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| Seiberg–Witten map | |
|---|---|
| Name | Seiberg–Witten map |
| Field | Mathematical physics |
| Introduced | 1990s |
| Notable users | Nathan Seiberg, Edward Witten |
Seiberg–Witten map The Seiberg–Witten map is a correspondence between gauge fields on commutative and noncommutative spaces that arose in the context of string theory and quantum field theory, providing an explicit relation between ordinary gauge theory configurations and their counterparts in noncommutative geometry formulations. It connects constructions used by researchers at institutions such as Institute for Advanced Study, Harvard University, and Princeton University with techniques developed in studies of D-brane dynamics, M-theory, and Matrix theory, and it has influenced work by figures including Nathan Seiberg, Edward Witten, Alain Connes, and Michael R. Douglas.
The map was introduced during investigations linking open string theory in the presence of a B-field to noncommutative gauge theories on brane worldvolumes, tying together insights from Polchinski, Strominger, and the String theory landscape program. It provides an explicit functional relation between a commutative gauge potential used in perturbative Yang–Mills theory calculations and a noncommutative gauge potential appearing in deformations studied by Seiberg and Witten, clarifying equivalences invoked in the AdS/CFT correspondence literature and echoing constructions from the Kontsevich formality theorem.
Formally, the map associates a commutative connection A on a principal bundle for a Lie group such as U(1), SU(N), or SO(N) to a noncommutative connection  defined on a deformation quantization of the base manifold, typically defined by a Poisson tensor θ drawn from models used by Maxim Kontsevich and Murray Gerstenhaber. The defining relation enforces that gauge transformations g in the commutative theory, elements of groups like Gauge group or Lie group representations, correspond to noncommutative gauge transformations ĝ via a homomorphism preserving curvature F and its star-product–deformed counterpart F̂, in analogy with structures appearing in BRST cohomology and the Batalin–Vilkovisky formalism.
Physically, the map explains how low-energy effective actions for D-branes in backgrounds with nonzero Kalb–Ramond field are equivalent to noncommutative DBI actions, a point emphasized in the work of Seiberg and Witten and followed up by groups at CERN, SLAC National Accelerator Laboratory, and University of California, Berkeley. Applications include reinterpretations of instantons and monopoles in noncommutative gauge theory, analyses of UV/IR mixing highlighted by Heinrich Grosse and Harald Steinacker, and constructions relevant to phenomenology attempts by collaborations at DESY and Fermilab seeking signals of spacetime noncommutativity.
Practically, the Seiberg–Witten map is constructed as a formal power series in the noncommutativity parameter θ, employing recursive formulas akin to those used in the Moyal–Weyl star product expansion and techniques familiar from Feynman diagram expansions and renormalization group methods developed by Kenneth G. Wilson and Gerard 't Hooft. The perturbative expansion yields order-by-order expressions for Â[A] and ĝ[A,g] that satisfy consistency conditions comparable to identities in Noether's theorem derivations and algebraic constraints studied in Hochschild cohomology.
Explicit constructions have been worked out for abelian gauge groups like U(1) and for nonabelian groups such as SU(2), SU(3), and U(N), with concrete formulae appearing in papers by researchers at MIT, Yale University, and University of Chicago. Example solutions include expressions for noncommutative electromagnetism potentials, noncommutative instanton moduli expressed using methods from ADHM construction, and explicit mappings for solitonic solutions that parallel analyses by Andrew Strominger and Cumrun Vafa.
The construction admits ambiguities tied to field redefinitions and gauge choices reminiscent of choices encountered in the Seiberg duality context and in treatments of anomalies by Alberto Sirlin and Gerard 't Hooft. These ambiguities are constrained by consistency conditions such as closure of gauge algebras and satisfaction of deformed Bianchi identities, with cohomological classifications related to work by Pierre Deligne and Jean-Luc Brylinski on deformation theory and by Daniel Quillen on algebraic K-theory.
Generalizations incorporate curved backgrounds, nonconstant Poisson structures studied by Maxim Kontsevich and Konstantin Nest, applications to supersymmetric theories influenced by Nathan Seiberg and Edward Witten's supersymmetry research, extensions to finite temperature field theory contexts explored at Brookhaven National Laboratory, and relations to approaches in noncommutative gravity pursued by teams at Perimeter Institute and Institute of Physics (Czech Academy of Sciences). Connections to the Kontsevich formality theorem, Fedosov quantization, and categorical viewpoints resonant with Gérard Laumon and Alexander Beilinson further broaden its mathematical reach.