Narrow Results

A family of quasi-$1$D Schrödinger operators is investigated through scattering theory. The continuous spectrum of these operators exhibits changes of multiplicity, and some of these operators possess resonances at thresholds. It is shown that the corresponding wave operators belong to an explicitly constructed $C^{\ast }$-algebra. The quotient of this algebra by the ideal of compact operators is studied, and an index theorem is deduced from these investigations. This result corresponds to a topological version of Levinson’s theorem in the presence of embedded thresholds, resonances, and changes of multiplicity of the scattering matrices. In the last two sections of the paper, the $K$-theory of the main $C^{\ast }$-algebra and the dependence on an external parameter are carefully analyzed. In particular, a surface of resonances is exhibited, probably for the first time. The contents of these two sections are of independent interest, and the main result does not depend on them.

We present the spectral and scattering theory of the Casimir operator acting on radial functions in $L^2(\mathrm{\text{SL}}(2,ℝ))$. After a suitable decomposition, these investigations consist in studying a family of differential operators acting on the half-line. For these operators, explicit expressions can be found for the resolvent, for the spectral density, and for the Møller wave operators, in terms of the Gauss hypergeometric function. An index theorem is also introduced and discussed. The resulting equality, generically called Levinson’s theorem, links various asymptotic behaviors of the hypergeometric function. This work is a first attempt to connect group theory, special functions, scattering theory, $C^{\ast }$-algebras, and Levinson’s theorem.

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

Non-Abelian strings are considered in non-supersymmetric theories with fermions in various appropriate representations of the gauge group U$(N)$. We derive the electric charge quantization conditions and the index theorems counting fermion zero modes in the string background both for the left-handed and right-handed fermions. In both cases we observe a non-trivial $N$ dependence.

By using the Atiyah–Singer theorem through some similarities with the instanton and the antiinstanton moduli spaces, the dimension of the moduli space for two- and four-dimensional BF theories valued in different background manifolds and gauge groups scenarios is determined. Additionally, we develop Dirac's canonical analysis for a four-dimensional modified BF theory, which reproduces the topological YM theory. This framework will allow us to understand the local symmetries, the constraints, the extended Hamiltonian and the extended action of the theory.

We outline a comprehensive and first-principle solution to the wall-crossing problem in D = 4N = 2 Seiberg–Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and of how this allows them to disappear abruptly as parameters or vacuum moduli are continuously changed. This means that the wall-crossing problem is really a bound state formation/dissociation problem. A low energy dynamics for arbitrary collections of dyons is derived, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We discover that the low energy dynamics of such BPS dyons cannot be reduced to one on the classical moduli space, , yet the index can be phrased in terms of . The so-called rational invariant, first seen in Kontsevich–Soibelman formalism of wall-crossing, is shown to incorporate Bose/Fermi statistics automatically. Furthermore, an equivariant version of the index is shown to compute the protected spin character of the underlying D = 4N = 2 theory, where isometry of is identified as a diagonal subgroup of rotation SU(2)_L and R-symmetry SU(2)_R.

We consider a graphene sheet folded in an arbitrary geometry, compact or with nanotube-like open boundaries. In the continuous limit, the Hamiltonian takes the form of the Dirac operator, which provides a good description of the low energy spectrum of the lattice system. We derive an index theorem that relates the zero energy modes of the graphene sheet with the topology of the lattice. The result coincides with analytical and numerical studies for the known cases of fullerene molecules and carbon nanotubes, and it extends to more complicated molecules. Potential applications to topological quantum computation are discussed.

We exploit an index theorem and a high degree of symmetry to understand unusual quantum Hall effects of the n = 0 Landau level in graphene. The high symmetry such as the pseudospin symmetry of Fermi points in a graphene sheet cannot couple to an external magnetic field. In the absence of the magnetic field, the index theorem provides a relation between the zero-energy state of the graphene sheet and the topological deformation of the compact lattice. Under the topological deformation, the zero-energy state emerges naturally without the Zeeman splitting at the Fermi points in the graphene sheet. This results in the fact that the pseudospin is an exact symmetry. In the case of nonzero energy, the up-spin and down-spin states have the exact high symmetry of spin, forming the pseudospin singlet pairing. We describe the peculiar and unconventional quantum Hall effects of the n = 0 Landau level in graphene on the basis of the index theorem and the high degree of symmetry.

This paper is devoted to state and prove a Digital Index Theorem for digital (n - 1)-manifolds in a digital space (Rⁿ, f), where f belongs to a large family of lighting functions on the standard cubical decomposition Rⁿ of the n-dimensional Euclidean space. As an immediate consequence we obtain the corresponding theorems for all (α, β)-surfaces of Kong–Roscoe, with α, β ∈ {6, 18, 26} and (α, β) ≠ (6, 6), (18, 26), (26, 26), as well as for the strong 26-surfaces of Bertrand–Malgouyres.

From the perspective of topological field theory we explore the physics beyond instantons. We propose the fluctons as nonperturbative topological fluctuations of vacuum, from which the self-dual domain of instantons is attained as a particular case. Invoking the Atiyah–Singer index theorem, we determine the dimension of the corresponding flucton moduli space, which gives the number of degrees of freedom of the fluctons. An important consequence of these results is that the topological phases of vacuum in non-Abelian gauge theories are not necessarily associated with self-dual fields, but only with smooth fields. Fluctons in different scenarios are considered, the basic aspects of the quantum mechanical amplitude for fluctons are discussed. A possible application of fluctons in the N = 4 Topologically Twisted Supersymmetric Yang–Mills Theory is explored and the case of gravity is discussed briefly.

The paper is a presentation of recent investigations on potential scattering in ℝ³. We advocate a new formula for the wave operators and deduce the various outcomes that follow from this formula. A topological version of Levinson's theorem is proposed by interpreting it as an index theorem.

In the present paper, we introduce spin groupoids associated with the degree two Stiefel-Whitney classes, and we also introduce spinor torsors equipped with certain connections compatible with an action of spin groupoid. Using the connection, we propose a Dirac-like operator acting on the space of smooth sections of the spinor torsor. We can show that its Dirac-Laplacian admits a heat kernel in a formal sense.

We explain various results on the asymptotic expansion of the Bergman kernel on Kähler manifolds and also on symplectic manifolds. We also review the "quantization commutes with reduction" phenomenon for a compact Lie group action, and its relation to the Bergman kernel.

The index theorem, discovered by Atiyah and Singer in 1963,⁶ is one of most important results in the twentieth century mathematics. It has found numerous applications in analysis, geometry and physics. Since it was discovered numerous generalizations have been made, see for example^3–5,13,17 to mention a few; some of these generalizations gave rise to new very productive areas of mathematics. In these lectures we first review the classical Atiyah-Singer index theorem and its generalization to so called transversally elliptic operators³ due to Atiyah and Singer. Then we discuss the recent developments aimed at generalization of the index theorem for transversally elliptic operators to non-compact manifolds,^11,25