The Mathai-Quillen Formalism and Topological Field Theory

Abstract

These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the Mathai-Quillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the Atiyah-Jeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections.

34 a4.sty pages (Notes of lectures given at the Karpacz Winter School on `Infinite Dimensional Geometry in Physics', 17-27 February 1992)

References in corpus (3)

• Topological Field Theory and Rational Curves
• N=2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant
• Topological gauge theories from supersymmetric quantum mechanics on spaces of connections

Cited by in corpus (36)

• Measuring Small Distances in N=2 Sigma Models
• Large N 2D Yang-Mills Theory and Topological String Theory
• Aspects of N_{T}\geq 2 Topological Gauge Theories and D-Branes
• Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories
• Chern-Simons and Twisted Supersymmetry in Higher Dimensions
• N=2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant
• A Worldsheet Theory for Supergravity
• Mathai-Quillen Formulation of Twisted N=4 Supersymmetric Gauge Theories in Four Dimensions
• Topological Rigid String Theory and Two Dimensional QCD
• Equivariant Localization of Path Integrals
• Les Houches Lectures on Fields, Strings and Duality
• Superconnections and the Higgs Field
• New Results in Topological Field Theory and Abelian Gauge Theory
• Duality in Topological Quantum Field Theories
• 2D Yang-Mills Theory and Topological Field Theory
• Criticality or Supersymmetry Breaking ?
• Noncommutative Cohomological Field Theory and GMS soliton
• Analogues of Mathai-Quillen forms in sheaf cohomology and applications to topological field theory
• On the Mathai-Quillen Formalism of Topological Sigma Models
• Ubiquitous order known as chaos
• A String Theory for Two Dimensional Yang-Mills Theory I
• Path integrals, SUSY QM and the Atiyah-Singer index theorem for twisted Dirac
• A Rigorous Path Integral for Supersymmetric Quantum Mechanics and the Heat Kernel
• Holomorphic Yang-Mills Theory and Variation of the Donaldson Invariants
• Basic and Equivariant Cohomology in Balanced Topological Field Theory
• Mathematical structures of cohomological field theories
• Transfer operators and topological field theory
• Supersymmetric field theories and the elliptic index theorem with complex coefficients
• Topological symmetry-restored phase of gravity
• Chaos, Ito-Stratonovich dilemma, and topological supersymmetry
• Equivariant BRST quantization and reducible symmetries
• General relativity as a two-dimensional CFT
• Representatives of the Thom class of a vector bundle
• A topological quantum field theory for $\mathrm{Spin}(7)$-instantons
• Evaluation of Euler Number of Complex Grassmann Manifold G(k, N) via Mathai-Quillen Formalism
• A Rigorous Path Integral for N=1 Supersymmetic Quantum Mechanics on a Riemannian Manifold