Spin Structures and Exact Dualities in Low Dimensions

TL;DR

The paper develops a systematic framework to derive exact lattice dualities in 1D and 2D, extending known KW and JW dualities and introducing twists to generate a broad web that includes parafermionic generalizations (FK) and $\,\mathbb{Z}_K$ theories. A central result is treating spin structures as $\oldsymbol{\mathbb{Z}_2}$ gauge fields generated by fermion parity, which unifies spin and para-spin data across dualities and motivates paraspin structures for parafermions. It analyzes the role of one-form and zero-form symmetries, flux attachment, and anomalies in higher dimensions, and connects these dualities to dimer/Kasteleyn formalisms and Arf invariants in 2D. The work highlights new fractional-statistics dualities, proposes a lattice path toward discretized Chern-Simons-type theories, and outlines open problems in extending these ideas to nonabelian and higher-dimensional settings.

Abstract

This paper derives a large web of exact lattice dualities in one and two spatial dimensions. Some of the dualities are well-known, while others, such as two-dimensional boson-parafermion dualities, are new. The procedure is systematic, independent of specific Hamiltonians, and generalizes to higher dimensions. One important result is a demonstration that spin structures in arbitrary lattice fermion theories can always be simply defined as topological gauge fields whose gauge group is the fermion number parity. This definition agrees with other expected properties of spin structures, and it motivates the introduction of "paraspin structures" that serve the same role in parafermion theories.

Figures (2)

• Figure 1: A periodic lattice with $N = 9$ sites (black) and $N$ edges between them. In this section, each site $v$ of such a lattice hosts a two-dimensional Hilbert space of an Ising spin, and the algebra of operators at that site is generated by the Pauli matrices $X_v$ and $Z_v$. The dual (red) lattice has a site corresponding to each of the edges of the original lattice, and an edge corresponding to each vertex of the dual lattice. The edge-vertex duality is the $d = 1$ avatar of the more general Poincaré duality.
• Figure 2: A rectangular lattice $\mathbb M$ (black) and its dual lattice $\mathbb M^\vee$ (red). The figure depicts examples of an oriented link $\ell$ and its endpoints $\ell_{1/2}$, a site $v \in \mathbb M$ and its dual face, and a face $f$ and its dual site. In this section, $\mathbb Z_2$ degrees of freedom can be associated either to vertices, edges, or faces of both $\mathbb M$ and $\mathbb M^\vee$.