Bulk flows in Virasoro minimal models with boundaries

TL;DR

This work analyzes how a genuinely relevant bulk perturbation $\phi_{(1,3)}$ in Virasoro minimal models induces coupled bulk-boundary RG flows. By combining perturbative RG for $\lambda$ and $\mu$, perturbations of the boundary entropy $g$, and topological defect techniques, the authors identify three perturbative fixed points and the non-perturbative endpoint of generic flows. They derive a precise end-point mapping: $(a_1,a_2)_m \to (1,a_1)_{m-1}$ if $a_2=1$, $(a_2-1,a_1)_{m-1}$ if $1<a_2<m$, and $(1, m-a_1)_{m-1}$ if $a_2=m$, and support this with defect-consistency arguments and numerical checks. The results organize boundary flows into coherent chains and connect perturbative and non-perturbative pictures, aligning with known TCSA/TBA findings in specific cases and extending understanding across the A-series.

Abstract

The behaviour of boundary conditions under relevant bulk perturbations is studied for the Virasoro minimal models. In particular, we consider the bulk deformation by the least relevant bulk field which interpolates between the mth and (m-1)st unitary minimal model. In the presence of a boundary this bulk flow induces an RG flow on the boundary, which ensures that the resulting boundary condition is conformal in the (m-1)st model. By combining perturbative RG techniques with insights from defects and results about non-perturbative boundary flows, we determine the endpoint of the flow, i.e. the boundary condition to which an arbitrary boundary condition of the mth theory flows to.

Figures (5)

• Figure 1: The combined flow diagram for $(a_{1},a_{2})=(2,3)$ and $m=100$ (for which $\alpha=-4$; see section 2.2 for details). We have magnified the vectors $(\dot{\mu},\dot{\lambda})$ by a factor $2.5$. The horizontal arrow indicates the pure boundary flow to the perturbative fixed-point (I) in the $m^{\rm th}$ minimal model, the vertical arrow describes the flow of the boundary condition (I) to the boundary condition (II) in the $(m-1)^{\rm st}$ minimal model. The three other flows that are depicted are generic bulk-boundary RG-flows that at the end tend towards the fixed point (IV) where $\mu =+\infty$ in the $(m-1)^{\rm st}$ minimal model ($\lambda =\lambda_{*}$).
• Figure 2: A segment of the chain of flows for small values of $a_{1},a_{2}$. In the upper line we have boundary conditions of the $m^{\rm th}$ and in the lower line of the $(m-1)^{\rm st}$ minimal model.
• Figure 3: The right end of the chain of flows. Starting from the boundary condition $(a_{1},2)_{m}$, in one direction the flow decomposes into only two separate flows ($\to (a_{1},1)_{m}\to (1,a_{1})_{m-1}$) instead of three separate flows in the generic case.
• Figure 4: The middle of the chain of flows for $m$ odd. The fixed point starting from the exact middle $(a_{1},\frac{m+1}{2})_{m}$ is reached for $\lambda =\lambda_{*}$, $\mu =0$ (flow 1 in the figure). When we go slightly away from the middle (flow 2), the flow can still be treated perturbatively.
• Figure 5: The flow diagram for \ref{['RGeqmiddle']} for $a_{2}=1$ and $m=101$. The vectors $(\dot{\mu},\dot{\lambda})$ have been magnified by a factor $2.5$. A generic flow (described by the solution \ref{['RGsolutionmiddle']} of \ref{['RGeqmiddle']}) reaches the perturbative fixed-point.