Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories

Abstract

We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, cohomological field theories, phase space path integrals, and two-dimensional Yang-Mills theory.

72 pages (60 A4 pages), LaTeX (to appear in the Journal of Mathematical Physics Special Issue on Functional Integration (May 1995))

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• Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD
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• Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model
• Supersymmetric Yang-Mills Theory On A Four-Manifold
• Coherent states, Path integral, and Semiclassical approximation
• On Diagonalization in Map(M,G)
• Duistermaat-Heckman measures in a non-compact setting
• An Integration Formula for the Moment Maps of Circle Actions
• Symplectic Manifolds, Coherent States and Semiclassical Approximation
• Exact Path Integrals by Equivariant Cohomology
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• On Quantum Integrability and the Lefschetz Number
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Cited by in corpus (63)

• R-Twisting and 4d/2d Correspondences
• Comments on twisted indices in 3d supersymmetric gauge theories
• The equivariant A-twist and gauged linear sigma models on the two-sphere
• On Continuous 2-Category Symmetries and Yang-Mills Theory
• Late time physics of holographic quantum chaos
• Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories
• Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory
• Balanced Topological Field Theories
• Random matrix theory and symmetric spaces
• Exact results for supersymmetric abelian vortex loops in 2+1 dimensions
• Two-dimensional QCD, instanton contributions and the perturbative Wu-Mandelstam-Leibbrandt prescription
• Non-Abelian Localization for Supersymmetric Yang-Mills-Chern-Simons Theories on Seifert Manifold
• Generalized Indices for $\mathcal{N}=1$ Theories in Four-Dimensions
• Matrix ensembles with global symmetries and 't Hooft anomalies from 2d gauge theory
• Integrable Matrix Models in Discrete Space-Time
• Calculating the Prepotential by Localization on the Moduli Space of Instantons
• Twisting with a Flip (the Art of Pestunization)
• Chern-Simons Theory, 2d Yang-Mills, and Lie Algebra Wanderers
• Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
• Testing Seiberg-Witten Theory to All Orders in the Instanton Expansion
• Mass Perturbations in Twisted N=4 Supersymmetric Gauge Theories
• A-Model Correlators from the Coulomb Branch
• Loop Correlators and Theta States in 2D Yang-Mills Theory
• Duality in twisted N=4 supersymmetric gauge theories in four dimensions
• Superfield theories on $S^3$ and their localization
• Localization for Wilson Loops in Chern-Simons Theory
• The instanton contributions to Yang-Mills theory on the torus: localization, Wilson loops and the perturbative expansion
• Localization for Yang-Mills Theory on the Fuzzy Sphere
• Observables of the generalized 2D Yang-Mills theories on arbitrary surfaces: a path integral approach
• A criterion for lattice supersymmetry: cyclic Leibniz rule
• Batalin-Vilkovisky Integrals in Finite Dimensions
• Topological Aspects of Gauge Fixing Yang-Mills Theory on S4
• Duality in Topological Quantum Field Theories
• Symplectic and Semiclassical Aspects of the Schläfli Identity
• AKSZ constructions for topological membranes on $G_2$-manifolds
• Some General Aspects of Coset Models and Topological Kazama-Suzuki Models
• Polyakov-Loops and Fermionic Zero Modes in QCD2 on the Torus
• From multi-gravitons to Black holes: The role of complex saddles
• Anomaly and Sign problem in $\mathcal{N}=(2,2)$ SYM on Polyhedra : Numerical Analysis
• Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae
• Gauge and matter superfield theories on $S^2$
• The Volume of the Quiver Vortex Moduli Space
• Supersymmetric Gauge Theory on the Graph
• Exact Results in Discretized Gauge Theories
• Undular diffusion in nonlinear sigma models
• Holomorphic Yang-Mills Theory and Variation of the Donaldson Invariants
• K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra
• Topological Quantum Field Theory on Non-Abelian Gerbes
• Field Strength Correlators For 2D Yang-Mills Over Riemann Surfaces
• A supersymmetric hierarchical model for weakly disordered $3d$ semimetals
• Localization of Equivariant Cohomology - Introductory and Expository Remarks
• Reflections on Topological Quantum Field Theory
• The Master Field For 2D QCD On The Sphere
• Conformal Motions and the Duistermaat-Heckman Integration Formula
• Weak Coupling Expansion of Yang-Mills Theory on Recursive Infinite Genus Surfaces
• Localization for the norm-square of the moment map and the two-dimensional Yang-Mills integral
• Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, $3j$ Symbols, and Character Localization
• Multi-Periodic Coherent States and the WKB-Exactness
• The Duistermaat-Heckman formula with application to circle actions and Poincaré $q$-polynomials in twisted equivariant K-theory
• On Path Integral Localization and the Laplacian
• Non-Local Observables in the A-Model
• Equivariant localization in Batalin-Vilkovisky formalism
• Lattice defect networks in 2d Yang-Mills