Bundles, Bundles, Bundles, Connections and the Yang-Mills Field

An in depth on bundles and related concepts

10 min readOct 11, 2022

As a follow up to this previous post, and due to my own interest with bundles I figured it would be neat to make a brief and sequential exposition on bundles and afterwards derive the Yang-Mills field.

**Note: text contains plenty of prerequisite topological concepts (Lie Groups, Algebras, Forms, etc.) and will get a bit abstract**

Preliminary Bundle Stuff

Again, as a quick run down, a bundle is a triple (E, 𝝅, M) where E and M are topological manifolds, E being the total space, M being the base space, and 𝝅 being a continuous surjective map from the total to the base space.

Additionally, Fₚ is the fibre at the point p, and is the pre-image of p or 𝝅⁻¹(p), more intuitively, Fₚ are the set of points in E (in the above diagram, we have a line) attached to the point p, and the map 𝝅 sends all these points in E to the single point p in M.

Lastly, we have the section, which is a map 𝛔, that sends the point p from M into the fibre Fₚ, such that 𝝅 ￮ 𝛔 is equal to the identity map on M, idᴍ (𝝅 ￮ 𝛔 sends the point back to the point).

We can have an isomorphism between two bundles, (E, 𝝅, M) and (E, 𝝅’, M), which means that the diagram below commutes (v ￮ 𝝅 = 𝝅’ ￮ u, and 𝝅 ￮ u⁻¹ = v⁻¹ ￮ 𝝅’ ):

Isomorphism of Bundles (from: MathWithPhysics)

Product Bundle

A trivial example of a bundle. It is just the triple (M⨯N, 𝝅, M). Here M⨯N is the cartesian product of two manifolds M and N, and the projection 𝝅: M⨯N → M which takes (p, q) → p; p ∈ M & q ∈ N.

Sub Bundle

If one has a bundle (E, 𝝅, M) a sub bundle is (E’, 𝝅’, M’) where, E’ is a subset of E, and M’ is a subset of M, are both sub-manifolds, and 𝝅’ = 𝝅|ᴇ’ (meaning the map 𝝅 is restricted to E’).

Restricted Bundle

With the bundle (E, 𝝅, M) and a sub-manifold of M, we call N, we can have the restricted bundle (E, 𝝅’, N) where 𝝅’:= 𝝅|pre-image𝜋 (N)

Pull Back Bundle

With the bundle (E, 𝝅, M) and a map f: M’ → M from another manifold M’, then we have a pull back bundle (E’, 𝝅’, M’) induced by f. Here E′ is defined by {(m′, e) ∈ M′⨯E | f(m′) = 𝝅(e)}.

Fibre Bundle

If we have a bundle (E, 𝝅, M) and a manifold F, then ((E, 𝝅, M), F) is called a fibre bundle with (typical) fibre F, if ∀ p ∈ M the pre-image𝜋({p}) is topologically isomorphic to F. We can represent it as such:

Vector Fibre Bundle

Exactly the same as the previous definition, however every fibre is a vector space, such as the tangent bundle (see below), or the cotangent bundle.

Smooth Bundle

Similar to the canonical bundle defined above, with the exception that the map 𝝅 is a smooth map. A map f is said to be smooth (as a map between smooth manifolds M and N, f: M → N) if and only if the map idɴ ￮ f ￮ idᴍ⁻¹ = f is smooth (infinitely differentiable).

Tangent Bundles

This is an elegant way to combine all the tangent spaces to a manifold M. By definition it is the disjoint union of all the tangent spaces (TₚM) to M, which we call TM, equipped with a projection map 𝝅. Hence it is the triple (TM, 𝝅, M). We can see this in the diagram below, where M can be taken to be a sphere, and from any tangent space in TM we can use 𝝅 to project into p ∈ M.

Principal Fibre Bundles

With all that out of the way, we can now build to something more juicy. In a general sense, a principal fibre bundle is a bundle whose fibre at each point is a Lie group.

Principal G-bundle:

A smooth bundle (E, 𝝅, M) such that E is equipped with a free right G-action ⊲, where G is a Lie group. Hence, one has the following bundle isomorphism:

Here 𝜌 is the quotient map, that sends every p ∈ E to its equivalence class (E/G).

This is a notion of interest due to the fact that the right action ⊲ of G on E is free, and hence for every p from E , the pre-image𝜌(Gₚ) = Gₚ which is diffeomorphic to G (a diffeomorphism is a bijective map ϕ: M → N between smooth manifolds M and N, such that ϕ and ϕ⁻¹ are smooth, one says that two manifolds M and N are diffeomorphic if there exists this diffeomorphism between them :) )

Hence we have that a principal G-bundle is such that, it is isomorphic to a bundle whose fibres are the equivalence classes under the right action of G, and in addition these equivalence classes are isomorphic to G (since the action is free). It can be pictorially represented as such:

Principal G-Bundle (from: MathWithPhysics)

Frame Bundle:

This is a variation of a principal fibre bundle.

For its construction, we first need a space LₚM which is the set of basis vectors of the tangent space TₚM at point p of a smooth manifold M.

Then we take LM to be the disjoint union of LₚM spaces on M, similar to the tangent bundle.

We then equip LM with the projection map 𝝅: LM → M which sends each basis (e₁,…, e𝑑𝑖𝑚ᴍ) to the unique point p ∈ M, such that (e₁,…, e𝑑𝑖𝑚ᴍ) is a basis of TₚM, and we have the bundle (LM, 𝝅, M).

We then equip LM with a free right GL(dimM, ℝ)-action ⊲, where GL is the General Linear group, such that the diagram below commutes:

We have that this action is free since:

(e₁,…, e𝑑𝑖𝑚ᴍ) ⊲ g = (e₁,…, e𝑑𝑖𝑚ᴍ) ⟺ (gᵃ₁eₐ,…,gᵃ𝑑𝑖𝑚ᴍ eₐ) = (e₁,…, e𝑑𝑖𝑚ᴍ)

Above gᵃ𝑏 are components of g ∈ GL(dimM, ℝ), where g is an endomorphism.

Hence, similar to the previous section, we have that a bundle isomorphism between (LM, 𝝅,M) and (LM, 𝜌, LM/GL(dimM,ℝ)) exists. Thus we can say that (LM, 𝝅,M) is a principal G-bundle, namely the frame bundle.

Associated Fibre Bundle

As the name suggest, this is a bundle associated to a principal G-bundle.

If we have a principal G-bundle (P, 𝝅, M), and F is a smooth manifold equipped with a left action G-action ⊳, then we can define Pꜰ := (P⨯F)/~ɢ, where ~ɢ is an equivalence relation defined as:

(p, f) ~ɢ (p’, f’):⟺ ∃ g ∈ G such that p’ = p⊲g and f’ = g⁻¹ ⊳ f.

Hence, the elements of Pꜰ are the equivalence classes of (p, f) where p ∈ P, f ∈ F and we can label them as [p, f]. We can also define the map 𝝅ꜰ:= Pꜰ → M which takes [p, f] → 𝝅(p).

Finally, we have that the associated bundle to ((P, 𝝅, M), F, ⊳) is the bundle (Pꜰ, 𝝅ꜰ, M).

Connection to Connections

Vector Field

If we have the tangent bundle (TM, 𝝅,M), then a vector field is simply a smooth section 𝛔, of the tangent bundle. For the set of all vector fields on M, one writes Γ(TM). So every element of Γ(TM) is a 𝛔: M → TM such that 𝛔 is smooth, and 𝝅 ￮ 𝛔 = idᴍ (just the definition of a section).

For a Lie Group G with identity element e, we have that 𝓛(G) is isomorphic as a vector space to TₑG (where 𝓛 represents the Lie algebra of G).

Left Invariant Vector Field

If we have a Lie Group G, a vector field from Γ(TG), say X, is left invariant if ∀ g ∈ G: (L𝑔)⁎(X) = X, where (L)⁎ is the induced push-forward map of the left translation 𝓁𝑔 on G. Meaning, the vector field X is invariant under the induced push-forward of the left translation map.

A Map

By definition, any element of TₑG, say A, produces a left invariant vector field on G, which we call Xᴬ. If we have a principal bundle (P, 𝝅, M), we can turn the vector field on G into a vector field on P, which we will call Xᴬ ₚ with a map iₚ: TₑG → TₚP

Vertical Subspace

Then from the principal bundle (P, 𝝅, M) and at a given point p one can make a vertical subspace VₚP at p which is just a subspace of of the vector space TₚP, VₚP := ker((𝝅⁎)ₚ). Where the kernel just takes an Xₚ from TₚP such that (𝝅*)ₚ(X) = 0.

We can view this locally in the diagram below:

Notice that the tangent vector v1 to the curve f, is the tangent vector of the projected/mapped curve, it is the push-forward of v1, 𝝅⁎(v1). We can see that it is not the zero vector, hence is does not lie in the kernel of the 𝝅⁎. However if we look at the middle fibre, and take a vector like v2, it will always be projected into the zero vector. Hence we can conclude that the entire fibre of p is a vertical subspace.

We can also conclude that the previously defined map iₚ is a bijection.

Horizontal Subspace

From the principal bundle (P, 𝝅, M) and at a given point p one can also make a horizontal subspace HₚP at p which is just a complement to vertical subspace VₚP defined previously. Meaning TₚP = HₚP ⊕VₚP (direct sum). The choice of HₚP is not unique but from the choice one gets the unique decomposition of an Xₚ from TₚP, as Xₚ = ver(Xₚ) + hor(Xₚ) where both the vertical and horizontal components are from VₚP.

Connection

We are then ready to define a connection. On a principal G-bundle (P,𝝅,M) a connection is a choice or assignment of a horizontal subspace at each p from P such that the following conditions hold:

a) TₚP = HₚP ⊕ VₚP

b) ∀ g ∈ G, p ∈ P and Xₚ ∈ HₚP: ((⊲g)⁎)Xₚ ∈ H𝑝⊲𝑔 P

c) the unique decomposition Xₚ = hor(Xₚ) + ver(Xₚ), leads for every smooth vector field X ∈ Γ(TP) to two smooth vector fields hor(X) and ver(X)

We can see b) outlined in the diagram below:

One needs to recognize that ver(Xₚ) and hor(Xₚ) directly depend on the choice of HₚP:

We can see that at the same point and with the same vector X, but a different choice of HₚP, we get different ver(X) and hor(X) components.

This choice of HₚP is however encoded in the induced Lie-algebra-valued one-form ωₚ: TₚP → TₑG (linear map) where Xₚ → ωₚ(Xₚ):= iₚ⁻¹(ver(Xₚ)) which is an element of TₑG. We defined the map iₚ previously above. This form ωₚ is called the connection one-form with respect to the connection.

We can also then get back the HₚP, since HₚP = ker(ωₚ).

Now we will take this lie-algebra-valued one-form on P (the Principle Bundle) and express it locally as the Lie-algebra-valued one-form on the base manifold. This then, depending on context, are called the local connection coefficients (Christoffel symbols, Γ ⁱⱼₖ) in general relativity, or the Yang-Mills field in particle physics, as we shall do here.

Yang Mills Field

We now have the connection one form ω: Γ(TP) → TₑG (linear map), and it needing to satisfy ω(Xᴬ) = A and ((⊲g)*ω)(X) =(Ad𝑔⁻¹)⁎ (ω(X)), where Ad is the adjoint map. Where X ∈ Γ(TP), g ∈ G, and A ∈ TₑG.

So it is a map from the section of the tangent bundle into the Lie algebra. We then have the following commuting diagram, where u and f are maps and the rest is a standard principal G-bundle:

In other words, we have a principal bundle automorphism, or an isomorphism that goes back to itself. Since we know that ω lives on the above diagram, we should be able to pull it back via the map u. Therefore we would have u*ω, and we would apply it as (u*ω)(X):= ω(u⁎X), where X is a vector in P.

In practice (i.e., for calculations), one works with some subset U of the base manifold M, and it is better to work with a one-form that is not globally defined. For example, if one is dealing with spacetime as M, one deals with things(physics) locally, and then extrapolates the results out globally afterwards (local experiments then global laws). So we choose a local section 𝛔: U → P such that 𝝅 ￮ 𝛔 = idᴜ. This local section then induces the following:

i) a Yang Mills field ωᵁ:= 𝛔*ω

ii) a local trivialization of the principal bundle P, or a map h: U⨯G → P where (m, g) → 𝛔(m)⊲g

iii) a local representation of ω on U, h*ω: Γ(T(U⨯G)) → TₑG.

We then get the following diagram:

We have ωᵁ as the Yang Mills field, the local representation h*ω, and the connection one form ω. The ωᵁ may seem scarcer in information relative to h*ω, however this is not the case. Over a local patch, ωᵁ and h*ω carry the same informational content.

The end.