[Paper Review] Supersymmetry, Path Integration, and the Atiyah-Singer Index Theorem

Alí Mostafazadeh, Uni. Texas Dissertation+2 more|ArXiv.org|May 8, 1994

Noncommutative and Quantum Gravity Theories43 references3 citations

TL;DR

This paper presents a novel supersymmetric proof of the Atiyah-Singer index theorem using path integral quantization of a supersymmetric classical system associated with the twisted Dirac operator. By resolving factor ordering ambiguities and exactly computing the path integral measure and Feynman propagator, the authors derive the index formula via a Gaussian superdeterminant, confirming the presence of an ℏ²R/8 scalar curvature term in the Hamiltonian and validating the supersymmetric approach through agreement between loop and heat kernel expansions.

ABSTRACT

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 formulation of quantum mechanics is then used to obtain a formula for the index. For the first time, the path integral "measure" and the Feynman propagator of the system are exactly computed. The derivation of the index formula relies solely on the definition of a Gaussian superdeterminant. The two-loop analysis of the path integral is also carried out. The results of the loop and heat kernel expansions of the path interal are in complete agreement. This confirms the existence of the scalar curvature factor in the Schrödinger equation and validates the supersymmetric proof of the index theorem. Many other related issues are addressed. Finally, reviews of the index theorem and the supersymmetric quantum mechanics are presented.

Motivation & Objective

• To provide a new supersymmetric derivation of the Atiyah-Singer index theorem using path integral quantization.
• To resolve factor ordering ambiguities in the quantization of the supersymmetric classical system corresponding to the twisted Dirac operator.
• To exactly compute the path integral measure and Feynman propagator for the system, enabling a direct derivation of the index formula.
• To verify the presence of the ℏ²R/8 scalar curvature term in the Hamiltonian through consistency checks with loop and heat kernel expansions.
• To establish a rigorous link between supersymmetric quantum mechanics and the Atiyah-Singer index theorem via path integral methods.

Proposed method

• Employing the Peierls bracket quantization scheme to quantize the supersymmetric classical system associated with the twisted Dirac operator.
• Analyzing the unique quantum system that respects the index theorem, focusing on the Hamiltonian structure and factor ordering.
• Deriving the path integral formulation of the quantum system and explicitly computing the path integral measure and Feynman propagator.
• Using the definition of a Gaussian superdeterminant to derive the index formula without relying on perturbative approximations.
• Performing a two-loop analysis of the path integral and comparing results with the heat kernel expansion to validate consistency.
• Integrating results from supersymmetric quantum mechanics and differential geometry to establish the connection to the index theorem.

Experimental results

Research questions

• RQ1 How can the Atiyah-Singer index theorem be derived using a supersymmetric path integral formulation?
• RQ2 What is the role of factor ordering in the quantization of the supersymmetric system relevant to the index theorem?
• RQ3 Can the path integral measure and Feynman propagator be exactly computed in this supersymmetric quantum system?
• RQ4 Does the presence of the ℏ²R/8 scalar curvature term in the Hamiltonian survive in the path integral formulation?
• RQ5 Are the results of the two-loop path integral expansion consistent with the heat kernel expansion in this context?

Key findings

• The path integral measure and Feynman propagator for the supersymmetric system are exactly computed for the first time in this context.
• The Hamiltonian operator contains a scalar curvature term proportional to ℏ²R/8, confirming its presence in the quantum mechanical formulation.
• The index formula is derived solely from the definition of a Gaussian superdeterminant, without relying on perturbative or regularization schemes.
• The two-loop path integral expansion yields results that are in complete agreement with the heat kernel expansion, validating the consistency of the approach.
• The supersymmetric quantum mechanical system uniquely selected by the index theorem exhibits a non-trivial curvature coupling in its Hamiltonian, confirmed by the path integral analysis.
• The results provide a rigorous, non-perturbative proof of the Atiyah-Singer index theorem using supersymmetry and path integral methods.