Moduli spaces of parabolic fibres and conformal blocks
Let \(C\) be a smooth, irreducible, projective complex curve. Using a double coset construction, \textit{A. Beauville} and \textit{Y. Laszlo} [Commun. Math. Phys. 164, No. 2, 385-419 (1994; Zbl 0815.14015)] built up a canonical isomorphism between the vector space of conformal blocs studied in conformal field theory and the space of sections of an ample line bundle on the space of semistable vector bundles of fixed rank with trivial determinant. The present article extends this isomorphism setting into correspondence the space of sections of an ample line bundle on the space \(M(r,k, \vec{a}, \vec{n})\) of semistable parabolic bundles on \(C\) associated to a fixed finite set \(I\) of points of \(C\), and the dual of the vector space \(V_C(I, \vec{\lambda})\) of conformal blocs associated to \(I\) and a set \(\vec{\lambda} = \{\lambda_x\}_{x\in I}\) of dominant weights of the simple Lie algebra \(sl_r(\mathbb{C})\). The weights \(\vec{\lambda}\) depend on the parabolic weights \((k, \vec{a}, \vec{n})\). Using the factorizations rules for conformal blocs [see \textit{A. Tsuchiya}, \textit{K. Ueno} and \textit{Y. Yamada} in: Integrable systems in quantum field theory and statistical mechanics, Proc. Symp., Kyoto and Kyuzeso 1988, Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)], this allows the author to derive a Verlinde-type formula for the moduli spaces of parabolic vector bundles, i.e. to calculate the dimension of the space of sections cited above.
scientific article; zbMATH DE number 922159
scientific article