Topological field theory and rational curves

Abstract

We analyze the quantum field theory corresponding to a string propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten's topological non-linear σ-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.

Notes on Certain (0,2) Correlation Functions

“…If the cohomology of the restriction of E jumps, then surely the corresponding numerical factor jumps, and so the required integral over the moduli space of zero modes can not have nearly so simple a form as appeared in [14].…”

“…Just as in [14], we can soak up these excess zero modes by using the four-fermi term in the worldsheet action. Recall the four-fermi term has the form…”

Lectures on the Topological Vertex

“…Therefore, the invariant n 0 β corresponds to primitive holomorphic maps, and the non-integrality of genus-zero GromovWitten invariants is due to the effects of multicovering. The multicovering phenomenon in genus 0 was found experimentally in Candelas et al (1991) and later derived in the context of Gromov-Witten theory by Aspinwall and Morrison (1993). The structure result of Gopakumar and Vafa also predicts that the multicovering of degree d of a genus g curve contributes with a weight d 3−2g (coming from Li 3−2g ).…”

Gromov-Witten invariants via algebraic geometry

“…Degenerate instantons first appeared in [23] in the course of compactifying the instanton moduli space of multiple cover maps to a rational curve. On the other hand, the fact that the instantons were degenerate played no role in the calculation.…”

“…This is the approach taken in [23] and the appendix to [2]. This space of degenerate instantons is a subset of the Hilbert scheme [24] of C × X, (or a relative Hilbert scheme, if C can vary).…”