MIT Infinite Dimensional Algebra Seminar (Spring 2026)

Stabilizer codes form a distinguished class of exactly solvable quantum lattice models, originally arising in quantum error correction and now central to the study of topological phases. In this lecture, I will explain how abelian anyon theories arise as algebraic invariants of translation-invariant stabilizer codes. Focusing on 2+1-dimensional models, I will describe how bulk excitations and boundary degrees of freedom admit a purely algebraic formulation in terms of finitely generated modules over Laurent polynomial rings equipped with skew-Hermitian forms. In this framework, the abelian anyon content of a stabilizer code is captured by a finite metric group extracted from the lattice Hamiltonian. Viewed this way, abelian anyon theories appear not as emergent topological quantum field theories, but as computable algebraic invariants of quantum lattice models, providing a concrete bridge between stabilizer codes and TQFT. The talk will be largely self-contained.

Let D be the cluster symplectic double. It carries a symplectic form W. The cluster modular group M acts by its automorphisms. The space of its complex points has a real subspace U, the unitary double. It is Lagrangian for Re(W) and symplectic for Im(W). We quantize the symplectic space U. The quantization Hilbert space carries a unitary action of the group M, and a canonical collection of vectors, permitted by M. Here is a geometric application. Let S be a topological surface with an orientation reversing involution c. We show that the space of G-local systems on (S, c) carries a structure of a (twisted) unitary double. Therefore we can quantize it. Next, the Analytic Langlands Program of Etingof-Frenkel-Kazhdan studies the spectrum of quantized Hitchin hamiltonians in L_2(Bun(G,C)) for a real algebraic curve C. When C moves, the spectrum moves with it. So it is a discrete set with an action of the modular group M(C) of C. Assume that the surface (S,c) describes the topology of C. We conjecture that then the spectrum, as a discrete M(C)-set, is identified with the M(C)-set of canonical vectors in the unitary double quantization. This suggests that the former is a degeneration of the latter.