Non-Supersymmetric Dualities in Three Dimensions
Three-dimensional bosonization and particle–vortex dualities form a tightly connected web of proposed infrared equivalences. Their anomaly counterterms, massive phases, line operators, and many operator maps pass demanding checks. Yet most interacting nonsupersymmetric nodes are still conjectural at finite rank: many edges of the web are derived from a smaller set of seed dualities, so their agreement is correlated rather than independent proof.
Evidence cutoff. 11 August 2026.
Required background. Supersymmetric parents of mirror, particle–vortex, and bosonization webs supplies controlled ancestors and deformations; generalized symmetries, global forms, and anomalies under duality fixes the global dictionary. Helpful background. Parity anomalies and contact terms supplies half-integer Chern–Simons bookkeeping, global form and matter representations fixes genuine operators, and ‘t Hooft anomaly matching supplies an RG-invariant test.
The finite-rank duality web
Section titled “The finite-rank duality web”Normative question. Which proposed three-dimensional dualities survive tests of anomalies, global form, deformations, operator maps, and quantitative observables?
The scope is nonsupersymmetric relativistic QFT in dimensions, especially Abelian particle–vortex/fermion dualities and non-Abelian Chern–Simons matter bosonization. A duality claim includes the spin or non-spin manifold, background fields, Chern–Simons contact terms, gauge-group global form, flavor bounds, and the infrared fixed point reached after tuning. Equality of schematic Lagrangians without those data is not the same claim.
Evidence with its dependencies exposed
Section titled “Evidence with its dependencies exposed”| Test | What survives in well-specified proposals | Limitation |
|---|---|---|
| Anomalies and contact terms | Parity anomaly, time-reversal response, and background Chern–Simons terms match after the correct counterterms and spin-structure dependence are included | Counterterms can make naïvely identical formulations inequivalent |
| Relevant mass deformations | Dual pairs flow to matching topological phases and level/rank dualities on either side | Matching endpoints does not prove the critical fixed points coincide |
| Operator and line maps | Monopoles map to bosons or fermions with matching global charges; baryon/monopole and genuine-line maps close in many examples | Scaling dimensions and multiplicities are often known only at large rank or perturbatively |
| Quantitative observables | Large-N free energies, thermal quantities, and selected bootstrap/lattice data are compatible in bounded cases | Finite-N extrapolation and shared assumptions limit independence |
| Web closure | Gauging background symmetries and applying modular operations produces mutually consistent duality cycles | Descendants inherit the truth of the seed duality |
Karch and Tong derived a fermion–boson relation from particle–vortex duality and matched phases and responses (Karch and Tong 2016). Seiberg, Senthil, Wang, and Witten organized the modern Abelian web, explicitly describing its derivations as plausible rather than rigorous (Seiberg et al. 2016). Hsin and Seiberg clarified level/rank duality, transparent spin theories, and background-field contact terms that are essential when moving around the web (Hsin and Seiberg 2016).
Non-Abelian Chern–Simons–matter dualities add baryon–monopole maps and large-rank evidence (Aharony 2016), but extrapolation to finite rank and strongly coupled operator data remains a separate assumption.
Non-Abelian bosonization relates or Chern–Simons gauge theories with fundamental fermions to rank/level-transformed theories with critical scalars, subject to a flavor/rank domain. Large-N calculations supply a controlled expansion, but a finite-N equality at the interacting critical point is not a theorem. Supersymmetric dualities and flows motivate many proposals, yet a flow can generate relevant operators or first-order behavior that invalidates an assumed fixed point.
Negative and qualifying evidence
Section titled “Negative and qualifying evidence”Several apparent contradictions in early formulations were resolved by specifying spin connections, gravitational Chern–Simons terms, or quotients of the gauge group. That history is itself a negative result: local operator matching alone cannot certify a duality. Other failure modes include an unaccounted emergent symmetry, monopole operators forbidden by the global form, a mismatch of one-form anomalies, and extrapolating a large-N dimension through a unitarity bound at small rank.
Assessment. The web is highly constrained and strongly supported, but generally unproved at finite rank. Anomaly, phase, and operator-map tests eliminate naïve variants and leave precise serious formulations. Exact topological level/rank limits and selected lattice particle–vortex transformations are established in their own domains; equality of the interacting nonsupersymmetric continuum fixed points remains conjectural for most headline pairs.
What would settle a dual pair?
Section titled “What would settle a dual pair?”A constructive resolution would give reflection-positive lattice regulators for both sides, prove their continuum limits, and identify their local and extended operator algebras. A sufficiently rich set of exact finite-rank correlators or partition functions, including backgrounds and defects, could also establish equality through reconstruction. Conversely, one verified mismatch in an anomaly, genuine-line spectrum, deformation endpoint, or universal observable—after conventions and accidental symmetries are exhausted—would refute that precise formulation.
Related routes are supersymmetry and duality, quantum matter and emergence, analytic and numerical conformal bootstrap, and the order of deconfined quantum critical transitions.
Evidence cutoff and source selection
Section titled “Evidence cutoff and source selection”The finite set emphasizes seed dualities, global/contact-term refinements, non-Abelian extensions, and tests with genuinely different inputs. Targeted arXiv, journal, INSPIRE, condensed-matter, and citation-chain searches covered public evidence through 11 August 2026. Descendants obtained only by gauging the same seed were not counted as independent confirmations.
References
Section titled “References”- Aharony, Ofer. “Baryons, Monopoles and Dualities in Chern–Simons-Matter Theories.” Journal of High Energy Physics 2016, no. 2 (2016): 093. DOI.
- Hsin, Po-Shen, and Nathan Seiberg. “Level/Rank Duality and Chern–Simons-Matter Theories.” Journal of High Energy Physics 2016, no. 9 (2016): 095. DOI.
- Karch, Andreas, and David Tong. “Particle–Vortex Duality from 3D Bosonization.” Physical Review X 6 (2016): 031043. DOI.
- Seiberg, Nathan, T. Senthil, Chong Wang, and Edward Witten. “A Duality Web in 2+1 Dimensions and Condensed Matter Physics.” Annals of Physics 374 (2016): 395–433. arXiv.