Infinite Dimensional Geometry Research Articles

A geometric approach to the Yang-Mills mass gap

I provide a new idea based on geometric analysis to obtain a positive mass gap in pure non-abelian renormalizable Yang-Mills theory. The orbit space, that is the space of connections of Yang-Mills theory modulo gauge transformations, is equipped with a Riemannian metric that naturally arises from the kinetic part of reduced classical action and admits a positive definite sectional curvature. The corresponding regularized Bakry-Émery Ricci curvature (if positive) is shown to produce a mass gap for 2+1 and 3+1 dimensional Yang-Mills theory assuming the existence of a quantized Yang-Mills theory on (ℝ1+2, η) and (ℝ1+3, η), respectively. My result on the gap calculation, described at least as a heuristic one, applies to non-abelian Yang-Mills theory with any compact semi-simple Lie group in the aforementioned dimensions. In 2+1 dimensions, the square of the Yang-Mils coupling constant gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} has the dimension of mass, and therefore the spectral gap of the Hamiltonian is essentially proportional to gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document} with proportionality constant being purely numerical as expected. Due to the dimensional restriction on 3+1 dimensional Yang-Mills theory, it seems one ought to introduce a length scale to obtain an energy scale. It turns out that a certain ‘trace’ operation on the infinite-dimensional geometry naturally introduces a length scale that has to be fixed by measuring the energy of the lowest glu-ball state. However, this remains to be understood in a rigorous way.

Soft particles and infinite-dimensional geometry

In the sigma model, soft insertions of moduli scalars enact parallel transport of S-matrix elements about the finite-dimensional moduli space of vacua, and the antisymmetric double-soft theorem calculates the curvature of the vacuum manifold. We explore the analogs of these statements in gauge theory and gravity in asymptotically flat spacetimes, where the relevant moduli spaces are infinite-dimensional. These models have spaces of vacua parameterized by (trivial) flat connections on the celestial sphere, and soft insertions of photons, gluons, and gravitons parallel transport S-matrix elements about these infinite-dimensional manifolds. We argue that the antisymmetric double-soft gluon theorem in d + 2 bulk dimensions computes the curvature of a connection on the infinite-dimensional space Map, where G is the global part of the gauge group. The analogous metrics in abelian gauge theory and gravity are flat, as indicated by the vanishing of the antisymmetric double-soft theorems in those models. In other words, Feynman diagram calculations not only know about the vacuum manifold of Yang–Mills theory, they can also be used to compute its curvature. The results have interesting implications for flat space holography.

Incompressible Euler equations with stochastic forcing: A geometric approach

We consider a stochastic version of Euler equations using the infinite-dimensional geometric approach as pioneered by Ebin and Marsden (1970). For the Euler equations on a compact manifold (possibly with smooth boundary) we establish local existence and uniqueness of a strong solution in spaces of Sobolev mappings (of high enough regularity). Our approach combines techniques from stochastic analysis and infinite-dimensional geometry and provides a novel toolbox to establish local well-posedness of stochastic non-linear partial differential equations.

Lie theory for asymptotic symmetries in general relativity: The NU group

We study the Newman–Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group into an infinite-dimensional Lie group. We then study the Lie theoretic properties of this group. Surprisingly, the group operations of the full NU group become discontinuous, whence the NU group does not support a Lie group structure. The NU group contains the Bondi–Metzner–Sachs (BMS) group as a subgroup, whose Lie group structure was constructed in a previous article. It is well known that the NU Lie algebra splits into a direct sum of Lie ideals of the Lie algebras of the BMS group and conformal rescalings of scri. However, the lack of a Lie group structure on the NU group implies that the BMS group cannot be embedded as a Lie subgroup therein.

Lie theory for asymptotic symmetries in general relativity: The BMS group

We study the Lie group structure of asymptotic symmetry groups in general relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group structure of the Bondi–Metzner–Sachs (BMS) group and discuss its Lie theoretic properties. We find that the BMS group is regular in the sense of Milnor, but not real analytic. This motivates us to conjecture that it is not locally exponential. Finally, we verify the Trotter property as well as the commutator property. As an outlook, we comment on the situation of related asymptotic symmetry groups. In particular, the much more involved situation of the Newman–Unti group is highlighted, which will be studied in future work.

Equifocal submanifolds in a symmetric space and the infinite dimensional geometry

Equifocal submanifolds in a symmetric space and the infinite dimensional geometry

Equifocal submanifolds in a symmetric space and the infinite dimensional geometry

Equifocal submanifolds in a symmetric space and the infinite dimensional geometry

Research of submanifolds in symmetric spaces by using the complexification and the infinite dimensional geometry

Research of submanifolds in symmetric spaces by using the complexification and the infinite dimensional geometry

Mini-Workshop: New Crossroads between Mathematics and Field Theory

In the last few years, it has been strongly emphasized the need to use new mathematical tools and structures which are not part of the tra- ditional pool of expertise of the community working on the analysis of the mathematical and structural properties of classical and quantum field theory. Goal of the workshop has been to bring together some of the major experts in these topics to discuss the latest results and the new insights brought to field theory by techniques, such as microlocal analysis, infinite dimensional geometry and homological algebra.

Surface Tension of Multi-phase Flow with Multiple Junctions Governed by the Variational Principle

We explore a computational model of an incompressible fluid with a multi-phase field in three-dimensional Euclidean space. By investigating an incompressible fluid with a two-phase field geometrically, we reformulate the expression of the surface tension for the two-phase field found by Lafaurie, Nardone, Scardovelli, Zaleski and Zanetti (J. Comp. Phys. \vol{113} \yr{1994} \pages{134-147}) as a variational problem related to an infinite dimensional Lie group, the volume-preserving diffeomorphism. The variational principle to the action integral with the surface energy reproduces their Euler equation of the two-phase field with the surface tension. Since the surface energy of multiple interfaces even with singularities is not difficult to be evaluated in general and the variational formulation works for every action integral, the new formulation enables us to extend their expression to that of a multi-phase ($N$-phase, $N\ge2$) flow and to obtain a novel Euler equation with the surface tension of the multi-phase field. The obtained Euler equation governs the equation of motion of the multi-phase field with different surface tension coefficients without any difficulties for the singularities at multiple junctions. In other words, we unify the theory of multi-phase fields which express low dimensional interface geometry and the theory of the incompressible fluid dynamics on the infinite dimensional geometry as a variational problem. We apply the equation to the contact angle problems at triple junctions. We computed the fluid dynamics for a two-phase field with a wall numerically and show the numerical computational results that for given surface tension coefficients, the contact angles are generated by the surface tension as results of balances of the kinematic energy and the surface energy.

Open problems in infinite dimensional geometry and topology

Open problems in infinite dimensional geometry and topology

Topology of the isometry group of the Urysohn space

Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space $\ell^2({\mathbb N})$. The proof is based on a lem

Deformation of Brody curves and mean dimension

AbstractThe main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

Poincaré’s program as an alternative to Klein’s (Centenary of the publication)

Poincare’s 1907 paper contained a new approach to the study of infinite-dimensional geometry as compared with Klein’s program. The approach is based on the distinguishing of a canonical object and on the reasoning from an object to a group rather than from a group to an object, as Klein had argued. The approach can be regarded as a method of finite-dimensional reduction which competes with the method of reduction of G-structures.

Three classes of smooth Banach submanifolds in B(E,F)

Let E,F be two Banach spaces, and B(E,F), Ф(E,F), SФ(E,F) and R(E,F) be the bounded linear, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. In this paper, using the continuity characteristics of generalized inverses of operators under small perturbations, we prove the following result: Let Σ be any one of the following sets: {T ∈ Ф(E,F) : IndexT = const. and dim N(T) = const.}, {T ∈ SФ(E,F) : either dim N(T) = const. < ∞ or codim R(T) = const. < ∞} and {T ∈ R(E,F) : RankT =const.< ∞}. Then Σ is a smooth submanifold of B(E,F) with the tangent space T AΣ = {B ∈ B(E,F) : BN(A) ⊂ R(A)} for any A ∈ Σ. The result is available for the further application to Thom’s famous results on the transversility and the study of the infinite dimensional geometry.

Integrable Systems in n-Dimensional Riemannian Geometry

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an n-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infinite dimensional geometry and integrable systems. Moreover one finds a Lax pair in on+1 with the vector modified Korteweg-De Vries equation (vmKDV) ut = uxxx + 3 ||u|| 2 ux as integrability condition. We indicate that other integrable vector evolution equations can be found by using a different Ansatz on the form of the Lax pair. We obtain these results by using the natural or parallel frame and we show how this can be gauged by a generalized Hasimoto ¯t = ¯

Infinite dimensional geometry and quantum field theory of strings. III. Infinite dimensional W-geometry of a second quantized free string

The present paper is devoted to various objects of the infinite dimensional W-geometry of a second quantized free string. Our purpose is to include the W-symmetries into the general infinite dimensional geometrical picture related to the quantum field theory of strings, which was described in part I of this series of papers. It is done by replacing the Lie algebra of all infinitesimal reparametrizations of a string world-sheet by the Lie quasi(pseudo)algebra of classical W-transformations (Gervais-Matsuo quasi(pseudo)algebra) as well as the Virasoro algebra by the central extended enlarged Gervais-Matsuo quasi(pseudo)algebra. A way to obtain W-algebras from the classical W-transformations (i.e. Gervais-Matsuo Lie quasi(pseudo)algebra) is proposed. A relationship between Gervais-Matsuo differential W-geometry and the Batalin-Weinstein-Karasev-Maslov approach to the geometry of nonlinear Poisson brackets as well as L.V. Sabinin's program of “nonlinear geometric algebra” is mentioned.

Spaces of Riemannian metrics on open manifolds

We define for the set M of metrics on an open manifold M n suitable uniform structures, obtain completed spaces b,m M or M r (I, B k ), respectively and calculate for each component of M r (I, B k ) the infinitedimensional geometry. In particular, we show that the sectional curvature is non positive.

Supersymmetric Dirichlet operators

A canonical supersymmetric extension of classical Dirichlet forms and operators given by measures on linear spaces is considered. In the case of log-concave measures, the supersymmetric Dirichlet operator generates a hypercontractive semigroup. This setting is appropriate for infinite-dimensional geometry and quantum field theory.

On the structure of L-algebra L( Cvir)

The scenery of the theory of groups, their representations, and homogeneous spaces in view of the intensive interaction with the mathematical physics deeply depends on changes in it. Thus, formation of the modern quantum field theory (string field theory, conformal field theory, etc.) provoked the elaboration of a new block including the infinite dimensional geometry, harmonic analysis on infinite dimensional manifolds, and the theory of representations of the infinite dimensional algebras and groups. At the same time, methods of the quantum field theory penetrated into abstract mathematics (topological quantum field theory [l, 23). There exist some arguments for the hypothesis that the conformal quantum field theory and its generalisations [3-91 would be useful in the singularity theory. One should suppose that the version of the operator formalism of the quantum field theory based on the local field algebras would play the role analogous to one of the coliomology theory in the topology. All these circumstances explain the appearance of the papers [IO, 1 l] devoted to the algebraic formulation of the described version based on the infinite dimensional geometry. In these papers we constructed the L-algebra L(Cvir) in which the model of the Verma modules over the Virasoro algebra is realised. The local held algebras are constructed from the following data: