Path Integral Methods in Index Theorems

Many introductory courses in quantum mechanics include Feynman's time-slicing definition of the path integral, with a complete derivation of the propagator in the simplest of cases. However, attempts to generalize this, for instance to non-quadratic potentials, encounter formidable analytic issues in showing the successive approximations in fact converge to a definite expression for the path integral. The present work describes how to carry out the analysis for a class of Lag...

These lectures are intended as an introduction to the technique of path integrals and their applications in physics. The audience is mainly first-year graduate students, and it is assumed that the reader has a good foundation in quantum mechanics. No prior exposure to path integrals is assumed, however. The path integral is a formulation of quantum mechanics equivalent to the standard formulations, offering a new way of looking at the subject which is, arguably, more intuit...

This expository paper contains a detailed introduction to some important works concerning the Gauss-Bonnet-Chern theorem. The study of this theorem has a long history dating back to Gauss's Theorema Egregium (Latin: Remarkable Theorem) and culminated in Chern's groundbreaking work [14] in 1944, which is a deep and wonderful application of Elie Cartan's formalism. The idea and tools in [14] have a great generalization and continue to produce important results till today. In th...

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, many-body physics and field theory and contain standard example...

In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.

We review equivariant localization techniques for the evaluation of Feynman path integrals. We develop systematic geometric methods for studying the semi-classical properties of phase space path integrals for dynamical systems, emphasizing the relations with integrable and topological quantum field theories. Beginning with a detailed review of the relevant mathematical background -- equivariant cohomology and the Duistermaat-Heckman theorem, we demonstrate how the localizatio...

These third-year lecture notes are designed for a 1-semester course in topological quantum field theory (TQFT). Assumed background in mathematics and physics are only standard second-year subjects: multivariable calculus, introduction to quantum mechanics and basic electromagnetism. Keywords: quantum mechanics/field theory, path integral, Hodge decomposition, Chern-Simons and Yang-Mills gauge theories, conformal field theory

A new supersymmetric proof of the Atiyah-Singer index theorem is presented. The Peierls bracket quantization scheme is used to quantize the supersymmetric classical system corresponding to the index problem for the twisted Dirac operator. The problem of factor ordering is addressed and the unique quantum system that is relevant to the index theorem is analyzed in detail. The Hamiltonian operator is shown to include a scalar curvature factor, $\hbar^2R/8$. The path integral fo...

In this short note we outline a simple probabilistic proof of the Gauss-Bonnet formula for compact Riemannian manifolds with boundary, which adapts to this setting an argument due to Hsu \cite{Hs1,Hs2} in the closed case. The new technical ingredient is the Feynman-Kac formula for differential forms satisfying absolute boundary conditions proved in \cite{dL}. Combined with the so-called supersymmetric aproach to index theory, this leads to a path integral representation of th...

We review the explicit derivation of the Gauss-Bonet and Hirzebruch formulae by physical model and give a physical proof of the Lefschetz fixed-point formula by twisting boundary conditions for the path integral.