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Minimal model boundary flows and c=1 CFT

K. Graham, I. Runkel, G. M. T Watts

TL;DR

This work analyzes perturbations of unitary Virasoro minimal models by boundary fields, focusing on the $c\to 1$ limit to gain insight into IR flows for $c<1$. It proposes that any boundary condition $(r,s)$ becomes a superposition of fundamental $c=1$ boundary conditions $\widehat{\alpha}$, with a commutative projector algebra governing weight-0 fields and boundary-condition changing operators governing weight-1 fields. Using the Truncated Conformal Space Approach, the authors test perturbations by $\phi_{rr}$ and show that IR endpoints match the $c=1$ boundary data in many cases, with strong evidence for integrable boundary flows in several examples. The results provide a coherent picture linking $c=1$ boundary data to IR endpoints of flows in $c<1$, suggesting avenues for exact treatment and extensions to other boundary types and bulk theories.

Abstract

We consider perturbations of unitary minimal models by boundary fields. Initially we consider the models in the limit as c -> 1 and find that the relevant boundary fields all have simple interpretations in this limit. This interpretation allows us to conjecture the IR limits of flows in the unitary minimal models generated by the fields φ_{rr} of `low' weight. We check this conjecture using the truncated conformal space approach. In the process we find evidence for a new series of integrable boundary flows.

Minimal model boundary flows and c=1 CFT

TL;DR

This work analyzes perturbations of unitary Virasoro minimal models by boundary fields, focusing on the limit to gain insight into IR flows for . It proposes that any boundary condition becomes a superposition of fundamental boundary conditions , with a commutative projector algebra governing weight-0 fields and boundary-condition changing operators governing weight-1 fields. Using the Truncated Conformal Space Approach, the authors test perturbations by and show that IR endpoints match the boundary data in many cases, with strong evidence for integrable boundary flows in several examples. The results provide a coherent picture linking boundary data to IR endpoints of flows in , suggesting avenues for exact treatment and extensions to other boundary types and bulk theories.

Abstract

We consider perturbations of unitary minimal models by boundary fields. Initially we consider the models in the limit as c -> 1 and find that the relevant boundary fields all have simple interpretations in this limit. This interpretation allows us to conjecture the IR limits of flows in the unitary minimal models generated by the fields φ_{rr} of `low' weight. We check this conjecture using the truncated conformal space approach. In the process we find evidence for a new series of integrable boundary flows.

Paper Structure

This paper contains 22 sections, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The boundary conditions of the unitary minimal models
  3. Some generalities
  4. Minimal models
  5. Boundary conditions
  6. The $c\to 1$ limit of the boundary unitary minimal models
  7. The $c\to 1$ limit of the cylinder partition functions
  8. The 'missing' fields
  9. The scalar fields of weight $0$
  10. Example: the $c\to 1$ limit of the (2,2) boundary
  11. The perturbations by ${\phi}_{rr}$
  12. The minimal models with ${c\,{<}1}$
  13. The scaling behaviour of the ground state energy
  14. The nonunitary models
  15. Conclusions
  16. ...and 7 more sections

Figures (3)

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