Gauge theory (mathematics)

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 in mathematical physics arose as a significant field of study with the seminal work of Robert Mills and Chen-Ning Yang on so-called Yang–Mills gauge theory, which is now the fundamental model that underpins the standard model of particle physics.

[1] The mathematical investigation of gauge theory has its origins in the work of Michael Atiyah, Isadore Singer, and Nigel Hitchin on the self-duality equations on a Riemannian manifold in four dimensions.

Around the same time Atiyah and Richard Ward discovered links between solutions to the self-duality equations and algebraic bundles over the complex projective space

[4] Another significant early discovery was the development of the ADHM construction by Atiyah, Vladimir Drinfeld, Hitchin, and Yuri Manin.

At this time the important work of Atiyah and Raoul Bott about the Yang–Mills equations over Riemann surfaces showed that gauge theoretic problems could give rise to interesting geometric structures, spurring the development of infinite-dimensional moment maps, equivariant Morse theory, and relations between gauge theory and algebraic geometry.

[7] The most significant advancements in the field occurred due to the work of Simon Donaldson and Edward Witten.

[11][12] Strong links to algebraic geometry were realised by the work of Donaldson, Uhlenbeck, and Shing-Tung Yau on the Kobayashi–Hitchin correspondence relating Yang–Mills connections to stable vector bundles.

[13][14] Work of Nigel Hitchin and Carlos Simpson on Higgs bundles demonstrated that moduli spaces arising out of gauge theory could have exotic geometric structures such as that of hyperkähler manifolds, as well as links to integrable systems through the Hitchin system.

The structures described here are standard within the differential geometry literature, and an introduction to the topic from a gauge-theoretic perspective can be found in the book of Donaldson and Peter Kronheimer.

The choice of which to study is essentially arbitrary, as one may pass between them, but principal bundles are the natural objects from the physical perspective to describe gauge fields, and mathematically they more elegantly encode the corresponding theory of connections and curvature for vector bundles associated to them.

However vector bundle connections admit a more powerful description in terms of a differential operator.

The most famous application of this idea is Donaldson's theorem, which uses the moduli space of Yang–Mills connections on a principal

The mathematical and physical fields of gauge theory involve the study of the same objects, but use different terminology to describe them.

(e.g. the covariant derivative applied to a section of an associated bundle, or a multiplication of two terms) As a demonstration of this dictionary, consider an interacting term of an electron-positron particle field and the electromagnetic field in the Lagrangian of quantum electrodynamics:[19] Mathematically this might be rewritten where

The second term is the regular Yang–Mills functional which describes the basic non-interacting properties of the electromagnetic field (the connection

, so connections which are critical points of this function are those with curvature as small as possible (or higher local minima of

Hodge theory provides a unique harmonic representative of every de Rham cohomology class

The theory of Yang–Mills equations when the base manifold is a compact Riemann surface was carried about by Michael Atiyah and Raoul Bott.

admits various rich interpretations, and the theory serves as the simplest case to understand the equations in higher dimensions.

Such connections are called projectively flat, and in the case where the vector bundle is topologically trivial (so

Simon Donaldson gave an alternative proof of the theorem of Narasimhan and Seshadri that directly passed from Yang–Mills connections to stable holomorphic structures.

[21] Atiyah and Bott used this rephrasing of the problem to illuminate the intimate relationship between the extremal Yang–Mills connections and the stability of the vector bundles, as an infinite-dimensional moment map for the action of the gauge group

This observation phrases the Narasimhan–Seshadri theorem as a kind of infinite-dimensional version of the Kempf–Ness theorem from geometric invariant theory, relating critical points of the norm squared of the moment map (in this case Yang–Mills connections) to stable points on the corresponding algebraic quotient (in this case stable holomorphic vector bundles).

The moduli space of Hitchin pairs naturally has (when the rank and degree of the bundle are coprime) the structure of a Kähler manifold.

Chern–Simons theory was used by Edward Witten to express the Jones polynomial, a knot invariant, in terms of the vacuum expectation value of a Wilson loop in

An extension of these ideas leads to Donaldson theory, which constructs further invariants of smooth four-manifolds out of the moduli spaces of connections over them.

These invariants are obtained by evaluating cohomology classes on the moduli space against a fundamental class, which exists due to analytical work showing the orientability and compactness of the moduli space by Karen Uhlenbeck, Taubes, and Donaldson.

on the underlying four-manifold, and choice of perturbing two-form, the moduli space of solutions is a compact smooth manifold.

In such theories the fields which act on strings live on bundles over these higher dimensional spaces, and one is interested in gauge-theoretic problems relating to them.

Non-trivial Z /2 Z principal bundle over the circle. There is no obvious way to identify which point corresponds to +1 or -1 in each fibre. This bundle is non-trivial as there is no globally defined section of the projection π .
The frame bundle of the Möbius strip is a non-trivial principal -bundle over the circle.
A vector bundle over a base with a section .
A principal bundle connection is required to be compatible with the right group action of on . This can be visualized as the right multiplication taking the horizontal subspaces into each other. This equivariance of the horizontal subspaces interpreted in terms of the connection form leads to its characteristic equivariance properties.
A principal bundle connection form may be thought of as a projection operator on the tangent bundle of the principal bundle . The kernel of the connection form is given by the horizontal subspaces for the associated Ehresmann connection .
The covariant derivative of a connection on a vector bundle may be recovered from its parallel transport. The values of a section are parallel transported along the path back to , and then the covariant derivative is taken in the fixed vector space, the fibre over .
Cobordism given by moduli space of anti-self-dual connections in Donaldson's theorem