Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model

Abstract

We give a derivation of the Verlinde formula for the $G_k$ WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function $Z_{\Sigma \times S^1}$ of $\Sigma \times S^1$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{\Sigma \times S^1}$ is reduced to the partition function of an Abelian topological field theory on $\Sigma$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $\Sigma \times S^1$. We derive the $G_k/G_k$ model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the $G_k/G_k$ path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the shift $k\ra k+h$ is given, based on the index of the twisted Dolbeault complex.

47 pages (in A4 format), LaTex file, (original was truncated by the mailer - apologies, m.b.), IC/93/83

References in corpus (4)

• Two Dimensional Gauge Theories Revisited
• A Large k Asymptotics of Witten's Invariant of Seifert Manifolds
• $G/G$--Topological Field Theories by Cosetting $G_k$
• Stable pairs, linear systems and the Verlinde formula

Cited by in corpus (112)

• Supersymmetric vacua and Bethe ansatz
• Integrating Over Higgs Branches
• $\mathcal{N}=1$ supersymmetric indices and the four-dimensional A-model
• A topologically twisted index for three-dimensional supersymmetric theories
• Symmetries and Strings of Adjoint QCD${}_2$
• Black Holes, q-Deformed 2d Yang-Mills, and Non-perturbative Topological Strings
• The Verlinde Algebra And The Cohomology Of The Grassmannian
• Issues in Topological Gauge Theory
• Branes and N=2 Theories in Two Dimensions
• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Comments on twisted indices in 3d supersymmetric gauge theories
• Relativistic Calogero-Moser model as gauged WZW theory
• The Thermal Free Energy in Large N Chern-Simons-Matter Theories
• Towards a 2d QFT Analog of the SYK Model
• Phases of large $N$ vector Chern-Simons theories on $S^2\times S^1$
• Localization and Diagonalization: A review of functional integral techniques for low-dimensional gauge theories and topological field theories
• Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
• Gauged WZW models and Non-abelian duality
• Supersymmetric partition functions and the three-dimensional A-twist
• Seifert fibering operators in 3d $\mathcal{N}=2$ theories
• Chern-Simons and Twisted Supersymmetry in Higher Dimensions
• Chern-Simons Theory on S^1-Bundles: Abelianisation and q-deformed Yang-Mills Theory
• Finite Factorization equations and Sum Rules for BPS correlators in N=4 SYM theory
• Bethe/Gauge correspondence on curved spaces
• IIB flux non-commutativity and the global structure of field theories
• BPS/CFT correspondence IV: sigma models and defects in gauge theory
• Higgs Bundles, Gauge Theories and Quantum Groups
• Disorder Operators in Chern-Simons-Fermion Theories
• The Topological G/G WZW Model in the Generalized Momentum Representation
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
• Duality and Higher Temperature Phases of Large $N$ Chern-Simons Matter Theories on $S^2\times S^1$
• On Diagonalization in Map(M,G)
• 5D SYM and 2D q-Deformed YM
• Level-rank duality of the U(N) WZW model, Chern-Simons theory, and 2d qYM theory
• 5D SYM on 3D Sphere and 2D YM
• Topological strings and large N phase transitions II: Chiral expansion of q-deformed Yang-Mills theory
• Topology of Future Infinity in dS/CFT
• From Quantum Groups to Liouville and Dilaton Quantum Gravity
• Chern-Simons theory on L(p,q) lens spaces and Localization
• The action of outer automorphisms on bundles of chiral blocks
• Chern-Simons Theory on Seifert 3-Manifolds
• Canonical Quantization of Two Dimensional Gauge Fields
• 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
• Smoothed Transitions in Higher Spin AdS Gravity
• Twenty-five years of two-dimensional rational conformal field theory
• Chern-Simons Theory in Lens Spaces from 2d Yang-Mills on the Cylinder
• G/G gauged WZW model and Bethe Ansatz for the phase model
• Integrable many-body problems from the field theories
• Large $N$ phase transition in $T\overline{T}$-deformed $2d$ Yang-Mills theory on the sphere
• Keeping matter in the loop in dS${}_3$ quantum gravity
• On the Boundary Dynamics of Chern-Simons Gravity
• Self-dual Strings and 2D SYM
• Refined Black Hole Ensembles and Topological Strings
• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• A Note on Knot Invariants and q-Deformed 2d Yang-Mills
• G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra
• Entropy of Quantum Black Holes
• Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques
• Anyonic statistics and large horizon diffeomorphisms for Loop Quantum Gravity Black Holes
• Equivariant Kaehler Geometry and Localization in the G/G Model
• Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM
• The Universal Perturbative Quantum 3-manifold Invariant, Rozansky-Witten Invariants, and the Generalized Casson Invariant
• The Classical $r$-Matrix for the Relativistic Ruijsenaars-Schneider System
• Unitary Chern-Simons matrix model and the Villain lattice action
• G/G models as the strong coupling limit of topologically massive gauge theory
• Chern-Simons Theory on Seifert Manifold and Matrix Model
• Quantum exponentials for the modular double and applications in gravity models
• Wormholes in Quantum Mechanics
• An analytic Approach to Turaev's Shadow Invariant
• Bosonization with a background $U(1)$ gauge field
• Two-dimensional Yang-Mills theory, Painleve equations and the six-vertex model
• Some General Aspects of Coset Models and Topological Kazama-Suzuki Models
• On Phases of 3d $\mathcal{N}=2$ Chern-Simons-Matter Theories
• From Phase Space to Integrable Representations and Level-Rank Duality
• 5D SYM on 3D Deformed Spheres
• Grassmannian Topological Kazama-Suzuki Models and Cohomology
• The Hilbert Space of large $N$ Chern-Simons matter theories
• Quantum Chern-Simons theories on cylinders: BV-BFV partition functions
• Localization and Non-Renormalization in Chern-Simons Theory
• BRST Cohomology and Vacuum Structure of Two-Dimensional Chromodynamics
• New Phase Transitions in Chern-Simons Matter Theory
• Localization on the D-brane, two-dimensional gauge theory and matrix models
• The Gibbs paradox, Black hole entropy and the thermodynamics of isolated horizons
• Large Gauge Invariance in NonAbelian Finite Temperature Effective Actions
• From simplicial Chern-Simons theory to the shadow invariant I
• From simplicial Chern-Simons theory to the shadow invariant II
• Defects in G/H coset, G/G topological field theory and discrete Fourier-Mukai transform
• Moduli space volume of vortex and localization
• A representation theoretic approach to the WZW Verlinde formula
• Aspects of quantum gravity
• New phase in Chern-Simons theory on lens space
• Chern-Simons Theory with Complex Gauge Group on Seifert Fibred 3-Manifolds
• One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories
• BF Theories and Group-Level Duality
• Explicit connection between conformal field theory and 2+1 Chern-Simons theory
• Abelian Varieties, RCFTs, Attractors, and Hitchin Functional in Two Dimensions
• 3D-3D Correspondence from Seifert Fibering Operators
• D-Branes in Topological Membranes
• Dimension of Conformal Blocks in Five Dimensional Kaehler-Chern-Simons Theory
• Quantum Gravity and Equivariant Cohomology
• Torus Knots and the Chern-Simons path integral: a rigorous treatment
• Five-dimensional fermionic Chern-Simons theory
• Solving Topological Field Theories on Mapping Tori
• Massive fields and Wilson spools in JT gravity
• Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids
• Topological phase, spin Chern-Simons theory and level rank duality on lens space
• Massive Ray-Singer Torsion and Path Integrals
• $U(1)$-Gauge Field Theories on $G_2$-Manifolds
• 3 Definitions of BF Theory on Homology 3-Spheres
• Some Lower Dimensional Quantum Field Theories Reduced from Chern-Simons Gauge Theories
• Residues and Topological Yang-Mills Theory in Two Dimensions