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| Polyakov action | |
|---|---|
| Name | Polyakov action |
| Field | Theoretical physics |
| Introduced by | Alexander Polyakov |
| Introduced | 1981 |
| Related | Nambu–Goto action, Conformal field theory, String theory |
Polyakov action
The Polyakov action is a formulation of relativistic string dynamics introduced by Alexander Polyakov that reformulates the Nambu–Goto action in a form amenable to quantization. It plays a central role in String theory, Conformal field theory, and studies of two-dimensional quantum field theory on curved worldsheet backgrounds. The action links techniques developed in Soviet and Western world theoretical physics and underpins connections between anomaly analysis, BRST quantization, and modern developments such as AdS/CFT correspondence.
The Polyakov action arose as an alternative to the geometric square-root formulation of the Nambu–Goto action and made explicit the coupling between the embedding maps and an auxiliary worldsheet metric. This construction enabled systematic treatments by methods familiar from path integrals and operator methods used in studies of Conformal invariance and Virasoro algebra. Historically it influenced research by groups around Princeton University, CERN, Institut des Hautes Études Scientifiques, and individual researchers including Michael Green, John Schwarz, David Gross, and Edward Witten.
The Polyakov action describes maps X^μ(σ^a) from a two-dimensional worldsheet Σ with coordinates σ^a into a target spacetime M with coordinates X^μ, coupled to an independent worldsheet metric h_{ab}. The classical functional can be written in a manifestly Lorentz-covariant way and is equivalent on shell to the Nambu–Goto action. Key ingredients and relations appear across literature on Calabi–Yau compactifications, Kaluza–Klein theory, and discussions of D-brane dynamics. The action couples to background fields such as the Kalb–Ramond field, the dilaton, and spacetime curvature tensors when generalized.
Variation with respect to X^μ yields the worldsheet wave equation (the two-dimensional analogue of the Klein–Gordon equation) for the embedding functions, while variation with respect to the metric h_{ab} produces the vanishing of the worldsheet energy–momentum tensor, giving the Virasoro constraints. These classical relations connect to the analysis performed in Noether theorem contexts and to conserved currents studied by researchers at Harvard University and Stanford University. Solutions include classical string configurations such as folded strings studied in the context of Anti-de Sitter space and rotating string solutions examined by groups at Cambridge University and University of Chicago.
The Polyakov action is invariant under worldsheet diffeomorphisms and local Weyl rescalings; these symmetries underlie the conformal invariance exploited in Conformal field theory and the derivation of the Virasoro algebra. Gauge choices commonly employed include the conformal gauge and the light-cone gauge, approaches used in canonical quantization treatments by Paul Dirac-inspired methods and in path integral computations by teams at SLAC National Accelerator Laboratory. Fixing these gauges requires careful handling of residual symmetries and introduces ghost systems such as the Faddeev–Popov and b–c ghost system familiar from BRST constructions pioneered by researchers at CERN and Rutgers University.
Quantization proceeds via canonical approaches or via the path integral over embeddings and the worldsheet metric, with gauge fixing yielding determinants expressed through ghost actions. The measure and regularization introduce anomalies whose cancellation leads to critical spacetime dimension conditions, a result obtained in seminal work by Miguel Virasoro-inspired analyses and later refined by Green and Schwarz in superstring contexts. Functional integral techniques relate to studies at Institute for Advanced Study and adaptations into modern treatments of Topological field theory and Matrix models. Renormalization of the worldsheet sigma model ties to beta-function calculations and low-energy effective actions like Einstein equations and dilaton equations.
The Polyakov action underlies perturbative string scattering amplitude computations used in explorations at CERN and by collaborations such as Superstring theory research groups. Its formulation facilitates connections between string dynamics and two-dimensional statistical mechanics models, lattice realizations investigated at Fermilab, and the study of nonperturbative phenomena like instantons and worldsheet solitons. The action is central to derivations of target-space effective actions, to analyses of T-duality and S-duality, and to conceptual frameworks used in the AdS/CFT correspondence proposed by Juan Maldacena.
Generalizations include coupling to background Ramond–Ramond fields leading to the Green–Schwarz action, inclusion of worldsheet supersymmetry producing the Ramond–Neveu–Schwarz action, and formulations on curved worldsheet topologies connecting to Teichmüller space and moduli integration performed by mathematical physics groups at IHES and Perimeter Institute. Further extensions link to Nonlinear sigma model renormalization, to doubled formulations exploring T-folds and generalized geometry, and to string field theory approaches developed by researchers at University of California, Berkeley and Yale University.