Algorithmic and Machine Learning Methods for Witten Genus Vanishing Theorems

Witten genus, a fundamental invariant in index theory, extends <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$A$</tex>-genus and plays a crucial role in differential topology and mathematical physics. A central problem concerns identifying conditions under which the Witten genus vanishes, with the Stolz conjecture proposing a connection to positive Ricci curvature. While classical vanishing results exist for specific classes of complete intersections, there is no systematic computational approach for generating and analyzing such manifolds. This work introduces an algorithmic framework for studying the vanishing properties of the Witten genus in string complete intersections. By encoding complete intersections as configuration matrices, we develop an efficient algorithm for generating string manifolds in direct products of projective spaces and Bott manifolds and computing their (mod 2) Witten genera as q-series. As an application, we implement a predictive model using a convolutional neural network, achieving 0.9757 accuracy. CNN consistently outperforms logistic regression, random forests, and XGBoost across most training ratios, demonstrating that structural patterns associated with vanishing behavior can be effectively captured through machine learning.