On the smooth Feshbach-Schur map

A new variant of the Feshbach map, called smooth Feshbach map, has been introduced recently by Bach et al., in connection with the renormalization analysis of non-relativistic quantum electrodynamics. We analyze and clarify its algebraic and analytic properties, and we generalize it to non-selfadjoint partition operators χ and .     

The renormalized electron mass in non-relativistic quantum electrodynamics

This work addresses the problem of infrared mass renormalization for a non-relativistic electron minimally coupled to the quantized electromagnetic field (the standard, translationally invariant system of an electron in non-relativistic QED). We assume that the interaction of the electron with the quantized electromagnetic field is subject to an ultraviolet regularization and an infrared regularization parametrized by σ>0. For the value p=0 of the conserved total momentum of electron and photon field, bounds on the renormalized mass are established which are uniform in σ→0, and the existence of a ground state is proved. For |p|>0 sufficiently small, bounds on the renormalized mass are derived for any fixed σ>0. A key ingredient of our proofs is the operator-theoretic renormalization group based on the isospectral smooth Feshbach map. It provides an explicit, finite algorithm for determining the renormalized electron mass at p=0 to any given precision.     

A new method of construction of resonances that applies to critical models

We introduce a new method of multi-scale analysis that can be used to study the spectral properties of operators in non-relativistic quantum electrodynamics with critical coupling functions. We utilize our method to prove the existence of resonances of nonrelativistic atoms which are minimally coupled to the quantized (ultraviolet-regularized) radiation field and construct them together with the corresponding resonance eigenvector in case of critical coupling, i.e., without any infrared regularization. This result was first proved in [19] with the ingredient of a suitable Pauli-Fierz transformation. The purpose of the present paper, however, is to demonstrate the power of our new method for the estimation of resolvents that is based on the isospectral Feshbach-Schur map [8]. The reconstruction formula for the resolvent of an operator in terms of the resolvent of its image under the Feshbach-Schur map allows us to use a fixed projection and to compare two resolvents without actually decimating the degrees of freedom. This is in contrast to the renormalization group based on Feshbach-Schur map, developed in [8], [5], that uses a decreasing sequence of ever-smaller projections and successively decimates the degrees of freedom. It is this new method that allows us to treat the critical and physically relevant Standard model of non-relativistic quantum electrodynamics [7] which is intractable by standard methods.     

An algorithm for studying the renormalization group dynamics in the projective space

The renormalization group dynamics is studied in the four-component fermionic hierarchical model in the space of coefficients that determine the Grassmann-valued density of the free measure. This space is treated as a two-dimensional projective space. If the renormalization group parameter is greater than 1, then the only attracting fixed point of the renormalization group transformation is defined by the density of the Grassmann δ-function. Two different invariant neighborhoods of this fixed point are described, and an algorithm is constructed that allows one to classify the points on the plane according to the way they tend to the fixed point.     

Contractibility of the space of rational maps

We define an algebro-geometric model for the space of rational maps from a smooth curve X to an algebraic group G, and show that this space is homologically contractible. As a consequence, we deduce that the moduli space $\operatorname {Bun}_{G}$ of G-bundles on X is uniformized by the appropriate rational version of the affine Grassmannian, where the uniformizing map has contractible fibers.     

Effect of compressibility on the annihilation process

Using the renormalization group in the perturbation theory, we study the influence of a random velocity field on the kinetics of the single-species annihilation reaction at and below its critical dimension d _c = 2. The advecting velocity field is modeled by a Gaussian variable self-similar in space with a finite-radius time correlation (the Antonov-Kraichnan model). We take the effect of the compressibility of the velocity field into account and analyze the model near its critical dimension using a three-parameter expansion in ∈, Δ, and η, where ∈ is the deviation from the Kolmogorov scaling, Δ is the deviation from the (critical) space dimension two, and η is the deviation from the parabolic dispersion law. Depending on the values of these exponents and the compressiblity parameter α, the studied model can exhibit various asymptotic (long-time) regimes corresponding to infrared fixed points of the renormalization group. We summarize the possible regimes and calculate the decay rates for the mean particle number in the leading order of the perturbation theory.     

Critical homoclinic orbits lead to snap-back repellers

When nondegenerate homoclinic orbits to an expanding fixed point of a map f:X→X,X⊆Rⁿ, exist, the point is called a snap-back repeller. It is known that the relevance of a snap-back repeller (in its original definition) is due to the fact that it implies the existence of an invariant set on which the map is chaotic. However, when does the first homoclinic orbit appear? When can other homoclinic explosions, i.e., appearance of infinitely many new homoclinic orbits, occur? As noticed by many authors, these problems are still open. In this work we characterize these bifurcations, for any kind of map, smooth or piecewise smooth, continuous or discontinuous, defined in a bounded or unbounded closed set. We define a noncritical homoclinic orbit and a homoclinic orbit of an expanding fixed point is structurally stable iff it is noncritical. That is, only critical homoclinic orbits are responsible for the homoclinic explosions. The possible kinds of critical homoclinic orbits will be also investigated, as well as their dynamic role.     

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory.     

A d-dimensional model of the canonical ensemble of open strings

We propose a d-dimensional model of the canonical ensemble of open self-avoiding strings. We consider the model of a solitary open string in the d-dimensional Euclidean space ? ^d, 2 ≤ d < 4, where the string configuration is described by the arc length L and the distance R between string ends. The distribution of the spatial size of the string is determined only by its internal physical state and interaction with the ambient medium. We establish an equation for a transformed probability density W(R,L) of the distance R similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations; this allows using the renormalization group method to investigate the asymptotic behavior of this density in the case where R→∞ and L→∞. We consider the model of an ensemble of M open strings with the mean string length over the ensemble given by \(\bar L\) , and we use the Darwin-Fowler method to obtain the most probable distribution of strings over their lengths in the limit as M →∞. Averaging the probability density W(R,L) over the canonical ensemble eventually gives the sought density 〈W(R, \(\bar L\) )〉.     

Zone Structure of the Renormalization Group Flow in a Fermionic Hierarchical Model

The Gaussian part of the Hamiltonian of the four-component fermion model on a hierarchical lattice is invariant under the block-spin transformation of the renormalization group with a given degree of normalization (the renormalization group parameter). We describe the renormalization group transformation in the space of coefficients defining the Grassmann-valued density of a free measure as a homogeneous quadratic map. We interpret this space as a two-dimensional projective space and visualize it as a disk. If the renormalization group parameter is greater than the lattice dimension, then the unique attractive fixed point of the renormalization group is given by the density of the Grassmann delta function. This fixed point has two different (left and right) invariant neighborhoods. Based on this, we classify the points of the projective plane according to how they tend to the attracting point (on the left or right) under iterations of the map. We discuss the zone structure of the obtained regions and show that the global flow of the renormalization group is described simply in terms of this zone structure.     

Arc Spaces and Equivariant Cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex G-variety X by its associated arc space J _∞ X, with its induced G-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $ \mathbb{Z} $ -bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $ \mathbb{Z} $ -bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).     

Infrared renormalization in non-relativistic QED and scaling criticality

We consider a spin- electron in the framework of non-relativistic Quantum Electrodynamics (QED). Let denote the fiber Hamiltonian corresponding to the conserved total momentum of the electron and the photon field, regularized by a fixed ultraviolet cutoff in the interaction term, and an infrared regularization parametrized by 0<σ?1. Ultimately, our goal is to remove the latter by taking σ↘0. We prove that there exists a constant 0<α₀?1 independent of σ>0 such that for all and all values of the finestructure constant 0<α<α₀, there exists a ground state eigenvalue of multiplicity two at the bottom of the essential spectrum. Moreover, we prove that the renormalized electron mass satisfies , uniformly in σ?0, in units where the bare mass has the value 1, and we prove the existence of the renormalized mass in the limit σ↘0. Our analysis uses the isospectral renormalization group method of Bach, Fröhlich, Sigal introduced in [V. Bach, J. Fröhlich, I.M. Sigal, Quantum electrodynamics of confined non-relativistic particles, Adv. Math. 137 (2) (1998) 299-395; V. Bach, J. Fröhlich, I.M. Sigal, Renormalization group analysis of spectral problems in quantum field theory, Adv. Math. 137 (1998) 205-298] and further developed in [V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, Smooth Feshbach map and operator-theoretic renormalization group methods, J. Funct. Anal. 203 (1) (2003) 44-92; V. Bach, T. Chen, J. Fröhlich, I.M. Sigal, The renormalized electron mass in non-relativistic QED, J. Funct. Anal. 243 (2) (2007) 426-535]. The limit σ↘0 determines a scaling-critical (or endpoint type) renormalization group problem, in which the interaction is strictly marginal (of scale-independent size). A main result of this paper is the development of a method that provides rigorous control of the renormalization of a strictly marginal quantum field theory characterized by a non-trivial scaling limit. The key ingredients entering this analysis include a hierarchy of exact algebraic cancelation identities exploiting the spatial and gauge symmetries of the model, and a combination of the isospectral renormalization group method with the strong induction principle.     

New perspectives on self-linking

We initiate the study of classical knots through the homotopy class of the nth evaluation map of the knot, which is the induced map on the compactified n-point configuration space. Sending a knot to its nth evaluation map realizes the space of knots as a subspace of what we call the nth mapping space model for knots. We compute the homotopy types of the first three mapping space models, showing that the third model gives rise to an integer-valued invariant. We realize this invariant in two ways, in terms of collinearities of three or four points on the knot, and give some explicit computations. We show this invariant coincides with the second coefficient of the Conway polynomial, thus giving a new geometric definition of the simplest finite-type invariant. Finally, using this geometric definition, we give some new applications of this invariant relating to quadrisecants in the knot and to complexity of polygonal and polynomial realizations of a knot.     

Renormalization group in a fermionic hierarchical model in projective coordinates

We study the renormalization group action in a fermionic hierarchical model in the space of coefficients determining the Grassmann-valued density of the free measure. This space is interpreted as the two-dimensional projective space. The renormalization group map is a homogeneous quadratic map and has a special geometric property that allows describing invariant sets and the global dynamics in the whole space.     

Ishikawa and Mann iteration methods for strongly accretive operators

LetE be a smooth Banach space. Suppose T:E →E is a strongly accretive map. It is proved that each of the two well known fixed point iteration methods (the Mann and Ishikawa iteration methods), under suitable conditions, converges strongly to a solution of the equationTx =f.     

C0 coarse geometry and scalar curvature

In this paper we introduce an alternative form of coarse geometry on proper metric spaces, which is more delicate at infinity than the standard metric coarse structure. There is an assembly map from the K-homology of a space to the K-theory of the C∗-algebra associated to the new coarse structure, which factors through the coarse K-homology of the space (with the new coarse structure). A Dirac-type operator on a complete Riemannian manifold M gives rise to a class in K-homology, and its image under assembly gives a higher index in the K-theory group. The main result of this paper is a vanishing theorem for the index of the Dirac operator on an open spin manifold for which the scalar curvature κ(x) tends to infinity as x tends to infinity. This is derived from a spectral vanishing theorem for any Dirac-type operator with discrete spectrum and finite dimensional eigenspaces.     

A generalized Harish-Chandra method of descent for reductive Lie algebras

Let G be a connected real reductive group and M a connected reductive subgroup of G with Lie algebras g and m, respectively. We assume that g and m have the same rank. We define a map from the space of orbital integrals of m into the space of orbital integrals of g which we call a transfer. We then consider the transpose of the transfer. This can be viewed as a map from the space of G-invariant distributions of g to the space of M-invariant distributions of m and can be considered as a restriction map from g to m. We prove that this map extends Harish-Chandra method of descent and we obtain a generalization of the radial component theorem. We give an application.     

Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable points with respect to some G-linearised Weil divisor on X. Applying this result to Hamiltonian actions on algebraic varieties, we prove that semistability with respect to a momentum map is equivalent to GIT-semistability in the sense of Mumford and Hausen. It follows that the number of compact momentum map quotients of a given algebraic Hamiltonian G-variety is finite. As further corollary we derive a projectivity criterion for varieties with compact Kähler quotient.     

On differentiable vectors for representations of infinite dimensional Lie groups

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations π:G→GL(V) of an infinite dimensional Lie group G on a locally convex space V. The first class of results concerns the space V^∞ of smooth vectors. If G is a Banach-Lie group, we define a topology on the space V^∞ of smooth vectors for which the action of G on this space is smooth. If V is a Banach space, then V^∞ is a Fréchet space. This applies in particular to C^∗-dynamical systems (A,G,α), where G is a Banach-Lie group. For unitary representations we show that a vector v is smooth if the corresponding positive definite function 〈π(g)v,v〉 is smooth. The second class of results concerns criteria for Ck-vectors in terms of operators of the derived representation for a Banach-Lie group G acting on a Banach space V. In particular, we provide for each k∈N examples of continuous unitary representations for which the space of Ck+1-vectors is trivial and the space of Ck-vectors is dense.