Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories

Abstract

These are expository lectures reviewing (1) recent developments in two-dimensional Yang-Mills theory, and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinite-dimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.

247 pages (280 pages "l" mode), 60 figures. Lectures presented at the 1994 Les Houches Summer School ``Fluctuating Geometries in Statistical Mechanics and Field Theory.'' (also available at http://xxx.lanl.gov/lh94/ ). replaced to correct inessential typos

References in corpus (8)

• Topological Amplitudes in String Theory
• Topological Field Theory and Rational Curves
• Twists and Wilson Loops in the String Theory of Two Dimensional QCD
• New Perspectives on the BRST-algebraic Structure of String Theory
• Interacting Fermion Systems from Two Dimensional QCD
• N=2 topological gauge theory, the Euler characteristic of moduli spaces, and the Casson invariant
• N=2 Topological Yang-Mills Theory on Compact Kähler Surfaces
• Two Dimensional String Theory and the Topological Torus

Cited by in corpus (182)

• Fermionic Localization of the Schwarzian Theory
• Local subsystems in gauge theory and gravity
• Gauge Theories and Macdonald Polynomials
• Asymptotic Fragility, Near $Ad{S}_2$ Holography and $T\bar{T}$
• Gauge/string duality for QCD conformal operators
• Solvable Models of Quantum Black Holes: A Review on Jackiw-Teitelboim Gravity
• The Quantum Gravity Dynamics of Near Extremal Black Holes
• Diagrams for Symmetric Product Orbifolds
• Entanglement entropy and nonabelian gauge symmetry
• Residues and World-Sheet Instantons
• Branes, Anti-Branes and Brauer Algebras in Gauge-Gravity duality
• On the Superconformal Index of Argyres-Douglas Theories
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model
• Brane Tilings and Their Applications
• All Loop Topological String Amplitudes From Chern-Simons Theory
• The Schwarzian Theory - A Wilson Line Perspective
• Two- and three-point Green's functions in two-dimensional Landau-gauge Yang-Mills theory
• An exact quantization of Jackiw-Teitelboim gravity
• Lectures on on Black Holes, Topological Strings and Quantum Attractors (2.0)
• Multiloop Superstring Amplitudes from Non-Minimal Pure Spinor Formalism
• Twisted Supersymmetric Gauge Theories and Orbifold Lattices
• Giant Graviton Correlators from Dual SU(N) super Yang-Mills Theory
• Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories
• Finite Factorization equations and Sum Rules for BPS correlators in N=4 SYM theory
• Enhanced symmetries of gauge theory and resolving the spectrum of local operators
• Non-perturbative Defect One-Point Functions in Planar $\mathcal{N}=4$ Super-Yang-Mills
• A double coset ansatz for integrability in AdS/CFT
• Higgs Bundles, Gauge Theories and Quantum Groups
• Matrix Models for Random Partitions
• Do gauge fields really contribute negatively to black hole entropy?
• Schwinger-Keldysh formalism II: Thermal equivariant cohomology
• Edge Dynamics from the Path Integral: Maxwell and Yang-Mills
• On F-term contribution to effective action
• On Irregular Singularity Wave Functions and Superconformal Indices
• Matrix Models for the Black Hole Information Paradox
• Quivers as Calculators: Counting, Correlators and Riemann Surfaces
• Quiver Yangian and Supersymmetric Quantum Mechanics
• Brownian Motion, Chern-Simons Theory, and 2d Yang-Mills
• From counting to construction of BPS states in N=4 SYM
• On the L^2-metric of vortex moduli spaces
• Phase transitions in q-deformed 2d Yang-Mills theory and topological strings
• Matrix ensembles with global symmetries and 't Hooft anomalies from 2d gauge theory
• Dynamical actions and q-representation theory for double-scaled SYK
• Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras
• Mirror symmetry and line operators
• Anomalies, Black strings and the charged Cardy formula
• Entanglement branes in a two-dimensional string theory
• From SO/Sp instantons to W-algebra blocks
• Taming Defects in $\mathcal{N}=4$ Super-Yang-Mills
• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• Gauging hidden symmetries in two dimensions
• Strings from Feynman Graph counting : without large N
• Edge State Quantization: Vector Fields in Rindler
• Undoing decomposition
• Twisted Superspace for N=D=2 Super BF and Yang-Mills with Dirac-Kähler Fermion Mechanism
• Permutations and the Loop
• Notes on nonabelian (0,2) theories and dualities
• Entanglement Entropy in 2D Non-abelian Pure Gauge Theory
• N=4 Twisted Superspace from Dirac-Kahler Twist and Off-shell SUSY Invariant Actions in Four Dimensions
• $T\bar{T}$ in JT Gravity and BF Gauge Theory
• Holomorphic maps and the complete 1/N expansion of 2D SU(N) Yang-Mills
• Yang--Mills Glueballs as Closed Bosonic Strings
• Quantum Entanglement of Baby Universes
• Supergroup Structure of Jackiw-Teitelboim Supergravity
• Cut-and-join operators and N=4 super Yang-Mills
• Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
• Quarter BPS classified by Brauer algebra
• A Simple Worldsheet Black Hole
• Large $N$ phase transition in $T\overline{T}$-deformed $2d$ Yang-Mills theory on the sphere
• Cech and de Rham Cohomology of Integral Forms
• Open Wilson Lines and Group Theory of Noncommutative Yang-Mills Theory in Two Dimensions
• On 't Hooft Defects, Monopole Bubbling and Supersymmetric Quantum Mechanics
• Dynamical Edge Modes and Entanglement in Maxwell Theory
• Nonplanar integrability at two loops
• Matrix Quantization of Gravitational Edge Modes
• Topological Vector Symmetry of BRSTQFT and Construction of Maximal Supersymmetry
• Comments on Half S-Branes
• Mass Perturbations in Twisted N=4 Supersymmetric Gauge Theories
• Complex matrix model duality
• A brief review of the 2d/4d correspondences
• Entanglement branes, modular flow, and extended topological quantum field theory
• A-Model Correlators from the Coulomb Branch
• Two-Dimensional Yang-Mills Theory on Surfaces With Corners in Batalin-Vilkovisky Formalism
• Chern-Simons matrix models, two-dimensional Yang-Mills theory and the Sutherland model
• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• Lectures on Two-Dimensional Noncommutative Gauge Theory 2: Quantization
• A Note on Knot Invariants and q-Deformed 2d Yang-Mills
• Unitarity of Maxwell theory on curved spacetimes in the covariant formalism
• Renormalization and BRST symmetry in Donaldson-Witten theory
• Morita Duality and Noncommutative Wilson Loops in Two Dimensions
• Circuit Complexity in Topological Quantum Field Theory
• The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction
• On the refined counting of graphs on surfaces
• 4d partition function on S^1 x S^3 and 2d Yang-Mills with nonzero area
• Exact $T\bar{T}$-deformation of two-dimensional Yang-Mills theory on the sphere
• A-branes, foliations and localization
• A criterion for lattice supersymmetry: cyclic Leibniz rule
• BPS Hall Algebra of Scattering Hall States
• $T\bar{T}$ deformed YM${}_2$ on general backgrounds from an integral transformation
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• BPS states, conserved charges and centres of symmetric group algebras
• Semifinite von Neumann algebras in gauge theory and gravity
• $T\overline{T}$-deformation of $q$-Yang-Mills theory
• A Note On The Semiclassical Formulation Of BPS States In Four-Dimensional N=2 Theories
• Unitary Chern-Simons matrix model and the Villain lattice action
• D-brane central charge and Landau-Ginzburg orbifolds
• On Exponential corrections to the 1/N expansion in two-dimensional Yang Mills theory
• Resurgence analysis of 2d Yang-Mills theory on a torus
• Tearing down spacetime with quantum disentanglement
• N=2 gauge theories, instanton moduli spaces and geometric representation theory
• Wilson surface observables from equivariant cohomology
• Two-dimensional Yang-Mills theory, Painleve equations and the six-vertex model
• Noncommutative Cohomological Field Theory and GMS soliton
• Perturbative and non-perturbative aspects of the two-dimensional string/Yang-Mills correspondence
• Integrality, Duality and Finiteness in Combinatoric Topological Strings
• Entanglement entropy and the large $N$ expansion of two-dimensional Yang-Mills theory
• Bosonization of non-relativistic fermions on a circle: Tomonaga's problem revisited
• The number of gauge singlets in supersymmetric Yang-Mills quantum mechanics
• The quantum detection of projectors in finite-dimensional algebras and holography
• String Theoretical Interpretation for Finite N Yang-Mills Theory in Two-Dimensions
• Permutation combinatorics of worldsheet moduli space
• Analogues of Mathai-Quillen forms in sheaf cohomology and applications to topological field theory
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Quantum Wilson surfaces and topological interactions
• BPS States Meet Generalized Cohomology
• 1/2 BPS Correlator and Free Fermion
• Decomposition and the Gross-Taylor string theory
• $T\bar{T}$-deformed 2D Yang-Mills at large N: collective field theory and phase transitions
• On Supersymmetric Interface Defects, Brane Parallel Transport, Order-Disorder Transition and Homological Mirror Symmetry
• Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae
• Triangulated Surfaces in Twistor Space: A Kinematical Set up for Open/Closed String Duality
• Spontaneous symmetry breaking in pure 2D Yang-Mills theory
• A String Theory for Two Dimensional Yang-Mills Theory I
• Large N Expansion of q-Deformed Two-Dimensional Yang-Mills Theory and Hecke Algebras
• Quivers, Line Defects and Framed BPS Invariants
• Higgs and Coulomb Branch Descriptions of the Volume of the Vortex Moduli Space
• Algorithms for representations of quiver Yangian algebras
• Supersymmetric Wilson Loops via Integral Forms
• Partition function of $\mathcal{N}=2$ supersymmetric gauge theory and two-dimensional Yang-Mills theory
• The gauging of BV algebras
• Gauge fixing and equivariant cohomology
• Localization on the D-brane, two-dimensional gauge theory and matrix models
• Basic and Equivariant Cohomology in Balanced Topological Field Theory
• The phase diagram of $T\bar{T}$-deformed Yang-Mills theory on the sphere
• Gravity Dual of Two-Dimensional $\mathcal{N}=(2,2{)}^{\ast }$ Supersymmetric Yang-Mills Theory and Integrable Models
• Wilson punctured network defects in 2D q-deformed Yang-Mills theory
• Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited
• Resurgence in 2-dimensional Yang-Mills and a genus altering deformation
• Functional renormalization of QCD in $1+1$ dimensions: four-fermion interactions from quark-gluon dynamics
• Matrix and tensor witnesses of hidden symmetry algebras
• Dilaton shifts, probability measures, and decomposition
• Branched Coverings and Interacting Matrix Strings in Two Dimensions
• Target Space Equivariant Cohomological Structure of the Poisson Sigma Model
• Chiral expansion and Macdonald deformation of two-dimensional Yang-Mills theory
• Loop Equation and Exact Soft Anomalous Dimension in N=4 Super Yang-Mills
• Algebras, traces, and boundary correlators in $\mathcal{N}=4$ SYM
• Symmetric Group Gauge Theories and Simple Gauge/String Dualities
• String hamiltonian from generalized YM gauge theory in two dimensions
• Supersymmetric Gauge Theories, Vortices and Equivariant Cohomology
• Vector Supersymmetry of 2D Yang-Mills Theory
• Tunnels Under Geometries (or Instantons Know Their Algebras)
• Ruelle zeta function from field theory
• Topological symmetry-restored phase of gravity
• Wilson loop expectations as sums over surfaces on the plane
• The 3BF theory as a TQFT
• Large-N transitions for generalized Yang-Mills theories in 1+1 dimensions
• Boson-Fermion Correspondence and Holomorphic Anomaly Equation in 2d Yang-Mills Theory on Torus
• Extended Connection in Yang-Mills Theory
• Weak Coupling Expansion of Yang-Mills Theory on Recursive Infinite Genus Surfaces
• Two-dimensional Yang-Mills Theory on Recursive Infinite Genus Surfaces
• Towards 2-dimensional non-commutative integrals
• Large-N limit of the two-dimensinal Non-Local Yang-Mills theory on arbitrary surfaces with boundary
• Equivariant Symplectic Geometry of Gauge Fixing in Yang-Mills Theory
• Lattice defect networks in 2d Yang-Mills
• Equivariant localization in Batalin-Vilkovisky formalism
• Cap amplitudes in random matrix models
• Quiver Yangian algebras associated to Dynkin diagrams of A-type and their rectangular representations
• Twisting BFSS & IKKT
• Transformations of Currents in Sigma-Models with Target Space Supersymmetry
• DGLA Dg and BV formalism
• Non-Local Observables in the A-Model