Supersymmetry and Duality
Supersymmetry and duality research uses graded spacetime symmetry and equivalences between descriptions to obtain exact or nonperturbative information. The field includes protected operator sectors, indices and partition functions, moduli spaces, supersymmetry breaking, strong–weak and infrared dualities, and non-Lagrangian theories. A duality is a scoped equivalence claim with an operator and global-data dictionary—not a visual similarity between Lagrangians or one matching number.
Evidence cutoff. 11 August 2026.
Required background. Graded spacetime symmetry and supersymmetry theorems fixes the algebraic possibilities; exact, infrared, and emergent equivalence claims separates distinct strengths of “duality.”
Helpful background. Exact-observable duality tests shows what localization can compare, BPS line observables carries global-form data, ‘t Hooft anomaly matching supplies a necessary infrared test, and anomaly/generalized-symmetry constraints narrows allowed phases.
Supersymmetric protected data and full-theory claims
Section titled “Supersymmetric protected data and full-theory claims”| Program | Robust object | What it can establish | What remains unprotected |
|---|---|---|---|
| representation and BPS analysis | shortening conditions, central charges, protected dimensions | exact multiplet structure and stability walls | generic long-multiplet dimensions and OPE data |
| indices and localization | supersymmetric partition functions, indices, defect expectation values | exact coupling/mass dependence under localization hypotheses | observables outside the chosen supercharge’s cohomology |
| infrared duality | anomaly data, moduli spaces, chiral rings, deformations, protected observables | a convergent network of necessary and quantitative tests | a general theorem of equality of all observables |
| Seiberg–Witten geometry | periods, monodromies, BPS spectra | exact low-energy data in controlled supersymmetric theories | arbitrary finite-energy dynamics |
| non-Lagrangian reconstruction | protected chiral algebras, defects, compactification data | partial specification and cross-checks | uniqueness and complete unprotected correlation functions |
Topological central charges sharpen supersymmetric mass bounds and identify protected sectors Witten and Olive 1978, foundational result. Seiberg’s electric–magnetic duality gave a sharply testable equivalence between distinct four-dimensional gauge descriptions in their infrared domains Seiberg 1995, foundational. Its evidential force comes from a network—global symmetries, anomalies, moduli, operator maps, deformations, and decoupling limits—not from any single match.
Localization can reduce a path integral to finite-dimensional data by adding a supercharge-exact deformation while preserving selected observables. Pestun’s sphere calculation is a benchmark because it makes the geometry, supercharge, saddle set, one-loop factors, and instanton contributions explicit Pestun 2012, method. The result does not imply that every observable localizes or that contour and global-form choices are irrelevant.
Duality validation and uncertainty
Section titled “Duality validation and uncertainty”Exactness shifts the error budget; it does not erase it. One must check the preserved supercharge, boundary conditions, anomalies, integration contour, zero modes, instanton sectors, global gauge group, line-operator lattice, and analytic continuation. Numerical evaluation of an exact integral still has truncation and precision errors. A superconformal index is protected but may exhibit cancellations Kinney et al. 2007, index construction: equality is strong evidence, while inequality refutes the proposed dictionary under matched conventions; equality alone need not distinguish two theories.
Duality tests are most independent when they arise from different structures: anomaly matching, relevant deformations, moduli-space geometry, line/defect spectra, localization, bootstrap data, and lattice or semiclassical limits. Two protected partition functions derived from the same integral identity share a derivational core.
A concrete duality check should map global symmetries, contact terms, operators, line defects, and relevant deformations before comparing observables. For SQCD-like phases, match all continuous and discrete anomalies and follow deformations to regimes with independently understood infrared behavior. In Seiberg–Witten theory, transport the periods around singularities, verify the integral electromagnetic monodromies, and recover the corresponding central charges and massless states. A black-hole index comparison must additionally control cancellations, ensemble and charge conventions, saddle selection, and the asymptotic step from a protected index to a degeneracy.
Obstructions and negative lessons
Section titled “Obstructions and negative lessons”A nonzero Witten index obstructs spontaneous supersymmetry breaking under its hypotheses; a zero index is inconclusive. Holomorphy constrains dependence but does not fix nonholomorphic observables. Anomaly matching is necessary for a proposed infrared phase, not sufficient to select it. Protected-sector agreement can coexist with disagreement in global form or line operators, and partition functions may require contact-term choices. Emergent infrared duality does not identify microscopic operators at arbitrary energy.
Non-Lagrangian theories demonstrate that Lagrangian data are not a prerequisite for physical definition. Compactification constructions organize families that include non-Lagrangian theories Gaiotto 2012, program synthesis, but they also expose a reconstruction problem: protected data and compactification ancestry may not uniquely determine the full theory. Proposed nonsupersymmetric descendants require fresh evidence because cancellations and exact controls can disappear.
Entering the field
Section titled “Entering the field”Choose one theory pair and build a typed dictionary: spacetime dimension, supersymmetry, gauge/global form, local and extended operators, anomalies, parameter map, vacuum branches, and energy regime. Reproduce one exact observable and one deformation test. The mathematical foundations pathway and scope and status module are good preparation.
This guide omits general string dualities unless a field-theory limit or observable dictionary is explicit. It also does not treat supersymmetry as phenomenological evidence for physics beyond the Standard Model.
Evidence scope and related assessments
Section titled “Evidence scope and related assessments”The finite search used INSPIRE, arXiv, journal/DOI records, exact-observable literature, and targeted searches for failed duality tests and global-form refinements. Sources public through 11 August 2026 were eligible. Reassess when a proposed duality fails an anomaly or defect test, a new exact observable distinguishes competing dictionaries, or a reconstruction result establishes uniqueness in a non-Lagrangian class.
Continue to three-dimensional nonsupersymmetric dualities, reconstructing non-Lagrangian QFTs, or semiclassics, resurgence, and transseries.
References
Section titled “References”- D. Gaiotto, “ Dualities,” JHEP 08 (2012) 034. arXiv.
- J. Kinney, J. Maldacena, S. Minwalla, and S. Raju, “An Index for 4 Dimensional Super Conformal Theories,” Communications in Mathematical Physics 275 (2007) 209–254. DOI.
- V. Pestun, “Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops,” Communications in Mathematical Physics 313 (2012) 71–129. DOI.
- N. Seiberg, “Electric–Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories,” Nuclear Physics B 435 (1995) 129–146. DOI.
- E. Witten and D. Olive, “Supersymmetry Algebras That Include Topological Charges,” Physics Letters B 78 (1978) 97–101. DOI.