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| large N expansion | |
|---|---|
| Name | Large N expansion |
| Caption | Diagrammatic representation of 1/N expansion |
| Field | Theoretical physics |
| Introduced | 1974 |
| Introduced by | 't Hooft |
large N expansion
The large N expansion is an asymptotic technique in theoretical physics and mathematical physics that organizes perturbative series by powers of 1/N, where N denotes the rank or number of components of a symmetry group such as SU(N), O(N), or Sp(N). It provides a controlled approximation for systems with a large internal symmetry and connects to methods used in Quantum Chromodynamics, Matrix models, and the AdS/CFT correspondence. The approach yields insights into nonperturbative phenomena, string-like topologies, and collective behavior in many-body systems.
The method reorders perturbation theory so that leading contributions at N → ∞ are identified with planar diagrams and subleading corrections with higher-genus surfaces; key contributors include Gerard 't Hooft, Alexander Polyakov, and Kenneth Wilson. It is closely related to developments in Random matrix theory, Planar limit, and the formulation of String theory via gauge/string dualities exemplified by the Maldacena conjecture. Practical implementations appear in studies of Quantum Chromodynamics, Sigma models, Gross–Neveu model, and models inspired by Sachdev–Ye–Kitaev.
The technique traces to work by Gerard 't Hooft in 1974, who introduced the planar expansion for SU(N) gauge theories and emphasized topological classification of Feynman diagrams. Subsequent developments involved Alexander Polyakov's work on Conformal field theory and Kenneth Wilson's renormalization group ideas, while later connections to String theory were highlighted by Juan Maldacena and collaborators in the late 1990s. Important milestones include applications to the Gross–Neveu model by David Gross and André Neveu, exploration of vector models by Stanley Coleman, and the resurrection of matrix model techniques in studies by Miguel Virasoro and Edouard Brézin.
Formally, one considers families of theories parameterized by N such as SU(N), O(N), or U(N), and rescales couplings (e.g., g^2 N fixed) to define the 't Hooft coupling. Diagrammatic weights are classified by genus using Euler characteristic arguments linked to results in Topology and Algebraic geometry frequently used by researchers at institutions like Institute for Advanced Study and CERN. The large N limit often converts interacting problems into saddle-point analyses of effective actions, invoking techniques from Functional integral methods, Riemann surfaces in matrix models, and semiclassical approximations exploited in work at Princeton University and Harvard University.
In Quantum Chromodynamics, the 1/N_c expansion (with N_c color) organizes meson and baryon spectra, motivates qualitative features of confinement, and underpins phenomenological models used by groups at SLAC National Accelerator Laboratory and Fermilab. The expansion underlies factorization in large-N gauge theories studied in contexts like Deep inelastic scattering and informs holographic constructions following Maldacena's proposal relating N=4 supersymmetric Yang–Mills theory to Type IIB string theory on AdS5 × S5. Large N methods also clarify critical behavior in Sigma models, inform renormalization group flows examined by Kenneth Wilson, and serve in analyses of anomalous dimensions pursued at CERN and DESY.
Vector large N limits apply to classical and quantum O(N), CP(N-1), and Gross–Neveu-type models used to study critical phenomena in systems investigated by researchers at Bell Labs and Bellcore. Matrix large N techniques inform eigenvalue distributions in Random matrix theory with applications to Quantum chaos and mesoscopic physics at institutions such as Los Alamos National Laboratory. The Sachdev–Ye–Kitaev inspired large N approaches have energized condensed matter studies of non-Fermi liquids and quantum criticality led by groups at Harvard University and MIT.
Computationally, the expansion employs diagrammatic classifications introduced by Gerard 't Hooft and algorithmic summations of leading planar graphs implemented in calculations at CERN and Brookhaven National Laboratory. Subleading 1/N corrections correspond to higher-genus surfaces and require techniques from Algebraic topology, advanced regularization schemes like those used in studies at Max Planck Institute for Physics, and numerical saddle-point evaluations akin to methods developed at Stanford University. Nonperturbative resumations, instanton analyses of suppressed contributions, and analytic continuation are standard tools used by theorists associated with institutions such as University of Cambridge and Caltech.
Criticisms concern the physical relevance of the N → ∞ limit for finite-N systems like Quantum Chromodynamics with N=3, and the convergence properties of 1/N series debated by researchers at Institute for Advanced Study and Perimeter Institute. Some phenomena—e.g., certain baryon dynamics, topology-changing transitions, and finite-N anomalies—may be nonuniform in 1/N and require complementary methods such as lattice simulations performed at CERN and Fermilab or nonperturbative bootstrap techniques advanced at KITP and Simons Foundation institutes.