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| Nonlinear sigma model | |
|---|---|
| Name | Nonlinear sigma model |
| Field | Theoretical physics |
| Introduced | 1960s |
| Notable | Alexander Polyakov, Kenneth G. Wilson, Sidney Coleman |
Nonlinear sigma model
The nonlinear sigma model is a class of quantum field theorys that describe maps from a source spacetime or worldsheet into a target Riemannian manifold or homogeneous space. Originating in studies of spin systems and pion dynamics, the model has become central to research linking condensed matter physics, statistical mechanics, high-energy physics, and differential geometry. It provides a framework for exploring symmetry breaking, topological solitons, and renormalization in low-dimensional systems.
The model was introduced in the context of pion phenomenology and low-energy nuclear physics research and later formalized in studies by figures such as Sigurd W. Bruun and Kenneth G. Wilson. A typical setup involves a map φ: Σ → M, with Σ a two-dimensional Riemann surface or d-dimensional Minkowski space and M a target manifold such as a sphere S^n, a group manifold G, or a homogeneous coset G/H related to symmetry breaking patterns. Early influential works include contributions by Alexander Polyakov, Sidney Coleman, and Ilya M. Gel'fand, establishing the model's role in renormalization group analyses and nonperturbative phenomena.
Classically the action is S[φ] = (1/2g^2) ∫_Σ d^dx √h h^{μν} ∂_μφ^a ∂_νφ^b g_{ab}(φ) where h_{μν} is a metric on Σ and g_{ab} is the Riemannian metric on M. For target spaces like S^n or SU(N) the action respects global Lie group symmetries and supports conserved currents via Noether's theorem. The Euler–Lagrange equations yield harmonic map equations studied by James Eells and J. H. Sampson in geometric analysis. Classical integrability arises in special cases such as the principal chiral model on SU(2), linked historically to work by Ludvig Faddeev and Leon Takhtajan on integrable hierarchies and Lax pairs.
Quantization introduces ultraviolet divergences and nontrivial renormalization group flows first analyzed by Kenneth G. Wilson and David J. Gross. In two dimensions the beta function to one loop scales like β(g) ∝ (n−2)g^3 for target S^{n}, leading to asymptotic freedom for certain targets as discussed by Alexander Polyakov and David Friedan. Perturbative expansions can be organized using background field methods pioneered by Julian Schwinger and refined by Steven Weinberg. Nonperturbative methods include the 1/N expansion initiated by Gerard 't Hooft and semiclassical instanton calculus leveraged by Edward Witten. Renormalization on curved Σ connects to anomalies studied by Alvarez-Gaumé and Luis Álvarez-Gaumé in the context of conformal field theory and the Virasoro algebra.
Nonlinear sigma models model low-energy excitations in systems with continuous symmetry breaking such as Heisenberg model magnets and antiferromagnets analyzed by Philip W. Anderson and P. A. Lee. In disordered systems they describe localization physics via replicas and supersymmetric formulations developed by K. B. Efetov and Igor Aleiner. The model underlies the theory of the Kosterlitz–Thouless transition originally studied by J. M. Kosterlitz and David J. Thouless, and appears in descriptions of quantum Hall plateau transitions investigated by B. I. Halperin and Shou-Cheng Zhang. In one dimension the mapping to Luttinger liquid or bosonization contexts connects with results by F. D. M. Haldane and John Cardy.
In high-energy theory the principal chiral sigma model and coset sigma models describe low-energy effective actions for Goldstone bosons in chiral perturbation theory formulated by Steven Weinberg and Howard Georgi. Two-dimensional sigma models serve as worldsheet theories for string propagation on target manifolds, central to formulations by Michael Green, John Schwarz, and Edward Witten. Supersymmetric sigma models including N=(2,2) constructions underpin mirror symmetry developed by Philip Candelas and Max Kreuzer, and contribute to topological field theories introduced by Edward Witten and Anton Kapustin. Target-space conditions derived from beta function vanishing relate to Einstein equations and Calabi–Yau geometry studied by Shing-Tung Yau.
Classical solutions include solitons like skyrmions examined by Tony Skyrme and vortices relevant to Nielsen–Olesen frameworks. Two-dimensional instantons contribute to nonperturbative dynamics, central to work by Alexander Belavin and A. M. Polyakov. Topological theta terms and Wess–Zumino–Witten terms introduce phases related to anomalies and level quantization, with seminal developments by Edward Witten and J. Wess and B. Zumino. Homotopy groups π_n(M) classify defects and textures, linking to mathematical results by Henri Poincaré and L. E. J. Brouwer.
The sigma model connects to harmonic map theory, Hodge theory, and Index theorems investigated by Atiyah–Singer collaborators such as Michael Atiyah and Isadore Singer. Generalizations include supersymmetric sigma models tied to Kähler geometry and generalized complex geometry developed by Nigel Hitchin and Marco Gualtieri. Poisson sigma models and topological sigma models relate to deformation quantization studied by Maxim Kontsevich and to categorical structures in derived algebraic geometry pursued by Jacob Lurie. Quantum groups and Yang–Baxter structures enter through integrable deformations connected to work by Vladimir Drinfeld and Leon Takhtajan.