Normal bundles to Laufer rational curves in local Calabi-Yau threefolds (Q2475257)

This paper proves a conjecture by \textit{F. Ferrari} [Adv. Theor. Math. Phys. 7, No. 4, 619--665 (2003; Zbl 1058.81057)] which predicts, by string-theoretic considerations, the structure of the normal bundle of certain rational curves in a local Calabi-Yau threefold (i.e.~a quasi-projective non-proper threefold with trivial canonical bundle). From a mathematical viewpoint, the problem can be stated as follows. One starts with a rank-2 vector bundle \(V\) on a rational curve \(\mathcal{C}\) whose total space has trivial canonical bundle and has a section. The bundle \(V\) is then isomorphic to \({\mathcal O}(-n-2)\oplus{\mathcal O}(n)\) for an integer \(n\geq 0\) and can be deformed into a fibration \(X\) over \(\mathcal{C}\) by adding a certain nonlinear term to the transition functions of \(V\). This term depends on a holomorphic function \(B\) on \(\mathbb{C}^\ast\times \mathbb{C}\), where \(\mathbb{C}^\ast\) is identified with the intersection of the two standard open sets of \({\mathcal C}\simeq \mathbb{P}^1\), and the sections of \(X\to \mathcal{C}\) are in one-to-one correspondence with the critical points of a function, called the superpotential, defined on the cohomology space \(H^0({\mathcal O},{\mathcal O}(n))\) as the integral of \(B(z,\omega(z))dz\). Ferrari's conjecture then says that the normal bundle to the image of the section of \(X\to \mathcal{C}\) corresponding to a critical point \(x\) is isomorphic to \({\mathcal O}(-r-1)\oplus{\mathcal O}(r-1)\), where \(r\) is the rank of the Hessian of \(W\) at \(x\). The proof is based on some explicit computations.