this construction in some detail and provide explicit examples for spherically symmetric su n monopoles with various symmetry breakings in particular we show how to obtain these monopoles from harmonic maps into complex projective spaces the approach extends in a natural way to monopoles in hyperbolic space and we use it to construct new spherically symmetric su n hyperbolic monopoles

monopoles and geodesics using the holomorphic geometry of the space of straight lines in euclidean 3 space it is shown that every static monopole of chargek may be constructed canonically from an algebraic curve by means of the atiyah ward ansatz a k

on a class of representations of the yangian and moduli space of monopoles a new class of infinite dimensional representations of the yangians yopen image in new window and yopen image in new window corresponding to a complex semisimple algebra open image in new window and its borel subalgebra open image in new window is constructed it is based on the generalization of the drinfeld realization of open image in new window in terms of quantum minors to the case of an arbitrary semisimple lie algebra open image in new window the poisson geometry associated with the constructed representations is described in particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of g monopoles defined as the components of the space of based maps of 1 into the generalized flag manifold open image in new window thus the constructed representations of the yangian may be considered as a quantization of the moduli space of the monopoles

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

asymptotic metrics for su n monopoles with maximal symmetry breaking we compute the asymptotic metrics for moduli spaces of su n monopoles with maximal symmetry breaking these metrics are exponentially close to the exact monopole metric1 as soon as for each simple root the individual monopoles corresponding to that root are well separated we also show that the estimates can be differentiated term by term in natural coordinates which is a new result even for su 2 monopoles

on computation of the compliance and stiffness contribution tensors of non ellipsoidal inhomogeneities we discuss computation of compliance stiffness contribution tensors of an inhomogeneity of non ellipsoidal shape and the difficulties that may be encountered in this context these issues are illustrated on the example of one specific shape factor concavity convexity of the inhomogeneity shape it is found in particular that the effect of the concavity factor depends on the specific constraint imposed if the inhomogeneity volume is kept constant then its compliance contribution rapidly increases with increasing concavity on the computational side it leads to difficulties related to unusually high requirements to the accuracy of the computed volume averages quantities if on the other hand the characteristic dimensions for example the distance between the farthest points are kept constant then the dependence of the compliance contribution on the concavity factor is almost linear

toda flows inverse spectral transform and realization theory abstract in this paper we establish some new relationships between the solution techniques for completely integrable nonlinear differential equations of the toda type and topics in realization theory

gelfand zeitlin actions and rational maps we observe that the analogue of the gelfand zeitlin action on mathfrak g mathfrak l n mathbb c which exists on any symplectic manifold m with an hamiltonian action of gl n mathbb c has a natural interpretation as a residual action after we identify m with a symplectic quotient of m times prod i 1 n t ast gl i mathbb c we also show that the gelfand zeitlin actions on t ast gl n mathbb c and on the regular part of mathfrak g mathfrak l n mathbb c can be identified with natural hamiltonian actions on spaces of rational maps into full flag manifolds while the gelfand zeitlin action on the whole mathfrak g mathfrak l n mathbb c corresponds to a natural action on a space of rational maps into the manifold of half full flags in mathbb c 2n

the geodesic approximation for the yang mills higgs equations in this paper we consider the dynamics of the monopole solutions of yang mills higgs theory on minkowski space the monopoles are solutions of the yang mills higgs equations on three dimensional euclidean space it is of interest to understand how they evolve in time when considered as solutions of the yang mills higgs equations on minkowski space i e the time dependent equations it was suggested by manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain metric on the space of guage equivalence classes of monopoles the moduli space the metric is defined by taking thel 2 inner product of tangent vectors to this space in this paper we will prove that manton s approximation is indeed valid in the right circumstances which correspond to the slow motion of monopoles the metric on the moduli space of monopoles was analysed in a book by atiyah and hitchin so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in yang mills higgs theory is obtained the strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions and then to use energy estimates to prove their validity for the case of monopoles to be considered here there is a technical obstacle to be overcome when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin this is overcome by using a norm introduced by taubes in a discussion of index theory for the yang mills higgs functional

nahm s equations and the classification of monopoles solutions of nahm s system of ordinary differential equations are produced by variational methods this leads to an explicit parametrisation of the solutions to the bogomolny equation over 3

closed geodesics on the space of stable two monopoles we show that the atiyah hitchin metric on the space of stable two monopoles admits closed geodesics