Quasi-Characters in  \(\widehat{su(2)}\) Current Algebra at Fractional Levels

The unitary highest weight representations of integral levels of \(\widehat{su(2)}\) current algebra conformal field theories (CFTs) satisfy all properties of a rational CFT (RCFT), but the story is not straightforward at admissible fractional levels. The admissible levels are labeled by two natural coprime numbers \((p \ge 2, u)\) such that the level is \(m=p/u-2\) . We show that almost every fractional admissible level \(\widehat{su(2)}_m\) current algebra exhibits one or more quasi-character(s). We find three special classes without quasi-characters: the sequence \((p = 2, u = 2N +1)\) , where the admissibility condition is saturated, at positive half-odd integer levels labeled by \((p = 2N +3, u = 2)\) , \(N\in \mathbb {N}\) , and an isolated point \((p = 3, u = 4)\) . We also relate the characters of these three classes with characters of RCFTs corresponding to integral levels of \(\widehat{su(2)}\) and \(\widehat{so(5)}\) . The sequence with \(u = 2\) is quite intriguing and seems to defy most of the usual CFT descriptions (except possibly the log CFT). We also report two criteria to eliminate character vectors of the fractional admissible level \(\widehat{su(2)}_m\) current algebra at \((p \in prime, u \in prime)\) and \( (p,Np-1)\) where \(N \in \mathbb {N}\) , admitting quasi-characters, as character vectors of an RCFT. Based on [2].