Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundary States and Anomalous Symmetries of Fermionic Minimal Models

Philip Boyle Smith

TL;DR

This work classifies and constructs the full set of conformal boundary states for fermionic minimal models, revealing that boundary states fall into two SPT-based classes and that inter-class pairs acquire a Majorana-related $\sqrt{2}$ factor in their interval partition functions. It systematically analyzes potential $\mathbb{Z}_2$ global symmetries, uncovering an anomalous symmetry that exchanges the two boundary-state classes in four models (extending Majorana-like physics to interacting theories) and showing this correlates with a vanishing Ramond-Ramond sector. A key theoretical ingredient is a conjecture about $\mathfrak{su}(2)$ affine parities, generalizing a Fermat-curve-related result, used to constrain possible parity structures. The results highlight deep connections between boundary-state categorization, global anomalies, and SPT physics in fermionic CFTs, and they establish a uniform framework across both infinite-series and exceptional fermionic minimal models.

Abstract

The fermionic minimal models are a recently-introduced family of two-dimensional spin conformal field theories. We determine all of their conformal boundary states and potentially anomalous $\mathbb{Z}_2$ global symmetries. The latter task hinges upon on a conjecture about $\mathfrak{su}(2)$ affine parities generalising an earlier result known to have an interpretation in terms of Fermat curves. Our results indicate a close connection between several properties of the models, including the matching of the sizes of the SPT classes of boundary states, the existence of anomalous $\mathbb{Z}_2$ symmetries, and the vanishing of the Ramond-Ramond sector, for which we provide an explanation.

Boundary States and Anomalous Symmetries of Fermionic Minimal Models

TL;DR

This work classifies and constructs the full set of conformal boundary states for fermionic minimal models, revealing that boundary states fall into two SPT-based classes and that inter-class pairs acquire a Majorana-related factor in their interval partition functions. It systematically analyzes potential global symmetries, uncovering an anomalous symmetry that exchanges the two boundary-state classes in four models (extending Majorana-like physics to interacting theories) and showing this correlates with a vanishing Ramond-Ramond sector. A key theoretical ingredient is a conjecture about affine parities, generalizing a Fermat-curve-related result, used to constrain possible parity structures. The results highlight deep connections between boundary-state categorization, global anomalies, and SPT physics in fermionic CFTs, and they establish a uniform framework across both infinite-series and exceptional fermionic minimal models.

Abstract

The fermionic minimal models are a recently-introduced family of two-dimensional spin conformal field theories. We determine all of their conformal boundary states and potentially anomalous global symmetries. The latter task hinges upon on a conjecture about affine parities generalising an earlier result known to have an interpretation in terms of Fermat curves. Our results indicate a close connection between several properties of the models, including the matching of the sizes of the SPT classes of boundary states, the existence of anomalous symmetries, and the vanishing of the Ramond-Ramond sector, for which we provide an explanation.

Paper Structure

This paper contains 19 sections, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Simple Illustration
  3. Summary of Results
  4. Infinite series
  5. Exceptionals
  6. The Plan of the Paper
  7. Review of Fermionic Minimal Models
  8. Boundary States
  9. Infinite Series
  10. Exceptionals
  11. Proof
  12. Proof
  13. Anomalous Symmetries
  14. The Case of No Majorana Mode
  15. The Case of a Majorana Mode
  16. ...and 4 more sections

Figures (3)

  • Figure 1: The classification of fermionic minimal models. Both the infinite series and exceptionals are labelled by a choice of integer $m$.
  • Figure 2: Boundary states for the infinite series of models are labelled by points in the bottom-left quadrant of the Kac table. The two classes are shown in blue and red. Note that for $m = 3, 4$ but no other values, the classes have equal sizes.
  • Figure 3: The models with an extra $\mathbb{Z}_2$ global symmetry, shown in red. Note that these are the same models which had matching class sizes earlier, as well as the models with vanishing RR sector.