Gauge theory (mathematics)

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. Gauge theory has its origins as far back as the formulation of Maxwell's equations describing classical electromagnetism, which may be phrased as a gauge theory with structure group the circle group. Work of Paul Dirac on magnetic monopoles and relativistic quantum mechanics encouraged the idea that bundles and connections were the correct way of phrasing many problems in quantum mechanics.

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Gauge group (mathematics)

A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle with a structure Lie group , a gauge group is defined to be a group of its vertical automorphisms. This group is isomorphic to the group of global sections of the associated group bundle whose typical fiber is a group which acts on itself by the adjoint representation. The unit element of is a constant unit-valued section of .

Adjoint bundle

In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge theory. Let G be a Lie group with Lie algebra , and let P be a principal G-bundle over a smooth manifold M. Let be the (left) adjoint representation of G.

Hitchin's equations

In mathematics, and in particular differential geometry and gauge theory, Hitchin's equations are a system of partial differential equations for a connection and Higgs field on a vector bundle or principal bundle over a Riemann surface, written down by Nigel Hitchin in 1987. Hitchin's equations are locally equivalent to the harmonic map equation for a surface into the symmetric space dual to the structure group.

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