Monotonicities of Tanaka-Nakayama flows

TL;DR

The paper proves that under the Tanaka-Nakayama RG flows $M(kq+I,q)+\phi_{1,2k+1}\to M(kq-I,q)$ between Virasoro minimal models, conformal dimensions of preserved symmetry objects monotonically decrease and satisfy a half-integer sum constraint. It derives explicit formulas for the conformal-dimension differences $h^{M(kq+I,q)}_{r,1}-h^{M(kq-I,q)}_{r,1}$ and their half-integer sums, as well as a global-dimension ratio $D^2$ between the UV and IR theories using the minimal-model $S$-matrix; these yield monotonic behavior in specified parameter regimes for both integer and half-integer $k$. The results rely on preserved Verlinde lines and explicit $M(p,p')$ data, and they discuss parameter regions where global dimensions can increase, highlighting nuanced aspects of RG flows and their relation to RCFT domain walls. Overall, the work advances understanding of monotonicity in nontrivial RG flows and clarifies conditions under which global dimensions decrease, with implications for domain-wall constructions in rational CFTs.

Abstract

We prove conformal and global dimensions monotonically decrease under the infinitely many Tanaka-Nakayama renormalization group flows between Virasoro minimal models. The flows also satisfy the half-integer condition.