The Topological Vertex

TL;DR

This paper introduces a cubic field theory that computes all genus amplitudes of the topological A-model for non-compact Calabi-Yau toric threefolds, linking Feynman diagrams to Calabi-Yau topology.

Contribution

It constructs a novel cubic field theory that encodes topological string amplitudes via Feynman diagrams and interprets it as an operatorial B-model mirror computation.

Findings

Provides a field theory encoding all genus amplitudes

Links Feynman diagram topology to Calabi-Yau topology

Identifies B-branes as fermions related to a chiral boson

Abstract

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact Calabi-Yau toric threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of Calabi-Yau. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the Kodaira-Spencer quantum theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.