on the layering transition of an sos surface interacting with a wall i equilibrium results we consider the model of a 2d surface above a fixed wall and attracted toward it by means of a positive magnetic field h in the solid on solid sos approximation when the inverse temperature fl is very large and the external field h is exponentially small in ft we improve considerably previous results by dinaburg and mazel on the competition between the external field and the entropic repulsion with the wall leading in this case to the phenomenon of layering phase transitions in particular we show using the pirogov sinai scheme as given by zahradnik that there exists a unique critical value h fl in the interval 88 4 k 4e 4 k such that for all h h i h and fl large enough there exists a unique infinite volume gibbs state the typical configurations are small perturbations of the ground state represented by a surface at height k 1 above the wall moreover for the same choice of the thermodynamic parameters the influence of the boundary conditions of the gibbs measure in a finite cube decays exponentially fast with the distance from the boundary when h h fl we prove instead the convergence of the cluster expansion for both k and k 1 boundary conditions this fact signals the presence of a phase transition in the second paper of this series we will consider a glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external field h using the results proven in this paper we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free absent

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

renormalization group transformations under strong mixing conditions gibbsianness and convergence of renormalized interactions in this paper we study a renormalization group map the block averaging transformation applied to gibbs measures relative to a class of finite range lattice gases when suitable strong mixing conditions are satisfied using a block decimation procedure cluster expansion and detailed comparison between statistical ensembles we are able to prove gibbsianness and convergence to a trivial i e gaussian and product fixed point our results apply to the 2d standard ising model at any temperature above the critical one and arbitrary magnetic field

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

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

the stable topology of self dual moduli spaces soit m une 4 variete de riemann compacte connexe orientee et soit g un groupe de lie connexe simple on fixe un entier q m q alors l application inclusion i induit un epimorphisme sur les groupes respectifs d homologie et d homotopie a toutes les dimensions inferieures ou egales a q

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

monopoles and maps from s 2 to s 2 the topology of the configuration space the configuration space for the su 2 yang mills higgs equations on 3 is shown to be homotopic to the space of smooth maps froms 2 tos 2 this configuration space indexes a family of twisted dirac operators the dirac family is used to prove that the configuration space does not retract onto any subspace on which the su 2 yang mills higgs functional is bounded

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 layering transition of an sos surface interacting with a wall ii the glauber dynamics we continue our study of the statistical mechanics of a 2d surface above a fixed wall and attracted towards it by means of a very weak positive magnetic fieldh in the solid on solid sos approximation when the inverse temperature is very large in particular we consider a glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions using the results proved in the first paper of this series we show that for allh h k 1 h k h k being the critical values of the magnetic field found in the previous paper the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions on the contrary forh h k and free boundary conditions we show that the gap in a cube of sidel is bounded from above and from below by a negative exponential ofl our results provide a strong indication that contrary to what happens in two dimensions for the three dimensional dynamical ising model in a finite cube at low temperature and very small positive external field with boundary conditions that are opposite to the field on one face of the cube and are absent free on the remaining faces the rate of exponential convergence to equilibrium which is positive in infinite volume may go to zero exponentially fast in the side of the cube

entropic repulsion of an interface in an external field we consider an interface above an attractive hard wall in the complete wetting regime and submitted to the action of an external increasing convex potential and study its delocalization as the intensity of this potential vanishes our main motivation is the analysis of critical prewetting which corresponds to the choice of a linear external potential we also present partial results on critical prewetting in the two dimensional ising model as well as a few weak results on pathwise estimates for the pure wetting problem for effective interface models

monopoles non linear σ models and two fold loop spaces in this paper we study the topology of hat m k the moduli spaces ofsu 2 monopoles associated with the yang mills higgs and bogomol nyi equations and m k non linear models from quantum field theory beautiful work of donaldson 18 19 hitchin 24 25 and taubes 37 39 40 shows that gauge equivalence classes of monopoles correspond to based rational self maps of the riemann sphere similarly the non linear models we consider here are based harmonic maps from the riemann sphere to complex projectivem space in seminal work segal 35 studied m k the space of based rational maps from the riemann sphere to complex projectivem space of a fixed degreek any element of m k is clearly an element of k2cp m the space of all based continuous maps from the riemann sphere to complex projectivem space of a fixed degreek and this assignment gives rise to the natural inclusion of m k in k2cp m segal showed that these natural inclusions are homotopy equivalences through dimensionk 2m 1 as the topology of the two fold loop space 2cp m is well understood segal s result gives a very efficient way to explicitly determine the low dimensional topology of m k thus iterated loop spaces have much to say about the topology of monopoles and non linear models

algebraic cycles and homotopy theory examine its homotopy groups which after a certain idealization of the space turn out to be astonishingly simple in fact for complex projective space pn the structure can be understood completely this yields new information about the topology of the classical chow varieties and establishes an explicit relationship with universal cohomology operations in the general case complete computations are difficult but we shall establish a complex suspension theorem and lay the foundations for a theory based on the homotopy groups of chow varieties to state the results we must give precise meaning to the space of subvarieties we begin with the fundamental case of subvarieties in complex projective n space pln for each pair of integers p and d with d 1 and 0 p n consider the set wp d pn of all finite formal sums

integrality of the monopole number in su 2 yang mills higgs theory on ℝ3 we prove that in classical su 2 yang mills higgs theories on 3 with a higgs field in the adjoint representation an integer valued monopole number magnetic charge is canonically defined for any finite actionl1 loc2 configuration in particular the result is true for smooth configurations the monopole number is shown to decompose the configuration space into path components

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

finite volume glauber dynamics in a small magnetic field we consider glauber dynamics on a finite cube in d dimensional lattice d 2 which is associated with basic ising model at temperature t 1 1 under a magnetic field h 0 we prove that if the effective magnetic field is positive then the relaxation of the glauber dynamics in the uniform norm is exponentially fast uniformly over the size of underlying cube the result covers the case of the free boundary condition with arbitrarily small positive magnetic field this paper is a continuation of an attempt initiated earlier by schonmann and yoshida to shed more light on the relaxation of the finite volume glauber dynamics when the thermodynamic parameter h is so near the phase transition line h c h 0 that the dobrushin shlosman mixing condition is no longer available

on the two dimensional dynamical ising model in the phase coexistence region we consider a glauber dynamics reversible with respect to the two dimensional ising model in a finite square of sidel in the absence of an external field and at large inverse temperature we first consider the gap in the spectrum of the generator of the dynamics in two different cases with plus and open boundary conditions we prove that when the symmetry under global spin flip is broken by the boundary conditions the gap is much larger than the case in which the symmetry is present for this latter we compute exactly the asymptotics of 1 l log gap asl and show that it coincides with the surface tension along one of the coordinate axes as a consequence we are able to study quite precisely the large deviations in time of the magnetization and to obtain an upper bound on the spin spin time correlation in the infinite volume plus phase our results establish a connection between the dynamical large deviations and those of the equilibrium gibbs measure studied by shlosman in the framework of the rigorous description of the wulff shape for the ising model finally we show that in the case of open boundary conditions it is possible to rescale the time withl in such a way that asl the finite dimensional distributions of the time rescaled magnetization converge to those of a symmetric continuous time markov chain on the two state space m m m being the spontaneous magnetization our methods rely upon a novel combination of techniques for bounding from below the gap of symmetric markov chains on complicated graphs developed by jerrum and sinclair in their markov chain approach to hard computational problems and the idea of introducing block glauber dynamics instead of the standard single site dynamics in order to put in evidence more effectively the effect of the boundary conditions in the approach to equilibrium

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

relaxation of disordered magnets in the griffiths regime we study the relaxation to equilibrium of discrete spin systems with random finite range not necessarily ferromagnetic interactions in the griffiths regime we prove that the speed of convergence to the unique reversible gibbs measure is almost surely faster than any stretched exponential at least if the probability distribution of the interaction decays faster than exponential e g gaussian furthermore if the interac tion is uniformly bounded the average over the disorder of the time autocorrelation function goes to equilibrium as exp k log t d d 1 in d 1 in agreement with previous results obtained for the dilute ising model

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

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

a remark on the low temperature behavior of the sos interface in half space we investigate the low temperature phase diagram of thed dimensional d 2 solid on solid sos interface constrained to lie above a rigid wall to which it is attracted by a constant force we prove uniqueness of the gibbs state and exponentially fast convergence in the diameter of the domain of the finite volume expectation of local observables for all values of parameters where uniqueness in the class of translation periodic limit gibbs states was established previously these domains of uniqueness are bordered by lines on which the system undergoes a layering transition

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

einstein solvmanifolds and the pre einstein derivation an einstein nilradical is a nilpotent lie algebra which can be the nilradical of a metric einstein solvable lie algebra the classification of riemannian einstein solvmanifolds possibly of all noncompact homogeneous einstein spaces can be reduced to determining which nilpotent lie algebras are einstein nilradicals and to finding for every einstein nilradical its einstein metric solvable extension for every nilpotent lie algebra we construct an essentially unique derivation the pre einstein derivation the solvable extension by which may carry an einstein inner product using the pre einstein derivation we then give a variational characterization of einstein nilradicals as an application we prove an easy to check convex geometry condition for a nilpotent lie algebra with a nice basis to be an einstein nilradical and also show that a typical two step nilpotent lie algebra is an einstein nilradical the theory of riemannian homogeneous spaces with an einstein metric splits into three very different cases depending on the sign of the einstein constant the scalar curvature among them the picture is complete only in the ricci flat case by the result of ak every ricci flat homogeneous space is flat the major open conjecture in the case of negative scalar curvature is the alekseevski conjecture al1 asserting that a noncompact einstein homogeneous space admits a simply transitive solvable isometry group this is equivalent to saying that any such space is a solvmanifold a solvable lie group with a left invariant riemannian metric satisfying the einstein condition by a deep result of j lauret la5 any einstein solvmanifold is standard this means that the metric solvable lie algebra s of such a solvmanifold has the following property the orthogonal complement to the derived algebra of s is abelian the systematic study of standard einstein solvmanifolds and the term standard originated from the paper of j heber heb on the lie algebra level all the metric einstein solvable lie algebras can be obtained as the result of the following construction heb la1 la5 lw one starts with the three pieces of data a nilpotent lie algebra n a semisimple derivationof n and an inner product h i n on n with respect to which is symmetric an extension of n byis a solvable lie algebra s rh n as a linear space with adh n the inner product on s is defined by h h ni 0 k hk 2 tr and coincides with the existing one on n the resulting metric solvable lie algebra s h i is einstein provided n is nice and the derivationand the inner product h i n are chosen in the correct way note however that these conditions are expressed by a system of algebraic equations which could hardly be analyzed directly see section 2 metric einstein solvable lie algebras of higher rank with the codimension of the nilradical greater than one having the same nilradical n can be obtained from s via a known procedure by further adjoining to n semisimple derivation commuting with it turns out that the structure of an einstein metric solvable lie algebra is completely encoded in its nilradical in the following sense given a nilpotent lie algebra n there is no more than one possibly none choice ofand of h i n up to conjugation by aut n and scaling which may result in an einstein metric solvable lie algebra s h i definition 1 a nilpotent lie algebra is called an einstein nilradical if it is the nilradical of an einstein metric solvable lie algebra a derivationof an einstein nilradical n and an inner product h i n for which the metric solvable lie algebra s h i is einstein are called an einstein derivation and a nilsoliton inner product respectively

einstein solvmanifolds are standard we study einstein manifolds admitting a transitive solvable lie group of isometries solvmanifolds it is conjectured that these exhaust the class of noncompact homogeneous einstein manifolds j heber has shown that under a simple algebraic condition he calls such a solvmanifold standard einstein solvmanifolds have many remarkable structural and uniqueness properties in this paper we prove that any einstein solvmanifold is standard by applying a stratification procedure adapted from one in geometric invariant theory due to f kirwan

the topology of rational functions and divisors of surfaces segal studied the homotopy types of the spaces of holomorphic functions of the 2 sphere s 2 of closed surfaces of higher genus and of the spaces of divisors of these surfaces we continue segal s program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces

monopoles braid groups and the dirac operator using the relation between the space of rational functions on the space ofsu 2 monopoles on 3 and the classifying space of the braid group see 10 we show how the index bundle of the family of real dirac operators coupled tosu 2 monopoles can be described using permutation representations of artin s braid groups we also show how this implies the existence of a pair consisting of a gauge fielda and a higgs field on 3 whose corresponding dirac equation has an arbitrarily large dimensional space of solutions

exponential relaxation of glauber dynamics with some special boundary conditions we consider attractive finite range glauber dynamics and show that if a certain mixing condition is satisfied then the system evolving on arbitrary subsets of the lattice with appropriate boundary conditions converges to equilibrium exponentially fast in the uniform sense uniformly over the subsets of the lattice this result applies for instance to the ferromagnetic nearest neighbor ising model in the so called basuev region where complete analyticity is expected to fail

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

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

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

moduli space of su 2 monopoles and complex cyclic toda hierarchy we study the problem of complete parametrization of the moduli space ofsu 2 yang mills higgs monopoles in terms of a nonlinear integrable system it is shown that the moduli space is homeomorphic to the solution space of a new generalization of finite nonperiodic toda equation called the complex cyclic toda hierarchy