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Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the ...
Through the process of dimensional reduction, the Yang–Mills equations may be used to derive other important equations in differential geometry and gauge theory. Dimensional reduction is the process of taking the Yang–Mills equations over a four-manifold, typically R 4 {\displaystyle \mathbb {R} ^{4}} , and imposing that the solutions be ...
N = 4 super Yang–Mills can be derived from a simpler 10-dimensional theory, and yet supergravity and M-theory exist in 11 dimensions. The connection is that if the gauge group U( N ) of SYM becomes infinite as N → ∞ {\displaystyle N\rightarrow \infty } it becomes equivalent to an 11-dimensional theory known as matrix theory .
A well understood and illustrative example of an instanton and its interpretation can be found in the context of a QFT with a non-abelian gauge group, [note 2] a Yang–Mills theory. For a Yang–Mills theory these inequivalent sectors can be (in an appropriate gauge) classified by the third homotopy group of SU(2) (whose group manifold is the ...
The notion of a Hermitian Yang–Mills connection is a specification of a Yang–Mills connection to the case of a Hermitian vector bundle over a complex manifold. It is possible to phrase the definition in terms of either the Hermitian metric itself, or its associated Chern connection , and the two notions are essentially equivalent up to ...
While sharing an office at Brookhaven National Laboratory, Frank Yang Chen-Ning and Robert Mills formulated in 1954 a theory now known as the Yang–Mills theory – "the foundation for current understanding of how subatomic particles interact, a contribution which has restructured modern physics and mathematics." [1]
In mathematics, and especially differential topology and gauge theory, Donaldson's theorem states that a definite intersection form of a compact, oriented, smooth manifold of dimension 4 is diagonalizable. If the intersection form is positive (negative) definite, it can be diagonalized to the identity matrix (negative identity matrix) over the ...
In gauge theory, topological Yang–Mills theory, also known as the theta term or -term is a gauge-invariant term which can be added to the action for four-dimensional field theories, first introduced by Edward Witten. [1]