An abelianization of the SU(2) WZW model (Q855255)

Let \(M_g\) be the moduli space of semi-stable holomorphic vector bundles of rank 2 on a Riemann surface \(C\) of genus \(g\). Denote by \(\mathcal{L}\) the determinant line bundle on \(M_g\). An element of the space of holomorphic sections \(H^0(M_g,\mathcal{L}^k)\) is called a a rank \(2\) theta function of level \(k\) or a nonabelian theta function. The space \(H^0(M_g,\mathcal{L}^k)\) can be identified (see the work of A. Beauville and Y. Laszlo) with the space of conformal block of level \(k\) of the SU(2) Wess-Zumino-Witten model for wich the author describes an abelianization procedure, carrying out the program of M.\ Atiyah and N.\ Hitchin. To be more precise, the author gives an explicit representation of a basis of \(H^0(M_g,\mathcal{L}^k)\) and its transformation formula in terms of classical Riemann theta functions of degree \(k+2\) with automorphic form coefficients, defined on the Prym variety associated with a two-fold branched covering surface. Using pants decomposition, the first section gives some topological properties of a family of 2-fold branched covering surfaces \(\widetilde{C}\) of \(C\) parametrized by the configuration space of \(4g-4\) unordered (mutually distinct) points. Now one considers the Prym varieties (more precisely the translated Prym varieties \(P'=\{ \widetilde{L}\in J^{2g-2}(\widetilde{C}) : \text{Nm}(\widetilde{L})=\omega_C\}\)) which can be equipped with natural coordinates coming from the pant decomposition. If we denote \(P\) the natural covering space of \(P'\), the author studies the morphism \(\pi:P\rightarrow M_g\) (which is a dominant map) and a pulled-back section of \(H^0(M_g,\mathcal{L}^k)\) can be expressed by Riemann theta functions of level \(2k\) on \(P\). A key point here is the description of the action induced by the fundamental group of the configuration space on the space of Riemann theta functions on \(P\). This leads to consider the notion of projectively invariant Riemann theta functions under this action. The map \(\pi\) is a holomorphic branched covering and its branching divisor is Div\((\Pi)\) where \(\Pi\) is the Pfaffian of \(\pi\), and can be expressed precisely using the coordinates on \(P\) as a Riemann theta function of level 4. Now, let's consider a family \(\widetilde{\psi}\) of holomorphic sections of line bundles on the family of Prym varieties \(P\) which are pull-back of a section \(\psi \in H^0(M,\mathcal{L}^k)\) by \(\pi\). Using the previous result, the author is able to construct a natural differential operator \(D\) on the space of the line bundles on \(P\) such that \(D\widetilde{\psi}=0\). In fact, as it is well explained by the author, the global symmetry (due to the action of the fundamental group of the configuration space) and the equation \(D\widetilde{\psi}=0\) characterize completely the pulled-back sections. The last step is now to use the previous characterization to build a basis of \(H^0(M_g,\mathcal{L}^k)\) in terms of the classical Riemann theta functions with automorphic form coefficients. The basis is indexed by the Quantum-Clebsch-Gordan condition of level \(k\), i.e the result gives directly that \(\dim H^0(M_g,\mathcal{L}^k)\) is equal to the number of admissible spin weights attached to a pant decomposition of the Riemann surface. Now, over the Teichmüller space \(\mathcal{T}\) and using the connection \(D\), the author is able to define a projectively flat connection on the vector bundle of fibre \(H^0(M_g,\mathcal{L}^k)_{\mathcal{T}}\). An important aspect of this approach is that one recovers naturally by this construction the connection introduced by \textit{N. J. Hitchin} [Commun. Math. Phys. 131, No. 2, 347--380 (1990; Zbl 0718.53021)] (a heat equation is also satisfied here). Moreover, the author obtains an hermitian product invariant with respect to this connection on \(H^0(M_g,\mathcal{L}^k)_{\mathcal{T}}\), that might be related to the one defined by A.\ Kirillov. In an other section, the author investigates how a symplectic linear transformation of the Prym variety \(P\) induces a transformation of the theta functions (of rank 2). It turns out to be related to the Maslov index. Finally a discussion about the case of genus 1 is included.