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| Closed string | |
|---|---|
| Name | Closed string |
| Type | Fundamental object |
| Field | Theoretical physics |
| Introduced | 1970s |
| Key people | Michael Green, John Schwarz, Edward Witten, Leonard Susskind, Yoichiro Nambu |
Closed string Closed string objects are one-dimensional loops that appear in Superstring theory, Bosonic string theory, Heterotic string constructions and related frameworks. They contrast with open strings in D-brane setups and play central roles in proposals by Edward Witten, Michael Green, John Schwarz, Cumrun Vafa and Leonard Susskind connecting perturbative and nonperturbative physics. In modern treatments closed strings mediate gravitational interactions in models studied by groups at CERN, Institute for Advanced Study, Princeton University and Caltech.
Closed strings were first recognized in early work on dual models and in analyses by Yoichiro Nambu, Gabriele Veneziano, Niels Bohr-era influenced researchers and later formalized by teams including Miguel Virasoro and Sergio Fubini. They are central to Type IIA string theory, Type IIB string theory, Bosonic string theory, Heterotic string formulations and to the discovery of T-duality and S-duality. Closed strings couple naturally to closed-form fields such as the graviton field in effective actions and are crucial in connecting perturbative expansions developed at institutions like Harvard University and University of Cambridge with nonperturbative dualities explored at Institute for Advanced Study.
Classically a closed string is described by a periodic embedding X^μ(σ, τ) mapping the circle S^1 parameterized by σ into a target spacetime used in General relativity-inspired backgrounds like Minkowski space or Anti-de Sitter space. The classical action is the Polyakov action developed by Alexander Polyakov and earlier Nambu–Goto formulations linked to Yoichiro Nambu and Tetsuo Goto. Boundaryless evolution enforces periodic boundary conditions akin to formulations used in analyses by Paul Dirac and methods employed at Max Planck Institute labs. Constraints from worldsheet reparameterization lead to Virasoro constraints derived by Miguel Virasoro and formal quantization paths explored by Richard Feynman style path integrals in works at CERN and Stanford University.
Quantization of closed strings proceeds via light-cone quantization, canonical quantization and covariant BRST quantization developed by researchers including Becchi, Rouet, Stora, Ivo Batalin and Gennady Vilkovisky. Critical dimensions such as 26 for Bosonic string theory and 10 for superstring variants emerged from anomaly cancellation studies by Michael Green and John Schwarz, and from computations formalized by Friedan and collaborators. Perturbative amplitudes use worldsheet genus expansions connected to mathematical work by Bernhard Riemann and to modular invariance properties explored by teams at Princeton University and Cambridge University.
The closed string spectrum decomposes into left-moving and right-moving sectors with massless states including the graviton, the Kalb–Ramond field (or B-field) and the dilaton that were characterized in analyses by John Wheeler, Steven Weinberg and later by Edward Witten. Excited modes correspond to massive Regge trajectories studied by Tullio Regge and appear in scattering amplitudes computed in the tradition of Veneziano-type dual amplitudes. Mode expansions employ oscillator algebras related to the Virasoro algebra introduced by Miguel Virasoro and are central to vertex operator constructions developed by Alexander Polyakov, Paul Ginsparg and groups at Harvard University.
Closed strings mediate gravitational interactions in string-derived effective field theories studied by Gerard 't Hooft and Stephen Hawking motivated research; their interactions are encoded in genus expansions of worldsheet surfaces classified by Bernhard Riemann moduli spaces and computed using conformal field theory methods pioneered by Belavin, Polyakov and Zamolodchikov. Closed string loops generate quantum corrections that relate to anomaly cancellation results by Green Schwarz and to dualities formulated by Edward Witten and Shamit Kachru. They interact with D-brane configurations analyzed by Joe Polchinski and can be emitted or absorbed by branes in setups studied at CERN and SLAC National Accelerator Laboratory.
When target space dimensions are compactified on manifolds like Calabi–Yau manifolds, Toruses or orbifolds studied by Shing-Tung Yau and Philip Candelas, closed strings acquire momentum and winding quantum numbers; winding modes play central roles in T-duality identified by Kikkawa and Yamasaki and in mirror symmetry studied by Kontsevich and Maxim Kontsevich-led programs. Compactified spectra determine low-energy phenomenology targeted in model-building by groups at CERN, Stanford University and University of Chicago and underlie flux compactification scenarios investigated by Joseph Polchinski and Shamit Kachru.
Mathematically, closed string theory connects to two-dimensional conformal field theory developed by Belavin, Alexander Zamolodchikov and Paul Ginsparg, to vertex operator algebra theory advanced by Richard Borcherds and to moduli of Riemann surfaces studied by Bernhard Riemann and Grothendieck-inspired mathematics. Dualities including T-duality, S-duality and the AdS/CFT correspondence proposed by Juan Maldacena relate closed string sectors to gauge theories such as N=4 supersymmetric Yang–Mills theory explored at Institute for Advanced Study and Princeton University. Developments in derived categories and mirror symmetry link work by Maxim Kontsevich and Philip Candelas to modern closed string compactification frameworks investigated at University of Cambridge and Harvard University.