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The Yang–Mills equations are gauge invariant. Mathematically, a gauge transformation is an automorphism of the principal bundle , and since the inner product on is invariant, the Yang–Mills functional satisfies. and so if satisfies (1), so does . There is a moduli space of Yang–Mills connections modulo gauge transformations.
The Yang–Mills existence and mass gap problem is an unsolved problem in mathematical physics and mathematics, and one of the seven Millennium Prize Problems defined by the Clay Mathematics Institute, which has offered a prize of US$1,000,000 for its solution. The problem is phrased as follows: [1] Yang–Mills Existence and Mass Gap.
Yang–Mills–Higgs equations. In mathematics, the Yang–Mills–Higgs equations are a set of non-linear partial differential equations for a Yang–Mills field, given by a connection, and a Higgs field, given by a section of a vector bundle (specifically, the adjoint bundle ). These equations are. with a boundary condition. where.
The Yang–Mills theory is a gauge theory based on a special unitary group SU (n), or more generally any compact Lie group. A Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U (1) × SU (2)) as ...
In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is ...
In mathematical physics, two-dimensional Yang–Mills theory is the special case of Yang–Mills theory in which the dimension of spacetime is taken to be two. This special case allows for a rigorously defined Yang–Mills measure, meaning that the (Euclidean) path integral can be interpreted as a measure on the set of connections modulo gauge transformations.
This theory is also important [1] in the context of the holographic principle. There is a duality between Type IIB string theory on AdS 5 × S 5 space (a product of 5-dimensional AdS space with a 5-dimensional sphere) and N = 4 super Yang–Mills on the 4-dimensional boundary of AdS 5. However, this particular realization of the AdS/CFT ...
Lipid raft organization, region (1) is a standard lipid bilayer, while region (2) is a lipid raft. The plasma membranes of cells contain combinations of glycosphingolipids, cholesterol and protein receptors organised in glycolipoprotein lipid microdomains termed lipid rafts. [1][2][3] Their existence in cellular membranes remains controversial.
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