Gauge theory (mathematics)

Gauge theory is an important area of mathematics, especially in differential geometry and mathematical physics. It studies special mathematical objects called connections on structures known as bundles. These bundles include vector bundles, principal bundles, and fibre bundles.

In mathematics, gauge theory looks at special equations called gauge-theoretic equations. These are differential equations related to connections on bundles. This connects gauge theory to geometric analysis. Some of these equations, like the Yang–Mills equations, are important in physics too. They relate to ideas such as quantum field theory and string theory.

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

History

Gauge theory began with ideas from Maxwell's equations. These equations explain how electric and magnetic fields work. Later, scientists like Paul Dirac and Chen-Ning Yang expanded on these ideas.

In mathematics, Michael Atiyah and others showed how these ideas could solve geometry problems. After that, Simon Donaldson used these ideas to find new facts about shapes in space. He received a big award in mathematics for this work.

Fundamental objects of interest

Gauge theory looks at special mathematical ideas called connections on things named vector bundles and principal bundles. These bundles are ways to add extra information to a space in a neat way.

Principal bundles use a group of symmetries to work, while vector bundles use vector spaces. Both help us understand how different parts of a space connect 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 an important idea in mathematical gauge theory. It studies special kinds of connections, called Yang–Mills connections. These connections are solutions to important equations known as the Yang–Mills equations. They 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 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 learn more about gauge theory in different dimensions.

Gauge theory in one and two dimensions

Gauge theory looks at special math structures called "connections" on bundles. When we study these in spaces with just one or two dimensions, the math becomes easier.

One big topic is the Yang–Mills equations. These describe connections on two-dimensional surfaces, like a sphere. Researchers like Michael Atiyah and Raoul Bott studied these equations. They found links to other math ideas, such as group representations and complex geometries. Another key concept is the Nahm equations, which relate to magnetic monopoles and can be solved with simple math. These equations also show how different parts of math are connected.

Gauge theory in three dimensions

Main article: Bogomolny equations

Gauge theory in three dimensions studies special kinds of math problems. One important idea is about monopoles. These are like imaginary magnetic particles that help us understand magnetic fields better. Monopoles come from solving equations called the Bogomolny equations. These equations come from a bigger theory in higher dimensions. When the equations use a special math group called the circle group, they describe a magnetic monopole. With another group called SU(2), the answers connect to other math ideas.

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

Main article: Chern–Simons theory

Gauge theory in four dimensions

Gauge theory is most often studied in four dimensions. In this area, it connects closely with physics, especially the Standard Model of particle physics. This model can be seen as a theory on a four-dimensional spacetime. Here, gauge theory helps us understand topological quantum field theory. This theory looks at properties of space that stay the same even when the shape changes.

One important part of gauge theory in four dimensions is the anti-self-duality equations. These equations make the Yang–Mills equations simpler and help find special solutions. They are very useful when studying simply connected spaces, like some four-dimensional shapes. This work shows how the shape of space can affect its smooth structures, which is different from lower dimensions where shape and smoothness are more closely related.

Gauge theory in higher dimensions

Main article: Hermitian Yang–Mills equations

Gauge theory in higher dimensions studies special math rules called Hermitian Yang–Mills equations. These rules help us understand complex shapes called Kähler and Hermitian manifolds. They link ideas from simpler spaces to those with many more dimensions.

New questions in gauge theory arise from ideas in superstring theory. This theory suggests our universe might have ten dimensions, with six tiny ones shaped like Calabi–Yau manifolds. The math from these extra dimensions uses gauge theory to explain how things act in these spaces.