Gauge theory (mathematics)

rdf:langString Gauge theory (mathematics)

rdf:langString The dx1⊗σ3 coefficient of a BPST instanton on the '-slice of R4 where σ3 is the third Pauli matrix . The dx2⊗σ3 coefficient . These coefficients determine the restriction of the BPST instanton A with g=2,ρ=1,z=0 to this slice. The corresponding field strength centered around z=0 . A visual representation of the field strength of a BPST instanton with center z on the compactification S4 of R4 . The BPST instanton is a classical instanton solution to the Yang–Mills equations on R'''4.

rdf:langString -y- plot; BPST instanton.png

rdf:langString BPST on sphere.png

rdf:langString Curvature of BPST Instanton.png

rdf:langString X- plot; BPST instanton.png

rdf:langString In mathematics, and especially differential geometry and mathematical physics, gauge theory is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics theory means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a mathematical model of some natural phenomenon. Gauge theory in mathematics is typically concerned with the study of gauge-theoretic equations. These are differential equations involving connections on vector bundles or principal bundles, or involving sections of vector bundles, and so there are strong links between gauge theory and geometric analysis. These equations are often physically meaningful, corresponding to important concepts in quantum field theory or string theory, but also have important mathematical significance. For example, the Yang–Mills equations are a system of partial differential equations for a connection on a principal bundle, and in physics solutions to these equations correspond to vacuum solutions to the equations of motion for a classical field theory, particles known as instantons. Gauge theory has found uses in constructing new invariants of smooth manifolds, the construction of exotic geometric structures such as hyperkähler manifolds, as well as giving alternative descriptions of important structures in algebraic geometry such as moduli spaces of vector bundles and coherent sheaves.