Supersymmetry and the Atiyah-Singer index theorem

Using a recently introduced index for supersymmetric theories, we present a simple derivation of the Atiyah-Singer index theorem for classical complexes and itsG-index generalization using elementary properties of quantum mechanical supersymmetric systems.     

An index theorem for super derivations

We show that the Chern character given by a super-KMS functional on a quantum algebra can be interpreted in terms of the index of a super derivation on a projection of the algebra.Dedicated to Roland DobrushinSupported in part by the National Science Foundation under Grant DMS/PHY 8816214     

Supersymmetric quantum mechanics and the index theorem

We consider the classical mechanics of the spinning particle and investigate which Abelian interactions can be added without breaking supersymmetry. A quantum theory is presented. The well known index theorem for the Dirac operator is extended to take into account the effect of anti-symmetric Abelian tensor fields. Furthermore interactions with non-Abelian anti-symmetric tensor fields are investigated. It turns out in both cases that these fields do not give any non-trivial contributions to the index.     

Gravitational anomalies and the family's index theorem

We discuss the use of the family's index theorem in the study of gravitational anomalies. The geometrical framework required to apply the family's index theorem is presented and the relation to gravitational anomalies is discussed. We show how physics necessitates the introduction of the notion oflocal cohomology which is distinct from the ordinary topological cohomology. The recent results of Alvarez-Gaumé and Witten are derived by using the family's index theorem.This work was supported in part by the National Science Foundation under Contracts PHY81-18547 and MCS80-23356; and by the Director, Office of High Energy and Nuclear Physics of the US Department of Energy under Contracts DE-AC03-76SF00098 and AT0380-ER10617Alfred P. Sloan Foundation Fellow     

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen’s form for families of Cauchy–Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new Kähler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local Kähler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg’s zeta function.

     

A focal index theorem for null geodesics

Employing techniques recently developed by D. Kalish for Riemannian manifolds, we obtain a focal Morse index theorem for a null geodesic segment initially and terminally perpendicular to spacelike submanifolds of arbitrary codimension in a general space-time.     

Callias' index theorem,elliptic boundary conditions,and cutting and gluing

It is shown that elliptic boundary conditions play the same role in Callias' index theorem as spectral boundary conditions do in the Atiyah-Patodi-Singer index theorem. This is used to generalize Callias' index theorem to arbitrary complete spin-manifolds.     

Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem

Fermionic quantization, or Clifford algebra, is combined with pseudodifferential operators to simplify the proof of the Atiyah-Singer index theorem for the Dirac operator on a spin manifold.     

A heat kernel proof of the index theorem for deformation quantization

Letters in Mathematical Physics - We give a heat kernel proof of the algebraic index theorem for deformation quantization with separation of variables on a pseudo-Kähler manifold. We use...     

Some families of generating functions and associated hypergeometric transformation and reduction formulas

Summation, transformation and reduction formulas for various families of hypergeometric functions in one, two and more variables are potentially useful in many diverse areas of applications. The main object of this paper is to derive several substantially more general results on this subject than those considered recently by Neethu et al. [7] in connection with Bailey’s transformation involving the Gauss hypergeometrc function 2F ₁ (see [1]). The methodology used here is based essentially on some families of hypergeometric generating functions. Relevant connections of the results presented in this paper with those in the earlier works are also pointed out.     

Exact formulas and phase-integral formulas,not involving wavefunctions,for expectation values pertaining to general potentials

An exact formula for quantal expectation values associated with bound states in a general potential is derived. The formula does not contain wavefunctions, but is expressed in terms of derivatives, with respect to an auxiliary parameter and with respect to the energy, of a function appearing in an exact quantization condition. Replacing the exact quantization condition by a phase-integral quantization condition (which in some cases may be exact as well), one obtains a useful formula for calculating quantal expectation values, without the use of wavefunctions, for any potential for which a phase-integral quantization condition is known. Explicit phase-integral formulas are given for the case of a single-well potential.     

Gauge fixing, families index theory, and topological features of the space of lattice gauge fields

The families index theory for the overlap lattice Dirac operator is applied to derive topological features of the space of SU(N) lattice gauge fields on the 4-torus: the topological sectors, specified by the fermionic topological charge, are shown to contain noncontractible even-dimensional spheres when N3, and noncontractible circles in the N=2 case. We describe how certain obstructions to the existence of gauge fixings without the Gribov problem in the continuum setting correspond on the lattice to obstructions to the contractibility of these spheres and circles. We also point out a canonical connection on the space of lattice gauge fields with monopole-like singularities associated with the spheres.     

Axial-vector anomalies and the index theorem in charged Schwarzschild and Taub-NUT spaces

The index theorem gives a topological expression for the excess of zero-eigenvalues of positive chirality over negative chirality solutions of the Dirac equation. These solutions are derived directly from the Dirac equation in charged Euclideanized Schwarzschild and Taub-NUT spaces, and the results are compared with the predictions of the index theorem.     

Zero modes of various graphene configurations from the index theorem

The diffusion of carbon dioxide in both NaX and NaY Faujasite systems is investigated by combining Quasi-Elastic Neutron Scattering (QENS) and Molecular Dynamics (MD) simulations. The transport diffusivity evaluated experimentally increases with the loading whereas the simulated self diffusivity decreases. This general behaviour is in good agreement with those previously reported in the literature for different gases in similar zeolites systems. It was also shown that the corrected diffusivity exhibits a significant concentration dependence. At low loading, the activation energies for diffusion derived from QENS and MD simulations are in agreement. They increase from NaY to NaX due to a stronger interaction between the CO₂ molecules and the extra-framework cations. The extrapolation of the transport and self diffusivities to zero coverage allowed us to emphasize a good agreement between experiment and simulation.     

Maslov distribution and formulas for the entropy

The Maslov distribution for a system of identical particles is used. The entropy and some other thermodynamical characteristics of this system are found for diverse fractal dimensions. A general formula for the entropy is established, which shows that the entropy is proportional to the derivative of the system energy with respect to the temperature. It is shown that a parastatistical parameter b, which is introduced formally, is related to the temperature of the system indeed. The nature of the phase transition in the system is studied in the two-dimensional case.     

Brillouin's theorem and the Hellmann-Feynman theorem for Hartree-Fock wave functions

It is shown that there is an intimate relation between the Hellmann-Feynman theorem and Brillouin's theorem. A more general form of Brillouin's theorem is provided, which applies to excited states of arbitrary symmetry and multiplicity. This new form leads to a simple proof of the Hellmann-Feynman theorem. This theorem is valid when all the orbitals that occur in the wave function are determined by a complete, and not a partial, variational procedure. Arguing in the opposite direction it is shown that the complete satisfaction of the generalized Brillouin's theorem provides an alternative scheme for obtaining the Hartree-Fock orbitals.