Compactified imaginary Toda theory

Based on the work of Guillarmou, Kupiainen, and Rhodes, we construct compactified imaginary Toda theory on closed Riemann surfaces, extending the rank-one construction to the higher-rank setting. This theory is expected to describe critical higher-rank models with extended symmetries, such as web models. We construct the correlation functions and prove that they satisfy the axioms of conformal field theory, including Segal's gluing axioms. On the Riemann sphere, we express the correlation functions as Dotsenko--Fateev type integrals. In the case $\mathfrak g=\mathfrak{sl}_n$, under a semidegenerate condition, we obtain a closed formula for the three-point structure constant.