| Pestun's localization | |
|---|---|
| Name | Pestun's localization |
| Field | Quantum field theory |
| Introduced | 2007 |
| Introduced by | Vasily Pestun |
| Notable results | Exact expectation values of supersymmetric Wilson loops on S^4 |
| Related | Supersymmetric gauge theory, Localization |
Pestun's localization
Pestun's localization is a technique introduced by Vasily Pestun that applies supersymmetric localization to certain four-dimensional supersymmetric gauge theorys, producing exact results for partition functions and protected observables. It matters in the context of Quantum field theory and Quantum Physics because it connects nonperturbative features of gauge theories to finite-dimensional integrals and to deep mathematical structures such as matrix models and equivariant cohomology.
Pestun's localization sits at the intersection of path integral methods in Quantum field theory and rigorous techniques from equivariant geometry. Prior work on localization in lower dimensions—most notably the work of Edward Witten on topological field theory and the Atiyah–Bott fixed-point theorem—provided conceptual foundations. The result fills a longstanding need for exact nonperturbative observables in four-dimensional N=2 supersymmetric gauge theories, complementing semiclassical expansions and lattice approaches championed by institutions such as CERN and research groups at universities like Princeton University and Harvard University.
Supersymmetric localization exploits a fermionic symmetry Q whose square generates compact bosonic transformations, typically including rotations and R-symmetry actions. By adding a Q-exact term to the action, path integrals localize to field configurations annihilated by Q. This technique draws on mathematical results from Equivariant cohomology and the Atiyah–Bott fixed-point theorem and builds on earlier applications in two and three dimensions by authors including Nikita Nekrasov and developments in the Omega background. Pestun adapted these principles to four dimensions, using rigid supersymmetry on curved backgrounds following formalisms developed by Festuccia and Seiberg and techniques connected to the study of supersymmetric backgrounds on spheres by groups at institutions such as IHES and Cambridge University.
Pestun formulated localization for N=2 theories on the round four-sphere S^4 by coupling the gauge theory to a specific background supergravity multiplet and choosing a supercharge Q with isolated fixed points. The localization locus reduces to solutions parameterized by constant values of the adjoint scalar in the vector multiplet and includes pointlike instanton contributions localized at the north and south poles; these instantons are evaluated via Nekrasov's instanton partition functions from the Nekrasov partition function framework. The central output is an exact expression for the partition function Z_{S^4} and for supersymmetric Wilson loop expectation values in representations of the gauge group such as SU(N), SO(N), or Sp(N). These results offered precise checks of proposed dualities including versions of the S-duality and contributed to tests of the AGT correspondence relating four-dimensional N=2 theories to two-dimensional Liouville field theory and Toda conformal field theories.
A hallmark of Pestun's result is the reduction of the infinite-dimensional path integral to a finite-dimensional matrix model: an integral over the Cartan subalgebra of the gauge group with a specific one-loop determinant and instanton factor. The matrix model captures expectation values of protected operators, notably supersymmetric Wilson loops (compare with tests in AdS/CFT correspondence where Wilson loops relate to string worldsheets in AdS_5 × S^5). The one-loop determinants connect to special functions studied in mathematical physics such as Barnes double gamma functions and to the Schur limit of superconformal indices. Observables computed via Pestun's localization provided data for checks against results from integrability in planar N=4 supersymmetric Yang–Mills theory and from strong–weak coupling dualities investigated by teams at Perimeter Institute and Simons Center for Geometry and Physics.
Following Pestun's work, numerous generalizations were developed: localization on squashed spheres and ellipsoids, incorporation of surface operators and defects, and applications to three-dimensional theories via dimensional reduction. Notable extensions include work by Hama, Hosomichi, and Lee on three-sphere localization, and by Drukker and collaborators on Wilson loop families. Applications span checks of the AGT correspondence, computations of supersymmetric indices, derivations of factorization formulas, and connections to topological string theory and matrix model dualities investigated at institutions like Caltech and MIT. Pestun-type results also informed exact computations in supersymmetric renormalization group flows and in studies of anomalies and central charges in superconformal field theories.
The formal structure of Pestun's localization links to rigorous mathematics: equivariant index theory, deformation quantization, and geometric representation theory. The instanton contributions invoke moduli spaces of framed instantons on R^4 and use localization in the sense of Nekrasov and Okounkov; the matrix model integrals relate to orthogonal polynomials and spectral theory. Efforts to place Pestun's derivation on fully rigorous footing involve analysis of determinant regularization, elliptic complexes on curved manifolds, and the role of boundary conditions. Collaborations between mathematical physicists at places such as MPI for Mathematics, IHES, and university research groups continue to clarify the link between Pestun's practical computations and deep results in modern geometry and representation theory.
Category:Quantum field theory Category:Supersymmetry Category:Mathematical physics