The Verlinde formula for Higgs bundles

Abstract

We propose and prove the Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. This generalizes the equivariant Verlinde formula for the case of $SU(n)$ proposed previously by the second and third author. We further establish a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. Finally, we prove that these dimensions form a one-parameter family of $1+1$-dimensional TQFT, uniquely classified by the complex Verlinde algebra, which is a one-parameter family of Frobenius algebras. We construct this one-parameter family of Frobenius algebras as a deformation of the classical Verlinde algebra for $G$.

arXiv comment39 pages; references added, misprints corrected and exposition improved

References in corpus (7)

• Geometric construction of modular functors from Conformal Field Theory
• Abelian Conformal Field theories and Determinant Bundles
• A new formulation of the Teichmüller TQFT
• Complex Quantum Chern-Simons
• Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality
• The equivariant Verlinde formula on the moduli of Higgs bundles
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• Argyres-Douglas Theories, Chiral Algebras and Wild Hitchin Characters
• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps
• Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality
• Verlinde formulae on complex surfaces: K-theoretic invariants
• 3d N=2 Chern-Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians
• Twisted argyle quivers and Higgs bundles
• Mirror symmetry with branes by equivariant Verlinde formulae
• Genus-one complex quantum Chern--Simons theory
• Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups
• Sheaves on surfaces and virtual invariants
• Level N Teichmüller TQFT and Complex Chern-Simons Theory
• Affine Pieri rule for periodic Macdonald spherical functions and fusion rings
• Elliptic Ruijsenaars difference operators, symmetric polynomials, and Wess-Zumino-Witten fusion rings