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 $\textit{strip algebra}$, $$ {\textbf{Str}}_{\mathcal{C}}\left(\mathcal{M}\right) $$ , which is defined in terms of the non-invertible symmetry, $$ \mathcal{C} $$ , a fusion category, and its action on boundary conditions encoded by a module category, $$ \mathcal{M} $$ . The strip algebra is a $\textit{C}$ -weak Hopf algebra, a fact which can be elegantly deduced by quantizing the three-dimensional Drinfeld center TQFT, $$ \mathcal{Z}\left(\mathcal{C}\right) $$ , 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 $$ {\mathcal{C}}_{\mathcal{M}}^{\ast } $$ . 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.

United States

Clay Córdova

Nicholas Holfester

Japan

Kantaro Ohmori

Representation theory of solitons

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.

Italy

Giovanni Rizi

Non-invertible symmetry in Calabi-Yau conformal field theories

We study non-invertible topological symmetry operators in massive quantum field theories in (1+1) dimensions. In phases where this symmetry is spontaneously broken we show that the particle spectrum often has degeneracies dictated by the non-invertible symmetry and we deduce a procedure to determine the allowed multiplets. These degeneracies are robust predictions and do not require integrability or other special features of renormalization group flows. We exhibit these conclusions in examples where the spectrum is known, recovering soliton and particle degeneracies. For instance, the Tricritical Ising model deformed by the subleading ℤ2 odd operator flows to a gapped phase with two degenerate vacua. This flow enjoys a Fibonacci fusion category symmetry which implies a threefold degeneracy of its particle states, relating the mass of solitons interpolating between vacua and particles supported in a single vacuum.

Diego García-Sepúlveda

Particle-soliton degeneracies from spontaneously broken non-invertible symmetry

Non-invertible global symmetry often predicts degeneracy in axion potentials and carries important information about the global form of the gauge group. When these symmetries are spontaneously broken they can lead to the formation of stable axion domain wall networks which support topological degrees of freedom on their worldvolume. Such non-invertible symmetries can be broken by embedding into appropriate larger UV gauge groups where small instanton contributions lift the vacuum degeneracy, and provide a possible solution to the domain wall problem. We explain these ideas in simple illustrative examples and then apply them to the Standard Model, whose gauge algebra and matter content are consistent with several possible global structures. Each possible global structure leads to different selection rules on the axion couplings, and various UV completions of the Standard Model lead to more specific relations. As a proof of principle, we also present an example of a UV embedding of the Standard Model which can solve the axion domain wall problem. The formation and annihilation of the long-lived axion domain walls can lead to observables, such as gravitational wave signals. Observing such signals, in combination with the axion coupling measurements, can provide valuable insight into the global structure of the Standard Model, as well as its UV completion.

South Korea

Sungwoo Hong

Lian-Tao Wang

Axion domain walls, small instantons, and non-invertible symmetry breaking

We describe the interplay between electric-magnetic duality and higher symmetry in Maxwell theory. When the fine structure constant is rational, the theory admits noninvertible symmetries which can be realized as composites of electric-magnetic duality and gauging a discrete subgroup of the one-form global symmetry. These noninvertible symmetries are approximate quantum invariances of the natural world which emerge in the infrared below the mass scale of charged particles. We construct these symmetries explicitly as topological defects and illustrate their action on local and extended operators. We also describe their action on boundary conditions and illustrate some consequences of the symmetry for Hilbert spaces of the theory defined in finite volume.

Quantum duality in electromagnetism and the fine structure constant

Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions of exactly marginal operators. We show that the curvature of such conformal manifolds can be computed using Euclidean and Lorentzian inversion formulae, which combine the operator content of the conformal field theory into an analytic function. Analogously, operators of fixed dimension define bundles over the conformal manifold whose curvatures can also be computed using inversion formulae. These results relate curvatures to integrated four-point correlation functions which are sensitive only to the behavior of the theory at separated points. We apply these inversion formulae to derive convergent sum rules expressing the curvature in terms of the spectrum of local operators and their three-point function coefficients. We further show that the curvature can smoothly diverge only if a conserved current appears in the spectrum, or if the theory develops a continuum. We verify our results explicitly in 2d examples. In particular, for 2d (2,2) superconformal field theories we derive a lower bound on the scalar curvature, which is saturated by free theories when the central charge is a multiple of three.

Bruno Balthazar

Geometry of conformal manifolds and the inversion formula

We explore the notion of approximate global symmetries in quantum field theory and quantum gravity. We show that a variety of conjectures about quantum gravity, including the weak gravity conjecture, the distance conjecture, and the magnetic and axion versions of the weak gravity conjecture can be motivated by the assumption that generalized global symmetries should be strongly broken within the context of low-energy effective field theory, i.e. at a characteristic scale less than the Planck scale where quantum gravity effects become important. For example, the assumption that the electric one-form symmetry of Maxwell theory should be strongly broken below the Planck scale implies the weak gravity conjecture. Similarly, the violation of generalized non-invertible symmetries is closely tied to analogs of this conjecture for non-abelian gauge theory. This reasoning enables us to unify these conjectures with the absence of global symmetries in quantum gravity.

Tom Rudelius

Generalized symmetry breaking scales and weak gravity conjectures

For any quantum system invariant under gauging a higher-form global symmetry, we construct a noninvertible topological defect by gauging in only half of the spacetime. This generalizes the Kramers-Wannier duality line in <math><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions to higher spacetime dimensions. We focus on the case of a one-form symmetry in <math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions, and determine the fusion rule. From a direct analysis of one-form symmetry protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. We give an explicit realization of this duality defect in the free Maxwell theory, both in the continuum and in a modified Villain lattice model. The duality defect is realized by a Chern-Simons coupling between the gauge fields from the two sides. We further construct the duality defect in non-Abelian gauge theories and the <math><msub><mi>Z</mi><mi>N</mi></msub></math> lattice gauge theory.

Yichul Choi

Po-Shen Hsin

Ho Tat Lam

Shu-Heng Shao

Noninvertible duality defects in <math><mrow><mn>3</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions

Axions, periodic scalar fields coupled to gauge fields through the instanton density, have a rich variety of higher-form global symmetries. These include a two-form global symmetry, which measures the charge of axion strings. As we review, these symmetries typically combine into a higher-group, a kind of non-abelian structure where symmetries that act on operators of different dimensions, such as points, lines, and strings, are mixed. We use this structure to derive model independent constraints on renormalization group flows that realize theories of axions at long distances. These give universal inequalities on the energy scales where various infrared symmetries emerge. For example, we show that in any UV completion of axion-Yang-Mills, the energy scale at which axion strings can decay is always larger than the mass scale of charged particles.

T. Brennan

Axions, higher-groups, and emergent symmetry

Using numerical methods for finding Ricci-flat metrics, we explore the spectrum of local operators in two-dimensional conformal field theories defined by sigma models on Calabi-Yau targets at large volume. Focusing on the examples of K3 and the quintic, we show that the spectrum, averaged over a region in complex structure moduli space, possesses the same statistical properties as the Gaussian orthogonal ensemble of random matrix theory.