Let \(X\) be a compact Riemann surface of genus \(g\), and denote by \({\mathcal U}_ X (d)\) the moduli space of semistable rank-2 vector bundles of degree \(d\). On \({\mathcal U}_ X (d)\) there is a natural polarizing line bundle \(\theta_ X\) which generalizes the line bundle associated with the Riemann theta divisor on the Jacobian of \(X\). For any positive integer \(k\), the space \(H^ 0({\mathcal U}_ X (d), \theta_ X^{\otimes k})\) is called the space of generalized theta functions of order \(k\) on \({\mathcal U}_ X (d)\). These spaces correspond to certain objects arising in conformal quantum field theory, namely to the so-called conformal blocks of level \(k\), and the arguments of physicists suggest that the spaces \(H^ 0({\mathcal U}_ X (d), \theta_ X^{ \otimes k})\) should be related to the spaces of generalized order-\(k\) theta functions for suitable curves \(Y\) of (the lower) genus \(g - 1\). In physics, such a relationship is known under the name “factorization rule” (or “glueing axiom”, respectively). The aim of the present paper is to establish such a relationship by rigorous algebro-geometric methods, i.e., to give an explicit geometric counterpart of that factorization principle in conformal quantum field theory. To this end, the authors consider degenerations of the given curve \(X\) into irreducible curves \(Y\) which are smooth except for a single node. Then the normalization \(\widetilde Y\) of \(Y\) is smooth of genus \(g - 1\). The authors’ main theorem states that, for such a curve \(Y\), the inequality \(g \geq 4\) implies \(H^ 1 ({\mathcal U}_ Y(d), \theta_ Y^{\otimes k}) = H^ 1 ({\mathcal U}_ X (d), \theta_ X^{\otimes k}) = 0\) which, in turn, means that the spaces of generalized theta functions on \({\mathcal U}_ X (d)\) and \({\mathcal U}_ Y (d)\) of order \(k\) are isomorphic.
The second part of the main theorem establishes a “factorization rule” by constructing an isomorphism \(H^ 0 ({\mathcal U}_ Y (d), \theta_ Y^{\otimes k}) \cong \bigoplus_ j H^ 0({\mathcal U}_{\widetilde Y}^ j (d), \theta_ j)\), where \(j\) runs through a certain index domain depending on \(k\), \({\mathcal U}^ j_{\widetilde Y} (d)\) denotes the moduli space of semistable rank-2 vector bundles of degree \(d\) on \(\widetilde Y\) with a certain parabolic structure [in the sense of V. B. Mehta and C. S. Seshadri, Math. Ann. 248, 205-239 (1980; Zbl 0454.14006)], also depending on \(k\), and \(\theta_ j\) is the natural polarizing line bundle (generalized theta bundle) on \({\mathcal U}^ j_{\widetilde Y} (d)\).
The proof of this important relation confirming the physicists’ arguments is rather involved and delicate, since the authors are dealing with nodal curves, for which the moduli spaces of parabolic vector bundles have not been constructed yet. Therefore a great part of the paper is devoted to establishing the existence of moduli spaces of semistable torsion-free sheaves of rank 2 on nodal curves and their properties, including the analysis of their generalized theta bundles. The authors announce that, in a subsequent work, they will remove the restrictional condition \(g \geq 4\) for the vanishing of \(H^ 1({\mathcal U}_ Y (d), \theta_ Y^{\otimes k})\). Then their results can be used to verify the Verlinde formula for the dimension of the spaces of generalized order-\(k\) theta functions on \({\mathcal U}_ X (d)\), which would give another approach in this case, different from the one made by A. Bertram [Invent. Math. 113, No. 2, 351-372 (1993)].