Topological Field Theory and Rational Curves

TL;DR

The paper develops a topological field theory framework for the $ ext{N}=2$ non-linear $ ext{sigma}$-model on a $ ext{Calabi--Yau}$ threefold and shows that genus-zero instantons contribute integer invariants $N^v_{abc}$ counting holomorphic maps to rational curves. By twisting to a topological theory and using a Mathai–Quillen construction on moduli spaces of holomorphic maps, it reduces the problem to a finite-dimensional Euler class computation, leading to explicit intersection-number formulas for both single and multiple covers. In the quintic example with $h^{1,1}=1$, the approach yields the $f$-polynomial $f = 5 + obreak obreak sum_{k eq 0} a_k k^3 q^k/(1-q^k)$, connecting the curve counts to mirror symmetry and confirming the conjectured structure of rational curves. Overall, the work provides a physically grounded derivation of integer curve-counting invariants and links topological field theory with algebraic geometry via BRST and Euler-class techniques.

Abstract

We analyze the superstring propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear sigma-model and the structure of rational curves on the Calabi-Yau manifold. We study in detail the case of the world-sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes.