Unphysical diagonal modular invariants

TL;DR

This paper establishes a group-theoretic criterion for when diagonal modular invariants in certain chiral conformal field theories are physically realizable, revealing that many such invariants are unphysical due to group automorphism properties.

Contribution

It provides a necessary and sufficient condition, based on double class inverting automorphisms, for the physicality of diagonal modular invariants in G-orbifolds of holomorphic CFTs.

Findings

Diagonal modular invariant is physical iff G has a double class inverting automorphism.

Groups lacking such automorphisms lead to unphysical diagonal invariants.

The criterion links group automorphisms to the physicality of modular invariants.

Abstract

A modular invariant for a chiral conformal field theory is physical if there is a full conformal field theory with the given chiral halves realising the modular invariant. The easiest modular invariants are the charge conjugation and the diagonal modular invariants. While the charge conjugation modular invariant is always physical there are examples of chiral CFTs for which the diagonal modular invariant is not physical. Here we give (in group theoretical terms) a necessary and sufficient condition for diagonal modular invariants of $G$-orbifolds of holomorphic conformal field theories to be physical. Mathematically a physical modular invariant is an invariant of a Lagrangian algebra in the product of (chiral) modular categories. The chiral modular category of a $G$-orbifold of a holomorphic conformal field theory is the so-called (twisted) Drinfeld centre ${\cal Z}(G,\alpha)$ of the finite group $G$. We show that the diagonal modular invariant for ${\cal Z}(G)$ is physical if and only if the group $G$ has a {\em double class inverting} automorphism, that is an automorphism $\phi:G\to G$ with the property that for any commuting $x,y\in G$ there is $g\in G$ such that $\phi(x) = gx^{-1}g^{-1},\ \phi(y) = gy^{-1}g^{-1}.$ Groups without double class inverting automorphisms are abundant and provide examples of chiral conformal field theories for which the diagonal modular invariant is unphysical.