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Fermionization and boundary states in 1+1 dimensions

Yoshiki Fukusumi, Yuji Tachikawa, Yunqin Zheng

TL;DR

The paper shows that in 1+1 dimensions every fermionic theory can be obtained from a bosonic theory with a non-anomalous Z2 symmetry via a generalized Jordan-Wigner construction. It develops a precise mapping of boundary states under Z2 gauging and fermionization (A↔D, A↔F, D↔F'), with explicit boundary-state formulas and normalization conditions, and interprets these mappings as fusion with the fermionization interface. Through detailed RCFT examples (Ising, SU(2)_k, Spin(N)_1) and Maldacena-Ludwig boundary states, it demonstrates how Majorana zero modes and spin structures organize boundary data across NS/R sectors. The work further constructs interfaces among A, D, F, and F', computes their actions on boundary states, and reveals how anomalous Z2 symmetry emerges when A≃D, linking duality defects to the fermionic boundary structure and quantum dimensions of interfaces.

Abstract

In the last few years it was realized that every fermionic theory in 1+1 dimensions is a generalized Jordan-Wigner transform of a bosonic theory with a non-anomalous $\mathbb{Z}_2$ symmetry. In this note we determine how the boundary states are mapped under this correspondence. We also interpret this mapping as the fusion of the original boundary with the fermionization interface.

Fermionization and boundary states in 1+1 dimensions

TL;DR

The paper shows that in 1+1 dimensions every fermionic theory can be obtained from a bosonic theory with a non-anomalous Z2 symmetry via a generalized Jordan-Wigner construction. It develops a precise mapping of boundary states under Z2 gauging and fermionization (A↔D, A↔F, D↔F'), with explicit boundary-state formulas and normalization conditions, and interprets these mappings as fusion with the fermionization interface. Through detailed RCFT examples (Ising, SU(2)_k, Spin(N)_1) and Maldacena-Ludwig boundary states, it demonstrates how Majorana zero modes and spin structures organize boundary data across NS/R sectors. The work further constructs interfaces among A, D, F, and F', computes their actions on boundary states, and reveals how anomalous Z2 symmetry emerges when A≃D, linking duality defects to the fermionic boundary structure and quantum dimensions of interfaces.

Abstract

In the last few years it was realized that every fermionic theory in 1+1 dimensions is a generalized Jordan-Wigner transform of a bosonic theory with a non-anomalous symmetry. In this note we determine how the boundary states are mapped under this correspondence. We also interpret this mapping as the fusion of the original boundary with the fermionization interface.

Paper Structure

This paper contains 36 sections, 136 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction and summary
  2. Mapping of the boundary states
  3. Gauging $\mathbb{Z}_2$: $\mathsf{A}\leftrightarrow \mathsf{D}$
  4. Setup:
  5. $\mathbb{Z}_2$ charge of $\ket{i}^\mathsf{A}$:
  6. $\mathbb{Z}_2$ charge of $\ket{i}_\mathrm{tw}^\mathsf{A}$:
  7. Rough properties of the boundary states:
  8. Fixing the normalization constants:
  9. Fermionizing $\mathbb{Z}_2$: $\mathsf{A}\to \mathsf{F}, \mathsf{F}'$
  10. Setup:
  11. Rough properties of the boundary conditions:
  12. Fixing the normalization constants:
  13. Comments
  14. On two common normalizations:
  15. Relation with the lattice Jordan-Wigner transformations:
  16. ...and 21 more sections

Figures (3)

  • Figure 1: $\mathbb{Z}_2$ defect, $\mathbb{Z}_2$ charge and the spin of the twisted Hilbert space. The $\mathbb{Z}_2$ charge of a state in the $\mathbb{Z}_2$ is given by the leftmost figure, which can be deformed to the middle figure. Therefore the charge is related to the spin of the state. When we put a $\mathbb{Z}_2$-invariant boundary condition on the top and bottom slices, we can always unwind the defect, and the spin is always integer valued.
  • Figure 2: Interfaces from the theory $\mathsf{A}$ to $\mathsf{D}$.
  • Figure 3: A closed loop of an interface without any operators in it can be evaluated to a number. This number can be called the quantum dimension of the interface, generalizing the same quantity for topological line operators of a single theory.