12.4 Gauge theory and fiber bundles

Gauge theory and fiber bundles are powerful tools in differential geometry, unifying various geometric structures through connections on principal bundles. They provide a framework for understanding Riemannian metrics, symplectic forms, and complex structures, with applications in mathematical physics.

Fiber bundles generalize product spaces, allowing fibers to vary smoothly over a base manifold. This concept is crucial for studying geometric structures like vector fields, differential forms, and metrics that change across a manifold's surface.

Gauge theory fundamentals

• Gauge theory is a fundamental tool in modern differential geometry that describes the geometry of fiber bundles and their connections
• It provides a unified framework for understanding various geometric structures, such as Riemannian metrics, symplectic forms, and complex structures, in terms of the properties of connections on principal bundles
• Gauge theory has important applications in mathematical physics, particularly in the study of Yang-Mills fields and the geometry of four-manifolds

Connections on principal bundles

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• A connection on a principal bundle is a way to specify how to lift paths from the base manifold to the total space of the bundle in a consistent manner
• Connections allow for the definition of parallel transport, which moves elements along paths in the base manifold while respecting the bundle structure
• The space of connections on a principal bundle is an affine space, with the difference between two connections being a tensorial 1-form with values in the Lie algebra of the structure group

Curvature of connections

• The curvature of a connection measures the extent to which parallel transport around infinitesimal loops fails to preserve the horizontal subspaces of the bundle
• Curvature is a tensorial 2-form with values in the Lie algebra of the structure group, satisfying the Bianchi identity
• Flat connections are those with vanishing curvature, and they correspond to locally trivial bundles with constant transition functions

Chern-Weil theory

• Chern-Weil theory is a powerful tool that relates the topology of a principal bundle to the curvature of a connection on the bundle
• It provides a way to construct characteristic classes, which are cohomology classes that measure the non-triviality of the bundle
• The Chern-Weil homomorphism maps invariant polynomials on the Lie algebra of the structure group to cohomology classes of the base manifold

Characteristic classes

• Characteristic classes are cohomological invariants associated to principal bundles, which provide obstructions to the existence of certain geometric structures on the bundle (flat connections, sections, etc.)
• Examples of characteristic classes include the Euler class for oriented vector bundles, Chern classes for complex vector bundles, and Pontryagin classes for real vector bundles
• Characteristic classes are natural under pull-backs and satisfy certain axioms, such as the Whitney product formula and the splitting principle

Fiber bundles

• Fiber bundles are a fundamental concept in differential geometry that generalize the notion of a product space by allowing the fibers to vary smoothly over the base manifold
• A fiber bundle consists of a total space, a base manifold, and a projection map satisfying the local triviality condition
• Fiber bundles provide a natural framework for studying geometric structures that vary smoothly over a manifold, such as vector fields, differential forms, and metrics

Definition and examples

• A fiber bundle is a tuple $(E,B,\pi,F)$ where $E$ is the total space, $B$ is the base manifold, $\pi:E\to B$ is a continuous surjection, and $F$ is the typical fiber, such that every point in $B$ has a neighborhood $U$ for which $\pi^{-1}(U)$ is homeomorphic to $U\times F$
• Examples of fiber bundles include the Möbius strip (non-orientable bundle over $S^1$ with fiber $\mathbb{R}$), the tangent bundle of a manifold (bundle over $M$ with fiber $\mathbb{R}^n$), and the frame bundle of a vector bundle (principal $GL(n,\mathbb{R})$-bundle)

Local trivializations

• A local trivialization of a fiber bundle is a collection of open sets $\{U_i\}$ covering the base manifold, together with homeomorphisms $\phi_i:\pi^{-1}(U_i)\to U_i\times F$ that commute with the projection onto $U_i$
• Local trivializations provide a way to work with fiber bundles locally as product spaces, which is often easier than working with the global bundle structure
• The existence of local trivializations is equivalent to the local triviality condition in the definition of a fiber bundle

Transition functions

• Transition functions are the data that encode how the local trivializations of a fiber bundle are related on the overlaps of their domains
• Given two local trivializations $\phi_i$ and $\phi_j$ over overlapping sets $U_i$ and $U_j$, the transition function $g_{ij}:U_i\cap U_j\to G$ is defined by $\phi_i\circ\phi_j^{-1}(x,f)=(x,g_{ij}(x)\cdot f)$, where $G$ is the structure group of the bundle acting on the fibers
• Transition functions satisfy the cocycle condition $g_{ij}g_{jk}=g_{ik}$ on triple overlaps, and they completely determine the bundle up to isomorphism

Associated bundles

• Given a principal $G$-bundle $P\to M$ and a left $G$-space $F$, the associated bundle is the fiber bundle $P\times_G F\to M$ obtained by taking the quotient of $P\times F$ by the diagonal $G$-action
• The fiber of the associated bundle over a point $x\in M$ is the orbit space $F_x=P_x\times_G F$, where $P_x$ is the fiber of $P$ over $x$ and $G$ acts on $F$ via the given left action
• Many important fiber bundles in differential geometry, such as vector bundles and principal bundles, can be constructed as associated bundles to a given principal bundle

Principal bundles

• Principal bundles are a special class of fiber bundles where the fibers are equipped with a free and transitive action of a Lie group $G$, called the structure group
• The total space of a principal bundle is a manifold $P$ on which $G$ acts freely and properly from the right, with orbit space isomorphic to the base manifold $M$
• Principal bundles provide a natural framework for studying geometric structures that are invariant under the action of a Lie group, such as Riemannian metrics, connections, and gauge fields

Definition and properties

• A principal $G$-bundle is a fiber bundle $\pi:P\to M$ together with a free right action of $G$ on $P$ that preserves the fibers and is compatible with the local trivializations
• The fibers of a principal bundle are diffeomorphic to the structure group $G$, and the action of $G$ on each fiber is free and transitive
• Principal bundles satisfy the local triviality condition, meaning that every point in the base manifold has a neighborhood $U$ such that $\pi^{-1}(U)$ is $G$-equivariantly diffeomorphic to $U\times G$

Principal G-bundles

• A principal $G$-bundle is completely determined by its transition functions, which are smooth maps $g_{ij}:U_i\cap U_j\to G$ satisfying the cocycle condition
• The set of isomorphism classes of principal $G$-bundles over a manifold $M$ is in bijection with the set of conjugacy classes of homomorphisms from the fundamental group of $M$ to $G$
• Principal $G$-bundles can be constructed by specifying a $G$-valued Čech 1-cocycle on a suitable open cover of the base manifold

Gauge transformations

• A gauge transformation of a principal $G$-bundle $P\to M$ is a $G$-equivariant diffeomorphism $f:P\to P$ that preserves the fibers and covers the identity map on $M$
• The group of gauge transformations, denoted $\mathcal{G}(P)$, is the group of smooth sections of the associated bundle $P\times_G G\to M$, where $G$ acts on itself by conjugation
• Gauge transformations act on the space of connections on $P$ by pulling back the connection 1-forms, and the curvature of a connection transforms in a covariant way under gauge transformations

Reduction of structure group

• A reduction of the structure group of a principal $G$-bundle $P\to M$ to a subgroup $H\subset G$ is a principal $H$-subbundle $Q\subset P$ such that $P\cong Q\times_H G$
• Reductions of structure group correspond to additional geometric structures on the bundle that are compatible with the $G$-action, such as orientation, Riemannian metrics, or almost complex structures
• The existence of a reduction of structure group can be characterized by the vanishing of certain characteristic classes of the bundle, such as the first Stiefel-Whitney class for orientability or the second Stiefel-Whitney class for spin structures

Vector bundles

• Vector bundles are fiber bundles whose fibers are vector spaces and whose transition functions are linear maps between the fibers
• They provide a natural framework for studying geometric objects that vary smoothly over a manifold, such as vector fields, differential forms, and metrics
• Many important constructions in differential geometry, such as the tangent bundle and the cotangent bundle, are examples of vector bundles

Definition and examples

• A vector bundle of rank $n$ over a manifold $M$ is a fiber bundle $\pi:E\to M$ such that each fiber $E_x=\pi^{-1}(x)$ is an $n$-dimensional vector space, and the transition functions are linear isomorphisms between the fibers
• Examples of vector bundles include the trivial bundle $M\times\mathbb{R}^n\to M$, the tangent bundle $TM\to M$, the cotangent bundle $T^*M\to M$, and the bundle of $k$-forms $\Lambda^k(T^*M)\to M$

Frame bundles

• The frame bundle of a vector bundle $E\to M$ is the principal $GL(n,\mathbb{R})$-bundle $F(E)\to M$ whose fiber over a point $x\in M$ consists of all ordered bases (frames) of the vector space $E_x$
• A section of the frame bundle is called a frame field, and it provides a way to trivialize the vector bundle locally
• The frame bundle of a manifold $M$ is the frame bundle of its tangent bundle, and it plays a fundamental role in the study of Riemannian geometry and general relativity

Sections of vector bundles

• A section of a vector bundle $E\to M$ is a smooth map $s:M\to E$ such that $\pi\circ s=id_M$, where $\pi:E\to M$ is the projection map
• The space of sections of a vector bundle, denoted $\Gamma(E)$, is an infinite-dimensional vector space over $\mathbb{R}$ (or $\mathbb{C}$ for complex vector bundles) and a module over the ring of smooth functions on $M$
• Many important geometric objects, such as vector fields, differential forms, and metrics, can be viewed as sections of appropriate vector bundles

Tangent and cotangent bundles

• The tangent bundle of a manifold $M$ is the vector bundle $TM\to M$ whose fiber over a point $x\in M$ is the tangent space $T_xM$
• The cotangent bundle of $M$ is the dual vector bundle $T^*M\to M$, whose fiber over $x$ is the dual space $(T_xM)^*$ of linear functionals on $T_xM$
• The tangent and cotangent bundles are fundamental objects in differential geometry, as they encode the infinitesimal structure of the manifold and allow for the definition of important geometric notions such as flows, Lie derivatives, and differential forms

Connections and curvature

• Connections on vector bundles provide a way to differentiate sections of the bundle and to compare fibers at different points of the base manifold
• The curvature of a connection measures the extent to which parallel transport around infinitesimal loops fails to preserve the fibers, and it encodes important geometric and topological information about the bundle
• Connections and curvature play a central role in gauge theory and the study of geometric structures on manifolds, such as Riemannian metrics and symplectic forms

Connections on vector bundles

• A connection on a vector bundle $E\to M$ is a linear map $\nabla:\Gamma(E)\to\Gamma(T^*M\otimes E)$ satisfying the Leibniz rule $\nabla(fs)=df\otimes s+f\nabla s$ for all smooth functions $f$ and sections $s$
• Connections allow for the definition of parallel transport, which is a way to move elements of the fibers along paths in the base manifold while preserving the linear structure of the fibers
• The space of connections on a vector bundle is an affine space, with the difference between two connections being a tensorial 1-form with values in the endomorphism bundle $End(E)$

Covariant derivatives

• The covariant derivative of a section $s$ of a vector bundle $E$ with respect to a connection $\nabla$ is the section $\nabla s$ of the tensor product bundle $T^*M\otimes E$
• In local coordinates, the covariant derivative is given by $\nabla_{\partial_i}s=\partial_i s+A_i s$, where $A_i$ are the connection coefficients (Christoffel symbols) and $\partial_i$ are the local coordinate vector fields
• The covariant derivative satisfies the Leibniz rule and is compatible with the tensor product and contraction operations on vector bundles

Parallel transport

• Parallel transport along a curve $\gamma:[0,1]\to M$ with respect to a connection $\nabla$ on a vector bundle $E\to M$ is a linear isomorphism $P_\gamma:E_{\gamma(0)}\to E_{\gamma(1)}$ between the fibers over the endpoints of the curve
• Parallel transport is defined by solving the linear ODE $\nabla_{\dot\gamma}s=0$ for sections $s$ of $E$ along $\gamma$, with initial condition $s(\gamma(0))=v$ for some $v\in E_{\gamma(0)}$
• Parallel transport is compatible with composition of curves and is invariant under reparametrization, making it a fundamental tool for comparing fibers at different points of the base manifold

Curvature tensor

• The curvature of a connection $\nabla$ on a vector bundle $E\to M$ is the tensorial 2-form $R\in\Omega^2(M,End(E))$ defined by $R(X,Y)s=\nabla_X\nabla_Y s-\nabla_Y\nabla_X s-\nabla_{[X,Y]}s$ for all vector fields $X,Y$ and sections $s$
• In local coordinates, the curvature tensor is given by $R_{ij}=\partial_i A_j-\partial_j A_i+[A_i,A_j]$, where $A_i$ are the connection coefficients and $[\cdot,\cdot]$ is the Lie bracket of matrices
• The curvature tensor measures the non-commutativity of the covariant derivative and the failure of parallel transport around infinitesimal loops to preserve the fibers

Bianchi identities

• The Bianchi identities are a set of relations satisfied by the curvature tensor of a connection on a vector bundle, expressing its algebraic and differential properties
• The first Bianchi identity states that the curvature tensor is antisymmetric in its last two arguments, i.e., $R(X,Y)=-R(Y,X)$ for all vector fields $X,Y$
• The second Bianchi identity, also known as the differential Bianchi identity, relates the covariant derivative of the curvature tensor to its cyclic permutations, i.e., $(\nabla_X R)(Y,Z)+(\nabla_Y R)(Z,X)+(\nabla_Z R)(X,Y)=0$ for all vector fields $X,Y,Z$

Yang-Mills theory

• Yang-Mills theory is a gauge theory that describes the dynamics of connections on principal bundles and their curvature, with applications in theoretical physics and geometry
• The Yang-Mills equations are a system of nonlinear partial differential equations for a connection on a principal bundle, generalizing Maxwell's equations and the equations of general relativity
• Solutions to the Yang-Mills equations, known as instantons and anti-instantons, have important topological and geometric properties and are used to define invariants of four-manifolds

Yang-Mills equations

• The Yang-Mills equations for a connection $A$ on a principal $G$-bundle $P\to M$ are the Euler-Lagrange equations for the Yang-Mills functional $S(A)=\int_M \langle F_A,F_A\rangle$, where $F_A$ is the curvature of $A$ and $\langle\cdot,\cdot\rangle$ is an invariant inner product on the Lie algebra of $