Logarithmic Gauged Linear Sigma Models

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Over the past few decades, deep connections between pure mathematics and physics have developed. One such area of fruitful interaction is between string theory, a branch of theoretical physics connected with the structure of elementary particles, and enumerative geometry, which studies counts of the number of solutions to geometric problems (how many lines, for example, pass through four fixed lines in space). This project will develop a new technique, which is inspired from the gauged linear sigma model (GLSM) in physics, to solve several enumerative problems.In more detail, the project will develop a new technique (log GLSM) in higher genus Gromov-Witten theory. The goal is to prove conjectures from physics such as the holomorphic anomaly equations for quintic threefolds and more general complete intersection Calabi-Yau 3-folds. For this, a new moduli space will be constructed whose localization formula gives a way of computing Gromov-Witten invariants of a convex complete intersection from those of its ambient space, and new "effective invariants". Different applications will exploit the structure of the localization formula, and correspondences between effective invariants.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.