Gauge theory (mathematics)

Gauge theory is an important area of mathematics, especially in the fields of differential geometry and mathematical physics. It studies special mathematical objects called connections on structures known as bundles, including vector bundles, principal bundles, and fibre bundles. While this concept shares a name with gauge theory in physics, in mathematics it refers to a whole theory that explores these connections and their properties.

In mathematics, gauge theory often involves studying special equations called gauge-theoretic equations. These are differential equations related to connections on bundles or to sections of these bundles. This connects gauge theory closely with geometric analysis. Some of these equations, like the Yang–Mills equations, are not just mathematical curiosities; they also relate to important ideas in physics, such as quantum field theory and string theory.

Beyond physics, gauge theory has many uses in pure mathematics. It helps mathematicians create new ways to measure and describe shapes called smooth manifolds. It also helps build unusual and interesting geometric structures, such as hyperkähler manifolds. Gauge theory even provides new ways to understand important concepts in algebraic geometry, like moduli spaces and coherent sheaves.

History

Gauge theory started with the ideas behind Maxwell's equations, which describe how electric and magnetic fields work. These ideas were later expanded by scientists like Paul Dirac and Chen-Ning Yang.

In mathematics, important work by Michael Atiyah and others showed how these ideas could help solve problems in geometry. Later, Simon Donaldson used these ideas to discover new facts about shapes in space, for which he received a major award in mathematics.

Fundamental objects of interest

Gauge theory studies important mathematical structures called connections on vector bundles and principal bundles. These bundles are ways of attaching extra data to a space in a consistent manner.

Principal bundles are built using a group of symmetries, while vector bundles use vector spaces. Both are central to understanding how different parts of a space relate to each other in modern geometry and physics.

Notational conventions

Different symbols are used in math to describe connections on bundles. The letter A often stands for a connection. If you pick a basic connection ∇₀, any other connection can be written as ∇₀ + A.

The symbol ∇ is also used for connections, acting like a math operator. Other notations like ∇ₐ or Dₐ may be used to show the connection depends on A.

The operator dₐ refers to the outer covariant derivative of a connection A. The symbol Fₐ or F∇ stands for the curvature of a connection. When the connection is written as ω, its curvature may be written as Ω.

The letter H is sometimes used for a special kind of connection called an Ehresmann connection. The math object ad(P) refers to the adjoint bundle, while Ad(P) refers to the adjoint bundle of a Lie group.

Dictionary of mathematical and physical terminology

Math and physics use different names for the same ideas in gauge theory. For example, in physics an interacting term in the Lagrangian of quantum electrodynamics might look like:

L = ψ̄( i γ^μ D_μ − m )ψ − 1/4 F_μν F^μν

In math, the same idea might be written as:

L = ⟨ψ, (D!/ₐ − m)ψ⟩ₗ² + ‖Fₐ‖ₗ²²

Here A is a connection on a special bundle U(1), ψ represents an electron-positron field, and Dₐ is linked to the connection ∇ₐ. The first part shows how the electron-positron field interacts with the electromagnetic field, while the second part describes the electromagnetic field’s basic properties.

Yang–Mills theory

Main article: Yang–Mills equations

See also: Yang–Mills theory

Yang–Mills theory is a key idea in mathematical gauge theory. It looks at special kinds of connections, called Yang–Mills connections, which are solutions to important equations known as the Yang–Mills equations. These connections help us understand geometry better by finding shapes where curvature is as small as possible.

When we work with four-dimensional spaces, we can simplify these equations using something called the Hodge star operator. This leads to self-dual and anti-self-dual connections, which are special solutions that make the equations easier to study. By looking at these simpler cases, mathematicians can explore many interesting properties of gauge theory in different dimensions.

Gauge theory in one and two dimensions

Gauge theory studies special kinds of mathematical structures called "connections" on various bundles. When these structures are examined in spaces with only one or two dimensions, the equations that describe them become much simpler.

One important area of study is the Yang–Mills equations. These equations describe connections on two-dimensional surfaces, like the surface of a sphere. Researchers like Michael Atiyah and Raoul Bott explored these equations, finding that they connect to other mathematical ideas such as representations of groups and complex geometries. Another important concept is the Nahm equations, which relate to magnetic monopoles and can be solved using simple mathematical methods. These equations also connect to other areas of mathematics, showing the deep links between different parts of the subject.

Gauge theory in three dimensions

Main article: Bogomolny equations

Gauge theory in three dimensions studies special kinds of mathematical equations. One important area is about monopoles, which are like imaginary magnetic particles that help scientists understand how magnetic fields work. These monopoles come from solving a set of equations called the Bogomolny equations, which come from a more general theory in higher dimensions. When the equations use a special kind of mathematical group called the circle group, they describe a magnetic monopole. With another group called SU(2), the solutions connect to other mathematical ideas.

Another important part of gauge theory in three dimensions is Chern–Simons theory. This theory looks at how certain mathematical objects behave on shapes with no edges, like a sphere. It helps connect geometry and topology, showing how knots can be described using these theories. This work showed that gauge theory could solve problems in shapes and spaces.

Main article: Chern–Simons theory

Gauge theory in four dimensions

Gauge theory has been most studied in four dimensions, where it connects closely with physics, especially the Standard Model of particle physics which can be seen as a theory on a four-dimensional spacetime. In this setting, gauge theory helps us understand topological quantum field theory, which looks at properties of space that don’t change with shape.

One important part of gauge theory in four dimensions involves the anti-self-duality equations. These equations simplify the Yang–Mills equations and help find special solutions. They are especially useful when studying simply connected spaces, like certain four-dimensional shapes. This work shows how the shape of space can affect what smooth structures it can have, unlike in lower dimensions where the shape and smoothness match more closely.

Gauge theory in higher dimensions

Main article: Hermitian Yang–Mills equations

Gauge theory in higher dimensions looks at special math rules called Hermitian Yang–Mills equations. These rules help us understand complex shapes called Kähler and Hermitian manifolds. They connect ideas from lower dimensions to work in many more dimensions.

New problems in gauge theory come from ideas in superstring theory. This theory imagines our universe as having ten dimensions, with six extra small ones shaped like Calabi–Yau manifolds. The math from these extra dimensions uses gauge theory to describe how things behave in these spaces.