On an equivalence of divisors on \(\overline{\mathrm{M}}_{0,n}\) from Gromov-Witten theory and conformal blocks (Q6594714)

The authors of this paper studied two families of base point-free divisors on the smooth projective variety, the moduli space \(\overline{\mathcal{M}}_{g,n}\) of \(n\)-pointed stable curves of genus \(g\). The first are obtained from the Gromov-Witten theory of Grassmannians, and the second are conformal blocks divisors, which are first Chern classes of globally generated vector bundles defined by resentations of a simple Lie algebra in type A. They are quite different in general. However, in some cases they are believed to be numerically equivalent with the same given data. So, they stated the \textbf{GW=CB} conjecture: Let \(\bar\lambda = (\lambda^1,\ldots, \lambda^n)\) be partitions corresponding to Schubert classes in \(Gr_{r,r+l}\) such that \(\sum_i|\lambda^i|=(r+1)(l+1)\). Then the GW divisor on \(\overline{\mathcal{M}}_{g,n}\) is numerically equivalent to the first Chern class of the critical level CB bundle. To convince the reader about the conjectures, they proved two theorems: This conjecture is true for any \(n\) if it is correct on \(\overline{\mathcal{M}}_{g,n}\); This conjecture is obeyed if the partition \(\bar\lambda\) satisfies the column condition.