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arXiv:2111.01139 · Physical review. D/Physical review. D.

Noninvertible duality defects in 3 + 1 dimensions

Yichul Choi, Clay Córdova, Po-Shen Hsin, Ho Tat Lam, Shu-Heng Shao

Published Jun 22, 2022• 282 citations
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Abstract

A novel kind of generalized global symmetries is uncovered in a large class of familiar 3+1-dimensional gauge theories, including the free Maxwell theory and Yang-Mills gauge theories. These new symmetries, known as the non-invertible global symmetries, do not have inverses and thus go beyond the traditional paradigm of (anti-)unitary transformations. They are implemented by topological duality defects, generalizing the Kramers-Wannier duality defects in 1+1 dimensions. Remarkably, the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase and hence gives new constraints on renormalization group flows.

DOI
10.1103/physrevd.105.125016

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A novel kind of generalized global symmetries is uncovered in a large class of familiar 3+1-dimensional gauge theories, including the free Maxwell theory and Yang-Mills gauge theories. These new symmetries, known as the non-invertible global symmetries, do not have inverses and thus go beyond the traditional paradigm of (anti-)unitary transformations. They are implemented by topological duality defects, generalizing the Kramers-Wannier duality defects in 1+1 dimensions. Remarkably, the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase and hence gives new constraints on renormalization group flows.
Global symmetries can be generalised to transformations generated by topological operators, including cases in which the topological operator does not have an inverse. A family of such topological operators are intimately related to dualities via the procedure of half-space gauging. In this work we discuss the construction of non-invertible defects based on T-duality in two dimensions, generalising the well-known case of the free compact boson to any Non-Linear Sigma Model with Wess-Zumino term which is T-dualisable. This requires that the target space has an isometry with compact orbits that acts without fixed points. Our approach allows us to include target spaces without non-trivial 1-cycles, does not require the NLSM to be conformal, and when it is conformal it does not need to be rational; moreover, it highlights the microscopic origin of the topological terms that are responsible for the non-invertibility of the defect. An interesting class of examples are Wess-Zumino-Witten models, which are self-dual under a discrete gauging of a subgroup of the isometry symmetry and so host a topological defect line with Tambara-Yamagami fusion. Along the way, we discuss how the usual 0-form symmetries match across T-dual models in target spaces without 1-cycles, and how global obstructions can prevent locally conserved currents from giving rise to topological operators.
As the understanding of global symmetry has been expanded, quantum phases of matter can have a refined classification, protected or enriched by noninvertible symmetry, a type of generalized symmetry whose transformation may not have an inverse. In this paper, I use the duality method, proposed in Ref. 1 , to classify noninvertible symmetry protected topological (NISPT) phases. Under the duality transformation, different NISPT phases are mapped to different classes of spontaneous symmetry breaking phases with ordinary group symmetry. I demonstrate this method with the simplest Kramers–Wannier duality transformation, also known as gauging [Formula: see text] symmetry, to classify Rep([Formula: see text]) NISPT in [Formula: see text]. I also construct lattice models realizing these NISPT on tensor product Hilbert space and study the projective algebras at the interfaces between different classes of NISPTs.
We study quantum field theories with boundaries by utilizing noninvertible symmetries. We consider three kinds of boundary conditions of the four dimensional Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theory at the critical point as examples. The weights of the elements on the boundary are determined so that these boundary conditions are related by the Kramers-Wannier-Wegner (KWW) duality. In other words, it is required that the KWW duality defects ending on the boundary are topological. Moreover, we obtain the ratios of the hemisphere partition functions with these boundary conditions; this result constrains the boundary renormalization group flows.
Non-invertible topological defects in 4-dimensional Z2\mathbb {Z}_2Z2​ pure lattice gauge theory
Abstract We explore topological defects in the 4D pure Z2\mathbb {Z}_2Z2​ lattice gauge theory. This theory has 1-form Z2\mathbb {Z}_{2}Z2​ center symmetry as well as Kramers–Wannier–Wegner (KWW) duality. We construct the KWW duality topological defects in a similar way to those constructed by Aasen et al. [J. Phys. A 49, 354001 (2016)] for the 2D Ising model. These duality defects turn out to be non-invertible. We also construct 1-form Z2\mathbb {Z}_{2}Z2​ symmetry defects as well as the junctions between the KWW duality defects and 1-form Z2\mathbb {Z}_{2}Z2​ center symmetry defects. The crossing relations between these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.
Recent advancements in generalized symmetries have drawn significant attention to gapped phases of matter exhibiting novel symmetries, such as noninvertible symmetries. By leveraging the duality transformations, the classification and construction of gapped phases with noninvertible symmetry can be mapped to those involving conventional group symmetries. We demonstrate this approach by classifying symmetry-protected-topological phases with a broad class of noninvertible symmetries in arbitrary spacetime dimensions. Our results reveal new classifications that extend beyond those based on group symmetries. Additionally, we construct lattice models in (1+1)D and (2+1)D that realize these new phases and explore their anomalous interfaces.