Noninvertible Peccei-Quinn Symmetry and the Massless Quark Solution to the Strong <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:mrow></mml:math> Problem

Physical Review X · 2025-07-10 · 4 citations

We consider theories of gauged quark flavor and identify noninvertible Peccei-Quinn symmetries arising from fractional instantons when the resulting gauge group has nontrivial global structure. Such symmetries exist solely because the standard model has the same number of generations as colors, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>N</a:mi><a:mi>g</a:mi></a:msub><a:mo>=</a:mo><a:msub><a:mi>N</a:mi><a:mi>c</a:mi></a:msub></a:math>, which leads to a massless down-type quark solution to the strong <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>C</c:mi><c:mi>P</c:mi></c:math> problem in an ultraviolet <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>S</e:mi><e:mi>U</e:mi><e:mo stretchy="false">(</e:mo><e:mn>9</e:mn><e:mo stretchy="false">)</e:mo></e:math> theory of quark color-flavor unification. We show how the Cabibbo-Kobayashi-Maskawa flavor structure and weak <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>C</i:mi><i:mi>P</i:mi></i:math> violation can be generated without upsetting our solution.

Representation theory of solitons

Journal of High Energy Physics · 2025-06-03 · 11 citations

A bstract Solitons in two-dimensional quantum field theory exhibit patterns of degeneracies and associated selection rules on scattering amplitudes. We develop a representation theory that captures these intriguing features of solitons. This representation theory is based on an algebra we refer to as the strip algebra , $$ {\textbf{Str}}_{\mathcal{C}}\left(\mathcal{M}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Str</mml:mi> <mml:mi>C</mml:mi> </mml:msub> <mml:mfenced> <mml:mi>M</mml:mi> </mml:mfenced> </mml:math> , which is defined in terms of the non-invertible symmetry, $$ \mathcal{C} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> , a fusion category, and its action on boundary conditions encoded by a module category, $$ \mathcal{M} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> . The strip algebra is a C ∗ -weak Hopf algebra, a fact which can be elegantly deduced by quantizing the three-dimensional Drinfeld center TQFT, $$ \mathcal{Z}\left(\mathcal{C}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> <mml:mfenced> <mml:mi>C</mml:mi> </mml:mfenced> </mml:math> , on a spatial manifold with corners. These structures imply that the representation category of the strip algebra is also a unitary fusion category which we identify with a dual category $$ {\mathcal{C}}_{\mathcal{M}}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>C</mml:mi> <mml:mi>M</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> . We present a straightforward method for analyzing these representations in terms of quiver diagrams where nodes are vacua and arrows are solitons and provide examples demonstrating how the representation theory reproduces known degeneracies and selection rules of soliton scattering. Our analysis provides the general framework for analyzing non-invertible symmetry on manifolds with boundary and applies both to the case of boundaries at infinity, relevant to particle physics, and boundaries at finite distance, relevant in conformal field theory or condensed matter systems.

Higgsing Transitions from Topological Field Theory &amp; Non-Invertible Symmetry in Chern-Simons Matter Theories

ArXiv.org · 2025-04-04 · 1 citations

Non-invertible one-form symmetries are naturally realized in (2+1)d topological quantum field theories. In this work, we consider the potential realization of such symmetries in (2+1)d conformal field theories, investigating whether gapless systems can exhibit similar symmetry structures. To that end, we discuss transitions between topological field theories in (2+1)d which are driven by the Higgs mechanism in Chern-Simons matter theories. Such transitions can be modeled mesoscopically by filling spacetime with a lattice-shaped domain wall network separating the two topological phases. Along the domain walls are coset conformal field theories describing gapless chiral modes trapped by a locally vanishing scalar mass. In this presentation, the one-form symmetries of the transition point can be deduced by using anyon condensation to track lines through the domain wall network. Using this framework, we discuss a variety of concrete examples of non-invertible one-form symmetry in fixed-point theories. For instance, $SU(k)_{2}$ Chern-Simons theory coupled to a scalar in the symmetric tensor representation produces a transition from an $SU(k)_{2}$ phase to an $SO(k)_{4}$ phase and has non-invertible one-form symmetry $PSU(2)_{-k}$ at the fixed point. We also discuss theories with $Spin(2N)$ and $E_{7}$ gauge groups manifesting other patterns of non-invertible one-form symmetry. In many of our examples, the non-invertible one-form symmetry is not a modular invariant TQFT on its own and thus is an intrinsic part of the fixed-point dynamics.

Non-invertible symmetries in finite-group gauge theory

SciPost Physics · 2025-01-16 · 2 citations

We investigate the invertible and non-invertible symmetries of topological finite-group gauge theories in general spacetime dimensions, where the gauge group can be abelian or non-abelian. We focus in particular on the 0-form symmetry. The gapped domain walls that generate these symmetries are specified by boundary conditions for the gauge fields on either side of the wall. We investigate the fusion rules of these symmetries and their action on other topological defects including the Wilson lines, magnetic fluxes, and gapped boundaries. We illustrate these constructions with various novel examples, including non-invertible electric-magnetic duality symmetry in 3+1d \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msub> </mml:math> gauge theory, and non-invertible analogs of electric-magnetic duality symmetry in non-abelian finite-group gauge theories. In particular, we discover topological domain walls that obey Fibonacci fusion rules in 2+1d gauge theory with dihedral gauge group of order 8. We also generalize the Cheshire string defect to analogous defects of general codimensions and gauge groups and show that they form a closed fusion algebra.

Generalized Level-Rank Duality, Holomorphic Conformal Field Theory, and Non-Invertible Anyon Condensation

arXiv (Cornell University) · 2025-12-30

We study the interplay between holomorphic conformal field theory and dualities of 3D topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the $c=24$ holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from $c=24$ to an infinite series. This includes the fact that Spin$(n^{2})_{2}$ is dual to a twisted dihedral group gauge theory. Finally, if $-1$ is a quadratic residue modulo $k$, we deduce the existence of a sequence of holomorphic CFTs at central charge $c=2(k-1)$ with fusion category symmetry given by $\mathrm{Spin}(k)_{2}$ or equivalently, the $\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category.

Generalized Level-Rank Duality, Holomorphic Conformal Field Theory, and Non-Invertible Anyon Condensation

ArXiv.org · 2025-12-30

We study the interplay between holomorphic conformal field theory and dualities of 3D topological quantum field theories generalizing the paradigm of level-rank duality. A holomorphic conformal field theory with a Kac-Moody subalgebra implies a topological interface between Chern-Simons gauge theories. Upon condensing a suitable set of anyons, such an interface yields a duality between topological field theories. We illustrate this idea using the $c=24$ holomorphic theories classified by Schellekens, which leads to a list of novel sporadic dualities. Some of these dualities necessarily involve condensation of anyons with non-abelian statistics, i.e. gauging non-invertible one-form global symmetries. Several of the examples we discover generalize from $c=24$ to an infinite series. This includes the fact that Spin$(n^{2})_{2}$ is dual to a twisted dihedral group gauge theory. Finally, if $-1$ is a quadratic residue modulo $k$, we deduce the existence of a sequence of holomorphic CFTs at central charge $c=2(k-1)$ with fusion category symmetry given by $\mathrm{Spin}(k)_{2}$ or equivalently, the $\mathbb{Z}_{2}$-equivariantization of a Tambara-Yamagami fusion category.

Non-invertible symmetry in Calabi-Yau conformal field theories

Journal of High Energy Physics · 2025-01-08 · 12 citations

A bstract We construct examples of non-invertible global symmetries in two-dimensional superconformal field theories described by sigma models into Calabi-Yau target spaces. Our construction provides some of the first examples of non-invertible symmetry in irrational conformal field theories. Our approach begins at a Gepner point in the conformal manifold where the sigma model specializes to a rational conformal field theory and we can identify all supersymmetric topological Verlinde lines. By deforming away from this special locus using exactly marginal operators, we then identify submanifolds in moduli space where some non-invertible symmetry persists. For instance, along ten-dimensional loci in the complex structure moduli space of quintic Calabi-Yau threefolds there is a symmetry characterized by a Fibonacci fusion category. The symmetries we identify provide new constraints on spectra and correlation functions. As an application we show how they constrain conformal perturbation theory, consistent with recent results about scaling dimensions in the K3 sigma model near its Gepner point.

Gapped theories have torsion anomalies

arXiv (Cornell University) · 2024-08-27

We prove special cases of a general conjecture: If an invertible field theory admits a projectively topological boundary theory, then it has finite order in the abelian group of invertible field theories. One can substitute `gapped' for `projectively topological'. Our proofs use evaluations of a field theory in parametrized families.

Anomalies of non-invertible symmetries in (3+1)d

SciPost Physics · 2024-11-12 · 51 citations

Anomalies of global symmetries are important tools for understanding the dynamics of quantum systems. We investigate anomalies of non-invertible symmetries in 3+1d using 4+1d bulk topological quantum field theories given by Abelian two-form gauge theories, with a 0-form permutation symmetry. Gauging the 0-form symmetry gives the 4+1d “inflow” symmetry topological field theory for the non-invertible symmetry. We find a two levels of anomalies: (1) the bulk may fail to have an appropriate set of loop excitations which can condense to trivialize the boundary dynamics, and (2) the “Frobenius-Schur indicator” of the non-invertible symmetry (generalizing the Frobenius-Schur indicator of 1+1d fusion categories) may be incompatible with trivial boundary dynamics. As a consequence we derive conditions for non-invertible symmetries in 3+1d to be compatible with symmetric gapped phases, and invertible gapped phases. Along the way, we see that the defects characterizing \mathbb{Z}_{4} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="double-struck"> <mml:mi>ℤ</mml:mi> </mml:mstyle> <mml:mn>4</mml:mn> </mml:msub> </mml:math> ordinary symmetry host worldvolume theories with time-reversal symmetry \mathsf{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> </mml:math> obeying the algebra \mathsf{T}^{2}=C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> or \mathsf{T}^{2}=(-1)^{F}C, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msup> <mml:mstyle mathvariant="sans-serif"> <mml:mi>𝖳</mml:mi> </mml:mstyle> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>F</mml:mi> </mml:msup> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> with C <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>C</mml:mi> </mml:math> a unitary charge conjugation symmetry. We classify the anomalies of this symmetry algebra in 2+1d and further use these ideas to construct 2+1d topological orders with non-invertible time-reversal symmetry that permutes anyons. As a concrete realization of our general discussion, we construct new lattice Hamiltonian models in 3+1d with non-invertible symmetry, and constrain their dynamics.

Deep learning lattice gauge theories

Physical review. B./Physical review. B · 2024-10-15 · 9 citations

Here, the authors use gauge-invariant neural network quantum states (NNQS) to study \ensuremath{\mathbb{Z}}${}_{N}$ lattice gauge theories. They compute ground states and map out phase transitions, identifying a continuous transition in \ensuremath{\mathbb{Z}}${}_{2}$ with Ising critical exponents and a weakly first-order transition in \ensuremath{\mathbb{Z}}${}_{3}$. Their work demonstrates how deep learning can overcome challenges in traditional lattice simulations, providing a novel approach for exploring complex quantum phases and topological order in condensed matter systems.