The \(SO(4)\) Verlinde formula using real polarizations
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].
scientific article; zbMATH DE number 7921289