Сейберг-Виттен теориясы: 4-массивтердегі қолданыстары

Theory in supersymmetric gauge theory
applications to 4 manifolds
In theoretical physics, Seiberg–Witten theory is an supersymmetric gauge theory with an exact low energy effective action (for massless degrees of freedom), of which the kinetic part coincides with the Kähler potential of the moduli space of vacua. Before taking the low energy effective action, the theory is known as supersymmetric Yang–Mills theory, as the field content is a single vector supermultiplet, analogous to the field content of Yang–Mills theory being a single vector gauge field (in particle theory language) or connection (in geometric language). The theory was studied in detail by Nathan Seiberg and Edward Witten .

The theory exhibits physical phenomena involving and linking magnetic monopoles, confinement, an attained mass gap and strong weak duality, described in section 5.6 of The study of these physical phenomena also motivated the theory of Seiberg–Witten invariants. The low energy action is described by the chiral multiplet with gauge group , the residual unbroken gauge from the original symmetry. This description is weakly coupled for large , but strongly coupled for small However, at the strongly coupled point the theory admits a dual description which is weakly coupled. The dual theory has different field content, with two chiral superfields , and gauge field the dual photon , with a potential that gives equations of motion which are Witten's monopole equations, also known as the Seiberg–Witten equations at the critical points where the monopoles become massless. In the context of Seiberg–Witten invariants, one can view Donaldson invariants as coming from a twist of the original theory at giving a topological field theory. On the other hand, Seiberg–Witten invariants come from twisting the dual theory at In theory, such invariants should receive contributions from all finite but in fact can be localized to the two critical points, and topological invariants can be read off from solution spaces to the monopole equations.

The special Kähler geometry on the moduli space of vacua in Seiberg–Witten theory can be identified with the geometry of the base of complex completely integrable system. The total phase of this complex completely integrable system can be identified with the moduli space of vacua of the 4d theory compactified on a circle. The relation between Seiberg–Witten theory and integrable systems has been reviewed by Eric D'Hoker and D. H. Phong. See Hitchin system.