Clay Córdova, Diego García-Sepúlveda, and Nicholas Holfester of the University of Chicago have demonstrated a surprising connection between particles and solitons in two-dimensional quantum chromodynamics. Their work reveals that a specific type of symmetry, described mathematically by a fusion category, can force particles and solitons to share the same mass, a result challenging the conventional understanding of these distinct quantum excitations.

This symmetry breaks down at long distances, creating multiple possible vacua, and the researchers map the resulting complex relationships using “quiver diagrams” where nodes represent vacua and arrows depict excited states. The findings offer new insight into strongly coupled physics and the infrared spectrum of gauge theories.

2D Quantum Chromodynamics & Spontaneous Symmetry Breaking

A specific symmetry within two-dimensional quantum chromodynamics dictates that certain particles and solitons must possess identical masses, a result challenging conventional expectations about these distinct quantum entities. This unexpected equivalence arises from a “finite non-invertible symmetry” described by a mathematical structure known as a fusion category, which governs the interactions within the theory. The research, focused on gapped theories, uses an interval compactification from a three-dimensional topological quantum field theory to identify the fusion category responsible for this symmetry.

This approach allows for the direct characterization of the spectrum of excitations within the system, revealing the conditions under which particles and solitons become mass-degenerate. The foundation of this work lies in understanding how symmetries break down at long distances, leading to multiple possible “vacua,” or ground states, for the system.

Researchers map the relationships between these vacua and the resulting particle and soliton excitations using a visual tool called quiver diagrams. In these diagrams, each node represents a distinct vacuum, while arrows indicate the presence of excited states connecting them. The structure of these quivers is directly dictated by the fusion category symmetry, providing a way to predict the allowed combinations of particles and solitons and their corresponding masses.

The team’s analysis demonstrates that the fusion category symmetry frequently implies the existence of stable states within the spectrum, further complicating the possible configurations of these excitations. The implications of this mass degeneracy extend to the broader understanding of strongly coupled physics, where traditional perturbative methods fail. By focusing on two-dimensional quantum chromodynamics, the researchers circumvent many of the complexities associated with higher-dimensional theories, allowing for exact analytical results.

This is achieved by using the representation theory of the fusion category, which provides a framework for classifying the allowed “multiplets” of particles and solitons. The work builds upon previous studies of non-invertible symmetries and their role in characterizing the vacuum structure of quantum chromodynamics, offering a new lens through which to examine confinement and other fundamental phenomena.

As noted in the paper, the fusion category is “fully spontaneously broken in the IR and hence characterizes the vacua of the gauge theory.” Consider the case of a theory without one-form symmetry; the stable solitons and particles must realize a representation furnishing a connected quiver, with all such quivers containing a specific sub-quiver. This constraint dictates that the theory must contain at least two stable particles over a given vacuum and two soliton-anti-soliton pairs.

The quiver structure is realized by an irreducible representation of the strip algebra, leading to the surprising conclusion that all these excitations share the same mass. The team emphasizes that the ability to characterize Lagrangian algebras, mathematical structures describing the possible interactions, provides a useful tool to characterize topological boundaries. The approach employed by the researchers also addresses the potential for a phenomenon where the theory breaks down into disconnected sectors.

By gauging the “common center” one-form symmetry, they ensure that all topological local operators correspond to vacua of a single universe, maintaining the consistency of the theory. This careful construction allows for a clear identification of the relevant fusion category, denoted as C, which then dictates the allowed symmetry-enforced degeneracies in the spectrum. The computation of these quivers, which visually represent the relationships between vacua and excitations, relies on understanding the fusion rules for the category C.

The findings demonstrate that the fusion category symmetry often implies that particle and solitons are in the same representation and hence necessarily have equal masses. This work provides a novel framework for understanding the complex interplay between symmetry, topology and particle physics in two-dimensional quantum chromodynamics, offering new insights into the fundamental nature of strongly coupled systems.

Anyon Condensation Identifies Particle & Soliton Multiplets

The spontaneous breaking of a finite non-invertible symmetry in two-dimensional quantum chromodynamics dictates the arrangement of distinct vacua within the theory, and new work reveals a surprising consequence. Researchers are using anyon condensation and fusion category representation theory to map the allowed combinations of these excitations, uncovering scenarios where fundamental particles and complex solitons share identical masses.

This challenges conventional expectations where solitons, as composite excitations, typically exhibit different properties than their fundamental particle counterparts. This symmetry isn’t merely present but actively shapes the possible configurations of the system at low energies, defining the arrangement of its vacua, the lowest energy states. By applying anyon condensation, a technique for understanding topological order, the team can determine the permissible groupings of particles and solitons, termed “multiplets,” that arise from this symmetry breaking.

The resulting degeneracy patterns are visualized using quiver diagrams, where nodes represent the distinct vacua and arrows denote the excited states connecting them, providing a visual map of the theory’s complex energy landscape. The implications extend beyond simply identifying possible states; the research demonstrates a direct link between symmetry and mass. The researchers state, highlighting a counterintuitive result. This equality isn’t a coincidence but a consequence of the underlying symmetry, forcing particles and solitons to behave similarly under certain transformations.

This feature, the team notes, is unique to spontaneously broken non-invertible symmetries and emphasises the intrinsically quantum nature of these systems. The team tested these findings in the SO(3) gauge theory with Majorana fermions in the five, finding that the theory possesses three lines with non-trivial fusions, leading to three possible irreducible multiplets.

The methodology employed relies on identifying a Lagrangian algebra, a mathematical structure defining the topological boundary of a three-dimensional topological quantum field theory. The team emphasizes that this approach is not limited to specific theories, but can be applied to a range of systems exhibiting similar symmetries. This connection provides a powerful constraint on the possible configurations of solitons, further refining the understanding of the system’s low-energy behavior. The researchers also note that the existence of certain stable states is frequently implied by the fusion category symmetry, adding another layer of predictability to the theory.

By applying these rules and using anyon condensation, they can determine the fusion categories present throughout the renormalization group flow, providing a comprehensive picture of the system’s evolution from high to low energies. The approach also allows for the calculation of vacuum expectation values for local operators, providing insights into the properties of the different vacua. The detailed computations, including the boundary condensation maps and allowed quivers for each theory, are included in an appendix, offering a transparent and reproducible account of the findings.

Mass Equivalence of Particles and Solitons via Symmetry

In two-dimensional quantum chromodynamics, the established distinction between fundamental particles and composite solitons is challenged by the unexpected finding that these excitations can share identical masses. The implications extend beyond simple mass equivalence, suggesting a deeper connection between the nature of vacuum states and the spectrum of excitations. The spontaneous breakdown of this symmetry at large distances is central to the observed phenomenon, creating multiple, distinct vacuum states within the theory.

When a mass gap exists, a minimum energy required to create a particle, the resulting spectrum consists of particle excitations originating from a single vacuum, alongside soliton sectors that connect these different vacuum states. This is not merely a mathematical curiosity. It’s a constraint on the possible configurations of these excitations, refining theoretical predictions. The ability to predict these states from purely kinematic considerations, without needing to solve the full dynamics, represents an advancement in understanding strongly coupled physics.

A particularly striking result is the observation that the mixing of particle and soliton states is driven by this non-invertible symmetry. This differs from traditional group-like symmetries, where the realization of a representation is a dynamical question; here, the symmetry itself can require the existence of certain stable states. “As will be demonstrated in the following examples, this additional information is often enough to require the existence of some stable particles or solitons from purely kinematic considerations,” the researchers state.

This means that the symmetry, in effect, dictates the presence of specific excitations, independent of the detailed dynamics of the theory. These quivers, visual representations of the relationships between vacua and excitations, contain a specific sub-quiver in all cases.

The existence of these states isn’t just a possibility. Further analysis of another theory showed similar constraints, requiring at least two stable particles, one above each of two distinct vacua, and a soliton-anti-soliton pair connecting them. Again, all these excitations are predicted to have degenerate mass, meaning they share the same energy level.

While symmetry alone doesn’t determine which specific quiver is realized, dynamics play a role in that, it does establish the minimum requirements for the system’s structure. The implications of this work extend beyond the specific theories examined.

Topological Cosets & the Mass Gap Criterion in QCD₂

The analysis of two-dimensional quantum chromodynamics reveals a surprising connection between particle and soliton masses, with calculations demonstrating scenarios where these excitations share identical energy levels. The central charge of a WZW model, a type of conformal field theory, determines whether a theory admits a mass gap. Specifically, the theory is gapped if and only if the coset Spin(dim)_1/G_(I) has vanishing central charge, a condition that constrains the allowed particle and soliton states.

This constraint arises from the application of anyon condensation, a technique used to directly determine the fusion category of lines at a topological boundary. The splitting rule governing how simple anyons transform into line operators at the boundary is important for understanding the fusion category.

Exploiting these constraints allows researchers to precisely determine the fusion ring of the fusion category, a key step in mapping the relationships between vacua and excitations. The connection between topological cosets and topological boundary conditions has been recently studied, revealing that a gauged WZW model with vanishing central charge admits a topological boundary when an embedding of chiral algebras exists. The branching rules of this conformal embedding, represented as * $χ_{Λ}^{Q_1} (q)$ = $s u m_{λ} b_{Λ, λ}$ $χ_{λ}^{G_k} (q) *$ , where q is the modular parameter and $b_{Λ, λ}$ are non-negative integers, induce the existence of a Lagrangian algebra.

This algebra defines the topological boundary of $Q_{1}$ × $G_{- k}$ , and dictates the allowed local operators, or vacua, in the infrared (IR) limit of the theory. Local operators in the 2D theory are constructed by stretching anyons between the ends of an interval, providing a physical interpretation of the mathematical formalism.

The topological OPE of these local operators, which governs their interactions, can be obtained by exploiting properties like commutativity, associativity and consistency with the Lagrangian algebra. The existence of an idempotent complete basis $v_{i}$ of local operators, in one-to-one correspondence with the clustering vacua of the theory, further refines the understanding of the system’s behavior.

The framework developed offers a novel approach to understanding the interplay between symmetry, topology and the spectrum of excitations in strongly coupled gauge theories. By using the representation theory of fusion categories, researchers can gain insights into the spectrum of these theories, and the constraints on particle and soliton masses provide a powerful tool for refining theoretical predictions.

👉 More information
🗞 Particle-Soliton Degeneracy in 2D Quantum Chromodynamics
✍️ Clay Córdova, Diego García-Sepúlveda and Nicholas Holfester
🧠 DOI: http://link.aps.org/doi/10.1103/jckx-z293

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