L2

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.

TBA

This is a joint OxPDE and Numerical Analysis seminar. 

Recent years have seen the expansion of the traditional notion of symmetry in quantum theory to so-called generalised or categorical symmetries, which may in particular be non-invertible. This seems to be at odds with Wigner's theorem, which asserts that quantum symmetries ought to be implemented by (anti)unitary -- and hence invertible -- operators on the Hilbert space. In this talk, we will try to resolve this puzzle for generalised symmetries that are described by (higher) fusion categories. After giving a gentle introduction to the latter, we will discuss how one can associate an inner-product-preserving operator to (possibly non-invertible) symmetry defects and illustrate our construction through concrete examples. Based on the recent work 2602.07110 with Gai and Schäfer-Nameki.

In this talk, I will examine the dynamics of the fermion–rotor system, originally introduced by Polchinski as a toy model for monopole–fermion scattering. Despite its simplicity, the system is surprisingly subtle, with ingoing and outgoing fermion fields carrying different quantum numbers. I will show that the rotor acts as a twist operator in the low-energy theory, changing the quantum numbers of excitations that have previously passed through the origin to ensure scattering consistent with all symmetries, thereby resolving the long-standing Unitarity puzzle. I will then discuss generalizations of this setup with multiple rotors and unequal charges, and demonstrate how the system can be viewed as a UV-completion of boundary states for chiral theories, establishing a connection to the proposed resolution of the puzzle using boundary conformal field theory.

In this seminar, I will explore how the sum over topologies in pure AdS_3​ quantum gravity furnishes a consistent statistical interpretation of the boundary CFT_2. By formulating a statistical version of the conformal bootstrap, which combines crossing symmetry with typicality at high energies, we will discover a large set of non-handlebody topologies in the bulk (of which I will give some examples) that are needed for consistency of the boundary description. Interestingly, this set contains only on-shell (i.e. hyperbolic) 3-manifolds, but not all of them. This suggests that the full sum over all on-shell saddles in 3d gravity may be a highly non-minimal solution of the statistical bootstrap. Based on the recent work 2601.07906 with Belin, Collier, Eberhardt and Liska.