Conformal Field Theory and Torsion Elements of the Bloch Group

TL;DR

The paper investigates a deep link between rational two-dimensional conformal field theories and torsion elements in algebraic K-theory, via the extended Bloch group and the Rogers dilogarithm. By analyzing integrable perturbations with elastic scattering, it derives modular sum representations characterized by a symmetric matrix $A$ and reduces the problem to algebraic equations $U=AV$ whose solutions live in $\hat{B}(\C)$, yielding finite dilogarithm values and torsion data that reflect conformal dimensions. The ADE-based constructions provide explicit cases where these algebraic numbers are roots of unity, supporting a conjectural K-theoretic classification of certain CFTs. The work lays out a program to formalize the connection and extend it beyond the explored ADE examples, inviting rigorous mathematical treatment of integrable QFTs through algebraic K-theory. In essence, it proposes a principled, computable bridge between 2D CFT data and torsion phenomena in $K_3(\C)$ via the extended Bloch group.

Abstract

We argue that rational conformally invariant quantum field theories in two dimensions are closely related to torsion elements of the algebraic K-theory group K_3(C). If such a theory has an integrable matrix perturbation with purely elastic scattering matrix, then the partition function has a canonical sum representation. Its asymptotic behaviour is given in terms of the solution of an algebraic equation which can be read off from the scattering matrix. The solutions yield torsion elements of an extension of the Bloch group which seems to be equal to K_3(C). These algebraic equations are solved for integrable models given by arbitrary pairs of equations are solved for integrable models given by arbitrary pairs of A-type Cartan matrices. The paper should be readable by mathematicians.