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Perturbed Defects and T-Systems in Conformal Field Theory

Ingo Runkel

Abstract

Defect lines in conformal field theory can be perturbed by chiral defect fields. If the unperturbed defects satisfy su(2)-type fusion rules, the operators associated to the perturbed defects are shown to obey functional relations known from the study of integrable models as T-systems. The procedure is illustrated for Virasoro minimal models and for Liouville theory.

Perturbed Defects and T-Systems in Conformal Field Theory

Abstract

Defect lines in conformal field theory can be perturbed by chiral defect fields. If the unperturbed defects satisfy su(2)-type fusion rules, the operators associated to the perturbed defects are shown to obey functional relations known from the study of integrable models as T-systems. The procedure is illustrated for Virasoro minimal models and for Liouville theory.

Paper Structure

This paper contains 18 sections, 80 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Functional equations for defects operators
  3. Unperturbed topological defects
  4. Chirally perturbed defects
  5. Perturbations by defect changing fields
  6. Fusion of chirally perturbed defects
  7. Commuting defect operators
  8. Defect fusion with one channel
  9. Defect fusion with two channels
  10. Defects with su(2)-type fusion
  11. Examples
  12. Virasoro minimal models
  13. Liouville theory
  14. Conclusions
  15. Appendix
  16. ...and 3 more sections

Figures (1)

  • Figure 1: In these pictures the lines $i\mathbb{R}$ and $i\mathbb{R} + L$ are identified. (i) The correlator of the two topological defects $a$, $b$ inserted on the cylinder at $ix$ and $iy$ does not depend on $x$ and $y$. (ii) In the limit $x \rightarrow y$ one obtains the fused defect $a \star b$.