The \(SO(4)\) Verlinde formula using real polarizations
type: Item
instance of: scholarly article
title: The \(SO(4)\) Verlinde formula using real polarizations
DOI: 10.1007/978-3-031-57204-3_10
author: Lisa C. Jeffrey
author: Kaidi Ye
publication date: 2024-10-02T00:00:00Z
review text: The Verlinde formula [\textit{E. Verlinde}, Nucl. Phys., B 300, 360--376 (1988; Zbl 1180.81120)] is a formula for the dimension of the geometric quantization of the space \(\mathcal{M}_g(G)\) of conjugacy classes of representations of the fundamental group of an oriented compact, closed 2-manifold of genus \(g\) into a compact Lie group \(G\). The aim of this paper is to adapt the construction of \textit{L. C. Jeffrey} and \textit{J. Weitsman} [Commun. Math. Phys. 150, No. 3, 593--630 (1992; Zbl 0787.53068)] to interpret the SO(4) Verlinde formula through a real polarization. The authors give an interpretation for the Verlinde formula for the group \(SO(4)\). The construction given extends the analogy with toric geometry to this case. The authors make use of the fact that \(SU(2)\times SU(2)\) is a double cover of \(SO(4)\). \N\NThe paper is organized as follows. Section 1 is an introduction to the subject. Section 2 describes real polarizations. Section 3 deals with some relations between several moduli spaces. Section 4 is devoted to the moduli space of a trinion. The authors describe the period of a Hamiltonian function in Section 5 and the Verlinde formula in Section 6.\N\NFor the entire collection see [Zbl 1540.57001].
reviewed by: Ahmed Lesfari
Mathematics Subject Classification ID: 53D55
zbMATH DE Number: 7921289
zbMATH Keywords: Verlinde formula
zbMATH Keywords: real polarizations
zbMATH Keywords: moduli spaces
MaRDI profile type: Publication
cites work: The Yang-Mills equations over Riemann surfaces
cites work: Hamiltonian Group Actions and Equivariant Cohomology
cites work: Invariant functions on Lie groups and Hamiltonian flows of surface group representations
cites work: Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula
cites work: ON THE VERLINDE FORMULAS FOR SO(3)-BUNDLES
cites work: Some comments on Chern-Simons gauge theory
cites work: Q4123957
cites work: Fusion rules and modular transformations in 2D conformal field theory
cites work: Real polarization of the moduli space of flat connections on a Riemann surface
cites work: Quantization via real polarization of the moduli space of flat connections and Chern-Simons gauge theory in genus one
cites work: Conformal field theory and the cohomology of the moduli space of stable bundles
cites work: Q4006890
description: scientific article; zbMATH DE number 7921289