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Two More Fermionic Minimal Models

Justin Kulp

TL;DR

By fermionizing the non-anomalous $\mathbb{Z}_2$ symmetries of the $m=11$ and $m=12$ exceptional unitary minimal models, the paper identifies two additional fermionic minimal models, completing the fermionic unitary minimal-model classification. It develops the fermionization framework via a generalized Jordan-Wigner transformation and Arf-weighted sums, connecting bosonic ADE-type minimal models through gauging and orbifold procedures. The main finding is the explicit identification and construction of the fermionic partners for the $m=11$ and $m=12$ exceptional models, including detailed partition functions and a vanishing RR sector, thereby completing the set of unitary fermionic minimal models. The work provides concrete lattice realizations and clarifies the role of the Arf phase and self-duality under $\mathbb{Z}_2$ orbifold in organizing the fermionic counterparts within the ADE classification.

Abstract

In this short note, we comment on the existence of two more fermionic unitary minimal models not included in recent work by Hsieh, Nakayama, and Tachikawa. These theories are obtained by fermionizing the $\mathbb{Z}_2$ symmetry of the m=11 and m=12 exceptional unitary minimal models. Furthermore, these should be the only missing cases.

Two More Fermionic Minimal Models

TL;DR

By fermionizing the non-anomalous symmetries of the and exceptional unitary minimal models, the paper identifies two additional fermionic minimal models, completing the fermionic unitary minimal-model classification. It develops the fermionization framework via a generalized Jordan-Wigner transformation and Arf-weighted sums, connecting bosonic ADE-type minimal models through gauging and orbifold procedures. The main finding is the explicit identification and construction of the fermionic partners for the and exceptional models, including detailed partition functions and a vanishing RR sector, thereby completing the set of unitary fermionic minimal models. The work provides concrete lattice realizations and clarifies the role of the Arf phase and self-duality under orbifold in organizing the fermionic counterparts within the ADE classification.

Abstract

In this short note, we comment on the existence of two more fermionic unitary minimal models not included in recent work by Hsieh, Nakayama, and Tachikawa. These theories are obtained by fermionizing the symmetry of the m=11 and m=12 exceptional unitary minimal models. Furthermore, these should be the only missing cases.

Paper Structure

This paper contains 10 sections, 14 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. MINIMAL MODELS
  3. Review and Classification
  4. Symmetries and Z2 Orbifold
  5. FERMIONIC MINIMAL MODELS
  6. Fermionization
  7. Fermionic Minimal Models
  8. The Exceptional E6 Cases
  9. m=11 Exceptional
  10. m=12 Exceptional

Figures (1)

  • Figure 1: Gauging the $(-1)^F$ symmetry of a spin theory $T_f$ produces a bosonic theory $T_b$ with $\mathop{\mathrm{\mathbb{Z}}}\nolimits_2$ symmetry. A different bosonic theory $T^\prime_b$ is produced if one first stacks with the $\mathop{\mathrm{\textrm{Arf}}}\nolimits$ theory. These two theories are related by $\mathop{\mathrm{\mathbb{Z}}}\nolimits_2$ orbifold.