Tensor Product CFTs and One-Character Extensions

TL;DR

This paper investigates one-character extensions of tensor product CFTs, providing explicit character formulas, classifying possible extensions, and conjecturing infinite series of such theories, with implications for the classification of c=24 CFTs.

Contribution

It introduces a framework for constructing and analyzing one-character extensions of tensor product CFTs, including explicit polynomial character expressions and classification results.

Findings

Explicit S-invariant polynomial bases for studied CFTs

Closed-form character expressions for high central charges

Identification of possible and impossible one-character extension CFTs

Abstract

We study one-character CFTs obtained as one-character extensions of the tensor products of a single CFT $\mathcal{C}$. The motivation comes from the fact that $28$ of the $71$ CFTs in the Schelleken's list of $c = 24$ CFTs are such CFTs. We study for $\mathcal{C}$ : (i) any two-character WZW CFT with vanishing Wronskian index, (ii) the Ising CFT, (iii) the infinite class of $D_{r,1}$ CFTs and the $A_{4,1}$ CFT. The characters being $S$-invariant homogenous polynomials of the characters of $\mathcal{C}$, when organised in terms of a $S$-invariant basis, take compact forms allowing for closed form answers for high central charges. We find a $S$-invariant basis for each of the CFTs studied. As an example, one can find an explicit expression for the character of the monster CFT as a degree-$48$ polynomial of the characters of the Ising CFT. In some CFTs, some of the $S$-invariant polynomials of characters compute, after using the $q$-series of the characters, to a constant value. Hence, the characters of one-character extensions are more properly elements of the quotient ring of polynomials (of characters) with the ideal needed for the quotient, generated by $S$-invariant polynomials that compute to a constant. In some cases, we are able to rule out the existence of one-character extension CFTs. In other cases, we predict their existence. We are able to conjecture a discrete set of six and four infinite series of one-character extension CFTs.