Riassunto analitico
Equivariant localization theory is a powerful tool that has been extensively used in the past thirty years to elegantly obtain exact integration formulas, in both mathematical and physical literature. This integration formulas are proved within the mathematical formalism of equivariant cohomology, a variation of standard cohomology theory that incorporates the presence of a symmetry group acting on the space at hand. A suitable infinite-dimensional generalization of this formalism provides the correct algebraic description of supersymmetry in Quantum Field Theories. In this context, the integrals of interests are the infinite-dimensional path integrals describing partition functions or correlation functions of physical observables, and the localization theory gives a systematic approach to understand when the saddle point approximation, or “semiclassical” approximation, of those path integrals can give an exact result for the full quantum dynamics, as in the well-known case of certain topological field theories.
In this work we review the equivariant localization formalism and some of its possible applications to Quantum Mechanics and Quantum Field Theory. We start from a description of the mathematical framework of equivariant cohomology and related theorems of localization for integrals in finite-dimensional geometry, also making contact with the case of Hamiltonian group actions on symplectic manifolds. From a physical point of view, a natural equivariant cohomological structure arises when the analyzed system is equipped with a graded symmetry action, so when we have a supersymmetric theory. In the case of topological field theories, or generically BRST-fixed theories, this supersymmetry is “hidden” and associated to the gauge symmetry, while in other cases we can have explicit supersymmetries, as the super-extension of the Poincaré symmetry, which results in an invariance of the physical system when there is exchange between bosonic and fermionic degrees of freedom.
We will describe the formal application of the localization principle to path integral in supersymmetric Quantum Mechanics and Quantum Field Theory when the considered group action is a fermionic U(1) symmetry, and give examples of localization in gauge theories with extended supersymmetries. Here one is able to give exact results for the partition function or the expectation value of certain supersymmetric physical operators, such as combination of local fields, supersymmetric Wilson loops or other non-local operators. Usually in those theories the localization principle is applicable when the path integral is free of infrared divergences, so when we consider the theory on a compact manifold. We are then forced to analyze those cases in which some amount of supersymmetry survives when the theory is formulated on curved space.
Finally, we consider the case in which the localization principle is extended to a bigger and possibly non-abelian group action. An important application of this principle gives explicitly the mapping between “topological” and “physical” Yang Mills theory in 2 dimensions.
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