Brownian Motion, Chern-Simons Theory, and 2d Yang-Mills

Abstract

We point out a precise connection between Brownian motion, Chern-Simons theory on S^3, and 2d Yang-Mills theory on the cylinder. The probability of reunion for N vicious walkers on a line gives the partition function of Chern-Simons theory on S^3 with gauge group U(N). The probability of starting with an equal-spacing condition and ending up with a generic configuration of movers gives the expectation value of the unknot. The probability with arbitrary initial and final states corresponds to the expectation value of the Hopf link. We find that the matrix model calculation of the partition function is nothing but the additivity law of probabilities. We establish a correspondence between quantities in Brownian motion and the modular S- and T-matrices of the WZW model at finite k and N. Brownian motion probabilitites in the affine chamber of a Lie group are shown to be related to the partition function of 2d Yang-Mills on the cylinder. Finally, the random-turns model of discrete random walks is related to Wilson's plaquette model of 2d QCD, and the latter provides an exact two-dimensional analog of the melting crystal corner. Brownian motion provides a useful unifying framework for understanding various low-dimensional gauge theories.

12 pages, v2: typos corrected, reference added

References in corpus (17)

• Knot Invariants and Topological Strings
• On Geometry and Matrix Models
• Chern-Simons theory, matrix integrals, and perturbative three-manifold invariants
• Quantum Foam and Topological Strings
• Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories
• Matrix Model as a Mirror of Chern-Simons Theory
• Chern-Simons Theory and Topological Strings
• Soft matrix models and Chern-Simons partition functions
• Seiberg-Witten Theory and Random Partitions
• Two Dimensional Yang-Mills, Black Holes and Topological Strings
• N=1 Supersymmetry, Deconstruction, and Bosonic Gauge Theories
• Calogero-Sutherland model and bulk-boundary correlations in conformal field theory
• Conformal Fractal Geometry and Boundary Quantum Gravity
• The String Partition Function for QCD on the Torus
• A Hexagon Model for 3D Lorentzian Quantum Cosmology
• D-branes as Defects in the Calabi-Yau Crystal
• Macdonald's identities and the large N limit of $Y{M}_2$ on the cylinder

Cited by in corpus (38)

• M5-branes from gauge theories on the 5-sphere
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Extremes of N vicious walkers for large N: application to the directed polymer and KPZ interfaces
• Matrix Models for Random Partitions
• Chern-Simons matrix models and Stieltjes-Wigert polynomials
• Derivation of Calabi-Yau Crystals from Chern-Simons Gauge Theory
• Reunion probability of N vicious walkers: typical and large fluctuations for large N
• Level-rank duality of the U(N) WZW model, Chern-Simons theory, and 2d qYM theory
• Topological strings and large N phase transitions II: Chiral expansion of q-deformed Yang-Mills theory
• Discrete and oscillatory Matrix Models in Chern-Simons theory
• Topological strings and large N phase transitions I: Nonchiral expansion of q-deformed Yang-Mills theory
• Chern-Simons Theory, 2d Yang-Mills, and Lie Algebra Wanderers
• Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
• Interacting Particles on the Line and Dunkl Intertwining Operator of Type A: Application to the Freezing Regime
• A Note on Knot Invariants and q-Deformed 2d Yang-Mills
• Diffusion of Wilson Loops
• Localization for Wilson Loops in Chern-Simons Theory
• Universal Correlators from Geometry
• Unitary Chern-Simons matrix model and the Villain lattice action
• Large N limit of 2D Yang-Mills Theory and Instanton Counting
• Multiple phases and meromorphic deformations of unitary matrix models
• Brownian Gibbs property for Airy line ensembles
• Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
• $T\bar{T}$-deformed 2D Yang-Mills at large N: collective field theory and phase transitions
• Instanton Counting and Dielectric Branes
• A Diffusion Model for SU(N) QCD Screening
• Matrix models and stochastic growth in Donaldson-Thomas theory
• Chern-Simons theory, exactly solvable models and free fermions at finite temperature
• From elongated spanning trees to vicious random walks
• Kramers-Moyall cumulant expansion for the probability distribution of parallel transporters in quantum gauge fields
• Crystal bases and three-dimensional $\mathcal{N}=4$ Coulomb branches
• Walking on Archimedean Lattices: Insights from Bloch Band Theory
• Complex (super)-matrix models with external sources and $q$-ensembles of Chern-Simons and ABJ(M) type
• Orientation Reversal and the Chern-Simons Natural Boundary
• Large-N transitions for generalized Yang-Mills theories in 1+1 dimensions
• Weak Coupling Expansion of Yang-Mills Theory on Recursive Infinite Genus Surfaces
• A new large N phase transition in YM2
• $T\bar{T}$ deformation of Calogero-Sutherland model via dimensional reduction