Noncommutative geometry and spacetime gauge symmetries of string theory

We illustrate the various ways in which the algebraic framework of noncommutative geometry naturally captures the short-distance spacetime properties of string theory. We describe the noncommutative spacetime constructed from a vertex operator algebra and show that its algebraic properties bear a striking resemblence to some structures appearing in M Theory, such as the noncommutative torus. We classify the inner automorphisms of the space and show how they naturally imply the conventional duality symmetries of the quantum geometry of spacetime. We examine the problem of constructing a universal gauge group which overlies all of the dynamical symmetries of the string spacetime. We also describe some aspects of toroidal compactifications with a light-like coordinate and show how certain generalized Kac—Moody symmetries, such as the Monster sporadic group, arise as gauge symmetries of the resulting spacetime and of superstring theories.     

Simplicial differential geometric theory for language cortical dynamics

This paper starts by reformulating synthetic differential geometry on abstract simplicial complexes. A new, multidimensional simplicial differential geometry is built by the authors: multidimensional vector fields, multidimensional distributions of vector fields, multidimensional Lie brackets, etc. New concepts of attractors, invariant distributions, and accessibility are formulated in this framework. Using these, control systems are for the first time introduced, in a global manner, on simplicial complexes in order to describe neurodynamics. Invariant distributions, attractors, etc. for cortical controlled systems form a rich picture of the neurodynamics of aphasia (deblocking or facilitation).     

Simplicial chains over a field andp-local homotopy theory

The author was partially supported by the National Science Foundation and the Alfred P. Sloan Foundation     

OnS-duality in Abelian gauge theory

U(1) gauge theory onR ⁴ is known to possess an electric-magnetic duality symmetry that inverts the coupling constant and extends to an action ofSL(2,Z). In this paper, the duality is studied on a general four-manifold and it is shown that the partition function is not a modular-invariant function but transforms as a modular form. This result plays an essential role in determining a new low-energy interaction that arises whenN=2 supersymmetric Yang-Mills theory is formulated on a four-manifold; the determination of this interaction gives a new test of the solution of the model and would enter in computations of the Donaldson invariants of four-manifolds with b ₂ ⁺ 1. Certain other aspects of abelian duality, relevant to matters such as the dependence of Donaldson invariants on the second Stieffel-Whitney class, are also analyzed.     

Fluctuation theory of media with pronounced spacetime inhomogeneity

A nonlinear determinstic thermodynamics is constructed for media with pronounced (r,t) inhomogeneity of the intensive variables or their derivatives. The balance equations in the theory are taken to be either the equations of an ideal fluid or an ideal fluid with heat conduction. The basic variables are taken to be (r,t) andT(r,t). The hypothesis of local equilibrium is represented in the form of the Gibbs-Duhem relation, the conjugate coordinates are (r,t) and (r,t), and the local potential isP(,T). The velocity potentialv _i (r,t) enters through the substantial derivative. A variational principle is formulated; in the case of an ideal fluid with heat conduction there arises naturally a local decrease of the entropy production:z ₂(t)=z ₀ ² exp(–2t/t), there0074 is the relaxation time.In memory of D. N. ZubarevJoint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 53–67, October, 1993.     

Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime

We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge–Wheeler and Zerilli equations obtained recently by Anderson et al. (Ann. Henri Poincaré 21:61–813, 2020) are sufficient to obtain the conformal scattering.

     

Ground state estimations in gauge theory

The lowest eigenvalue λ(F) of a vector bundle F over a compact Riemannian manifold M is estimated in terms of the curvature of M and of the connection Γ on F.     

Simplicial Endomorphisms

The monoids of simplicial endomorphisms, i.e., the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley–Lieb algebras, and as the monoids of Temperley–Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphisms are linked to arbitrary adjoint situations. This link is established through diagrams of the kind found in Temperley–Lieb algebras. Results about these matters, which were previously prefigured up to a point, are here surveyed and reworked. A presentation of monoids of simplicial endomorphisms by generators and relations has been given a long time ago. Here a closely related presentation is given, with completeness proved in a new and self-contained manner.     

Electrically charged solitons in gauge field theory

Monopoles and vortices are well known magnetically charged soliton solutions of gauge field equations. Extending the idea of Dirac on monopoles, Schwinger pioneered the concept of solitons carrying both electric and magnetic charges, called dyons, which are useful in modeling elementary particles. Mathematically, the existence of dyons presents interesting variational partial differential equation problems, subject to topological constraints. This article is a survey on recent progress in the study of dyons.     

Simplicial cross-polytopic d-arrangements

Branko Grünbaum has observed that the projective d-arrangements formed by the facet hyperplanes of a regular polytope together with some of its hyperplanes of mirror symmetry and possibly the hyperiplane at infinity are sometimes simplicial.We investigate the projective d-arrangements associated in this manner with a cross-polytope. Fourteen such simplicial arrangements are known: three 1-arrangements, four 2-arrangements, six 3-arrangements, and one 4-arrangement. In this paper we prove that no other arrangement so associated with a cross-polytope is simplicial.     

Simplicial nonpositive curvature

We introduce a family of conditions on a simplicial complex that we call local k-largeness (k≥6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that k-largeness implies non-positive curvature if k is sufficiently large. We also show that locally k-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.     

Colourful Simplicial Depth

Inspired by Barany’s Colourful Caratheodory Theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d² + 1 and that the maximum is d^d+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.