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Hodge Structures of Complex Multiplication Type from Rational Conformal Field Theories
Abstract
Gukov and Vafa have proposed that a conformal field theory describing a string compactification on a manifold is rational (an RCFT) if and only if the manifold admits complex multiplication (CM). We investigate and extend the Gukov-Vafa proposal by constructing Hodge structures of CM type using only RCFT data, without reference to a geometric interpretation.
We use the chiral and boundary states of the RCFT to construct the complex and rational vector spaces underlying the Hodge structure. Using the known notion of Galois symmetry of RCFTs and some elementary Galois theory, we are able to show that these Hodge structures are of CM-type, subject to some technical assumptions that can be verified explicitly for large classes of theories, including those without known geometric interpretation. We also discuss briefly the relation of complex multiplication to arithmetic geometry.
This talk is based on arXiv:2510.25708 with H. Jockers and M. Sarve.
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The Oxford Maths Festival is returning for 2026! 
For any students due to sit their exams in Weeks 4-9 of Trinity term in a 'small room' venue, this venue will be St Luke's Chapel, just outside the Mathematical Institute. The third annual Oxford Women and Non-Binary People in Mathematics Day (OxWIM Day) ran very successfully on the 28th of February in the Andrew Wiles Building. This year’s theme was ‘Shaping Success’ and explored the many ways underrepresented genders can define their own success within a mathematical career.