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Journal of High Energy Physics

Topological cosets via anyon condensation and applications to gapped QCD2

Clay Córdova, Diego García-Sepúlveda

Published Aug 17, 2026• 0 citations
Publisher

Abstract

A bstract The coset construction of two-dimensional conformal field theory (2D CFT) defines a 2D CFT by taking the quotient of two previously known chiral algebras. In this work, we use the methods of non-abelian (non-invertible) anyon condensation to describe 2D topological cosets, defined by the special case where the quotient of chiral algebras is a conformal embedding. In this case, the coset has zero central charge, and the coset theory is thus purely topological. Using non-abelian anyon condensation we describe in general the spectrum of line and local operators as well as their fusion, operator product expansion, and the action of the lines on local operators. An important application of our results is to QCD 2 with massless fermions in any representation that leads to a gapped phase, where topological cosets (conjecturally) describe the infrared fixed point. We discuss several such examples in detail. For instance, we find that the Spin (8) 1 / SU (3) 3 and Spin (16) 1 / Spin (9) 2 topological cosets appearing at the infrared fixed point of appropriate QCD 2 theories are described by ℤ 2 × ℤ 2 triality and ℤ 2 × Rep(S 3 ) fusion categories respectively. Additionally, using this setup, we argue that chiral Spin (8) QCD 2 with massless chiral fermions in the vectorial and spinorial representations is not only gapped, but moreover trivially gapped, with a unique ground state.

DOI
10.1007/jhep08(2026)141

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A bstract The coset construction of two-dimensional conformal field theory (2D CFT) defines a 2D CFT by taking the quotient of two previously known chiral algebras. In this work, we use the methods of non-abelian (non-invertible) anyon condensation to describe 2D topological cosets, defined by the special case where the quotient of chiral algebras is a conformal embedding. In this case, the coset has zero central charge, and the coset theory is thus purely topological. Using non-abelian anyon condensation we describe in general the spectrum of line and local operators as well as their fusion, operator product expansion, and the action of the lines on local operators. An important application of our results is to QCD 2 with massless fermions in any representation that leads to a gapped phase, where topological cosets (conjecturally) describe the infrared fixed point. We discuss several such examples in detail. For instance, we find that the Spin (8) 1 / SU (3) 3 and Spin (16) 1 / Spin (9) 2 topological cosets appearing at the infrared fixed point of appropriate QCD 2 theories are described by ℤ 2 × ℤ 2 triality and ℤ 2 × Rep(S 3 ) fusion categories respectively. Additionally, using this setup, we argue that chiral Spin (8) QCD 2 with massless chiral fermions in the vectorial and spinorial representations is not only gapped, but moreover trivially gapped, with a unique ground state.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.
Boundary conformal field theory is brought to bear on the study of topological insulating phases of non-Abelian anyonic chains. These phases display protected anyonic end modes. We consider spin-1/2 su(2)k{}_{k}k​ chains at any level kkk, focusing on the most prominent examples: the case k=2k=2k=2 describes Ising anyons (equivalent to Majorana fermions) and k=3k=3k=3 corresponds to Fibonacci anyons. The method we develop is quite general and rests on a deep connection between boundary conformal field theory and topological symmetry. This method tightly constrains the nature of the topological insulating phases of these chains for general kkk. Emergent anyons which arise at domain walls are shown to have the same braiding properties as the physical quasiparticles. This suggests a ``solid-state'' topological quantum computation scheme in which emergent anyons are braided by tuning the couplings of non-Abelian quasiparticles in a fixed network.
In this thesis we use a network of chiral junctions to realize non-Abelian topological phases in two dimensions. Our construction is based on boundary fixed points of low-energy field theories describing Y junctions of critical spin-chain models. To set the stage, we first study a single junction of gapless spin-1 chains belonging to the SU(2)2 universality class, whose spectrum includes fractional excitations such as Majorana fermions. We find that chiral fixed points appear as special points on a transition line that separates two regimes described by open boundary conditions. Remarkably, along this transition the junction behaves as a tunable spin circulator as the spin conductance varies continuously with the coupling constant of a marginal boundary operator. We then construct a family of topological chiral spin liquids departing from a honeycomb network made of critical spin-S chains. The chiral spin liquid phase harbors SU(2)k=2S anyons, which stem from the underlying WessZumino-Witten models that describe the constituent spin chains of the network. The network exhibits quantized spin and thermal Hall conductances. We illustrate our construction by inspecting the topological properties of the SU(2)2 model. We find that this model has emergent Ising anyons, with spinons acting as vortex excitations that bind Majorana zero modes. We also show that the ground state of this network is threefold degenerate on the torus, asserting its non-Abelian character. Our work provides a controllable analytical framework to study non-Abelian topological phases, sheding new light on the stability of such phases in artificial quantum materials.
Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2 + 1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.
Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2+1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.
Chiral edge states of [Formula: see text]-dimensional Abelian and non-Abelian topological phases can be represented by chiral conformal field theories with integer and noninteger values of central charge, respectively. In this work we describe certain edge states in terms of constrained fermionic fields that realize chiral coset CFT structures. This construction arises naturally in the so-called quantum wires approach for topological phases and allows for representing fractionalized edge states directly in terms of fermionic degrees of freedom. At the same time, the constrained fermions description introduces some subtleties concerning gauge anomalies since it involves the coupling of chiral fermions to gauge fields. We describe in this paper how to handle these issues.