Connections on principal bundles Santiago Quintero de los Ríos* December 16, 2020 Contents 1 Connections on principal bundles 1 1.1 Connections as horizontal distributions ........................ 1 1.2 Connections as 1-forms ................................ 3 1.3 Local expressions, or, why physicists did nothing wrong ............... 5 1.4 Horizontal lifts, parallel transport and holonomy ................... 8 2 Curvature 11 2.1 The curvature 2-form and structure equation ..................... 11 2.2 Local expressions (Curvature Edition) ......................... 14 2.3 The exterior covariant derivative ........................... 15 3 The relation with connections on vector bundles 17 3.1 From vector bundles to principal bundles ....................... 17 3.2 Interlude: Associated bundles ............................. 19 3.3 From principal bundles to vector bundles ....................... 22 3.4 In physics language .................................. 24 Notation These notes compile some general facts about connections on principal bundles, and their relation to connections on vector bundles. A few notes on notation: Weare working in the context of a principal 퐺-bundle 푃 over a manifold 휋 푀. This we denote as 퐺 ↪ 푃 → 푀; where 휋 ∶ 푃 → 푀 is the projection map. The right action of 퐺 on 푃 is denoted as 휎 ∶ 푃 ×퐺 → 푃. For any 푔 ∈ 퐺, we denote right multiplication by 푔 as 휎푔 ∶ 푃 → 푃; and for every 푝 ∈ 푃, we denote the orbit map as 휎푝 ∶ 퐺 → 푃. Given a smooth function 푓 ∶ 푀 → 푁 between manifolds, we denote the tangent map at some 푥 ∈ 푀 as 푇푥푓 ∶ 푇푥푀 → 푇푓(푥)푁. This is to explicitly show the functoriality of 푇푥. 1 Connections on principal bundles 1.1 Connections as horizontal distributions Recall that a vector 푣 ∈ 푇푝푃 is called vertical if 푇푝휋(푣) = 0. We denote the subspace of vertical vectors by 푉푝푃 ⊂ 푇푝푃. By definition, 푉푝푃 is nothing more than the kernel of 푇푝휋, so we have a short exact sequence 푇푝휋 0 푉푝푃 푇푝푃 푇휋(푝)푀 0 . *Please send corrections, suggestions, etc. to [email protected]. Latest version on homotopico.com/notes . 1 Since this is a sequence of vector spaces, it splits, and thus we have an isomorphism 푇푝푃 ≅ 푉푝푃 ⊕ 푇휋(푝)푀. However, the splitting (and thus the isomorphism) is not canonical: it depends on a choice of a sub- space 퐻푝 ⊂ 푇푝푃 that is complementary to 푉푝푃, and an isomorphism 푇휋(푝)푀 → 퐻푝. We call any complementary space to 푉푝푃 a horizontal space at 푝, such that: 푇푝푃 = 푉푝푃 ⊕ 퐻푝. Figure 1: A choice of a horizontal space 퐻푝 at 푇푝푃. There are many such choices (in dotted lines). Once we have chosen a single horizontal subspace 퐻푝 ⊂ 푇푝푃 at 푝, we can find horizontal subspaces for all points in the same fiber of 푝. This follows since the action of 퐺 on 푃, which we denote 휎푔(푝) = 푝 ⋅ 푔, is a fiber-preserving diffeomorphism, and thus 푇푝휎푔 is an isomorphism of tangent spaces that preserves the vertical subspace. This suggests that 푇푝휎푔(퐻푝) is a horizontal subspace at 푝 ⋅ 푔. Indeed, noting that 푇푝⋅푔휋 ∘ 푇푝휎푔 = 푇푝(휋 ∘ 휎푔) = 푇푝휋(푣), we see that 푇푝휋(푉푝푃) ⊆ 푉푝⋅푔푃. Similarly, if 푢 ∈ 푉푝⋅푔푃, we can write 푢 = 푇푝휎푔(푇푝⋅푔휎푔−1 (푢)) = 푇푝휎푔( ̃푢), where by the same argument above ̃푢= 푇푝⋅푔휎푔−1 (푢) ∈ 푉푝푃 is vertical. Therefore, we obtain that 푉푝⋅푔푃 = 푇푝휎푔(푉푝푃). Furthermore, since 푇푝휎푔 ∶ 푇푝푃 → 푇푝⋅푔푃 is an isomorphism, we obtain that 푇푝⋅푔푃 = 푇푝휎푔(푇푝푃) = 푇푝휎푔(푉푝푃) ⊕ 푇푝휎푔(퐻푝) = 푉푝⋅푔푃 ⊕ 푇푝휎푔(퐻푝), And so we have proved the following: Lemma 1.1 (Translation of horizontal subspaces). If 퐻푝 ⊂ 푇푝푃 is horizontal at 푝, then for all 푔 ∈ 퐺, 푇푝휎푔(퐻푝) is horizontal at 푝 ⋅ 푔. So far we have been working at a single point 푝 ∈ 푃. We can now consider a smooth choice of horizontal spaces above each element of 푃: Definition 1.2 ((Principal) Connection). A connection or Ehresmann connection on 푃 is a distribution 퐻 on 푃 such that for all 푝 ∈ 푃, 퐻푝 ⊂ 푇푝푃 is a horizontal subspace. We say that a connection 퐻 is principal if it is compatible with the group action in the sense that for all 푔 ∈ 퐺 and all 푝 ∈ 푃, 푇푝휎푔(퐻푝) = 퐻푝⋅푔. 2 The notion of connection is independent of the group action on the total space 푃, and indeed it ap- plies to general fiber bundles. The condition for a connection to be principal states that our choice of horizontal subspaces along a single fiber is consistent with the “translation” lemma 1.1. We think of a connection 퐻 as a preferred way of relating “neighboring” fibers of the bundle. Once we have 푝 ∈ 푃, we might think that the preferred way of moving to another fiber is along a “direction” (i.e. tangent vector) in the horizontal space 퐻푝. This gives us a little bit of intuition and (sort of) justifies (kind of) the name connection. In practice, however, working with distributions might be cumbersome. Fortunately for us, there are other (equivalent) presentations of connections. 1.2 Connections as 1-forms Let 픤 be the Lie algebra of 퐺. Recall that for all 푝 ∈ 푃, we have the infinitesimal action of 픤 on 푇푝푃, 푎푝 ∶ 픤 → 푇푝푃 given as d | 푎푝(푋) ∶= | 푝 ⋅ exp(푡푋). d푡 |푡=0 Writing 휎푝 ∶ 퐺 → 푃 as 휎푝(푔) = 푝 ⋅ 푔, we see that the infinitesimal action is simply the differential of 휎푝: 푎푝(푋) = 푇푒휎푝(푋). This infinitesimal action induces, for each 푋 ∈ 픤, a vector field 푋♯ called the f undamental vector field associated to 푋 given by ♯ 푋푝 ∶= 푎푝(푋). We have that 휎푝 is a diffeomorphism onto the fiber containing 푝, and thus 푎푝 = 푇푒휎푝 induces a 푎푝 linear isomorphism 픤 ≅ 푉푝푃. Suppose that we have a principal connection 퐻 on 푃. Then in particular, we have a subspace 퐻푝 ⊂ 푇푝푃 such that 푇푝푃 = 푉푝푃 ⊕ 퐻푝, and so we can construct a map 휔푝 ∶ 푇푝푃 → 픤 as 푉 퐻 −1 푉 휔푝(푣 + 푣 ) = 푎푝 (푣 ), 푉 퐻 where 푣 ∈ 푉푝푃 and 푣 ∈ 퐻푝. By construction, we have that 휔푝(푎푝(푋)) = 푋 for all 푋 ∈ 픤. Wecan also see how 휔푝 compares to 휔푝⋅푔, since we know that our horizontal distribution behaves nicely along the fibers of the action. For this, first note that for all 푔 ∈ 퐺, d | 푇푝휎푔(푎푝(푋)) = | 휎푔(푝 ⋅ exp(푡푋)) d푡 |푡=0 d | = | 푝 ⋅ exp(푡푋)푔 d푡 |푡=0 d | = | (푝 ⋅ 푔) ⋅ (푔−1 exp(푡푋)푔). d푡 |푡=0 Now we ask ourselves, do we know what the tangent vector of 푔−1 exp(푡푋)푔 is? Yes, yes we do: d | −1 d | | 푔 exp(푡푋)푔 = | Conj푔−1 (exp(푡푋)) = Ad푔−1 (푋), d푡 |푡=0 d푡 |푡=0 1 −1 where we have written Conj푔(ℎ) = 푔ℎ푔 , and Ad푔 = 푇푒 Conj푔. Then we have d | −1 푇푝휎푔(푎푝(푋)) = | (푝 ⋅ 푔) ⋅ (푔 exp(푡푋)푔) = 푎푝⋅푔(Ad푔−1 (푋)). d푡 |푡=0 푉 퐻 푉 With this, we can see that for 푣 ∈ 푇푝푃, which we write as 푣 = 푣 + 푣 with 푣 = 푎푝(푋) for some 푋 ∈ 픤: ∗ 푉 퐻 푉 퐻 (휎푔 휔)푝(푣 +푣 ) = 휔푝⋅푔(푇푝휎푔(푣 )+푇푝휎푔(푣 )) = 휔푝⋅푔(푇푔휎푔(푎푝(푋))) = Ad푔−1 (푋) = (Ad푔−1 ∘휔푝)(푣), 1https://xkcd.com/927/ 3 and so we conclude that ∗ (휎푔 휔) = Ad푔−1 ∘휔. Then we have proved, modulo the small detail of smoothness2, the following: 1 Proposition 1.3 ( -form induced by principal휋 connection). Let 퐻 be a principal connection on 퐺 ↪ 푃 → 푀. Then there exists a (unique) 픤-valued 1-form 휔 ∈ Ω1(푃, 픤), such that for all 푝 ∈ 푃, 푔 ∈ 퐺 and 푋 ∈ 픤: 1. 휔푝(푎푝(푋)) = 푋, ∗ 2. 휎푔 휔 = Ad푔−1 ∘휔, and 3. ker(휔푝) = 퐻푝. We call any 픤-valued 1-form satisfying these properties a connection 1-form: Definition 1.4 (Connection 1-form). A connection 1-form on 푃 is a 픤-valued 1-form 휔 ∈ Ω1(푃, 픤) such that for all 푝 ∈ 푃, 푔 ∈ 퐺 and 푋 ∈ 픤: 1. 휔푝(푎푝(푋)) = 푋, and ∗ 2. 휎푔 휔 = Ad푔−1 ∘휔. The converse to proposition 1.3 is also true: Proposition 1.5. Principal connection induced by connection 1-form Let 휔 ∈ Ω1(푃, 픤) be a connection 1-form. Then the distribution 퐻 defined pointwise as 퐻푝 = ker(휔푝) ⊂ 푇푝푃 is a principal connection on 푃. Proof. — First, let’s see that indeed 퐻푝 = ker(휔푝) is horizontal. If 푣 ∈ ker(휔푝)∩푉푝푃, then 푣 = 푎푝(푋) for some 푋 ∈ 픤, so that 0 = 휔푝(푣) = 휔푝(푎푝(푋)) = 푋, and thus 푣 = 0. Therefore ker(휔푝) ∩ 푉푝푃 = {0}. Now for an arbitrary 푣 ∈ 푇푝푃, set 푉 푣 = 푎푝(휔푝(푣)). 푉 푉 Then we have that 푇푝휋(푣 ) = 0, since it is in the image of 푎푝, and thus 푣 ∈ 푉푝푃. Finally, setting 푣퐻 = 푣 − 푣푉 , we have 퐻 휔푝(푣 ) = 휔푝(푣) − 휔푝(푎푝(휔푝(푣))) = 휔푝(푣) − 휔푝(푣) = 0, 퐻 푉 퐻 푉 퐻 and so 푣 ∈ ker 휔푝 = 퐻푝. We have then shown that 푣 = 푣 + 푣 , with 푣 ∈ 푉푝푃 and 푣 ∈ 퐻푝, and so 푇푝푃 = 푉푝푃 ⊕ 퐻푝. Thus 퐻푝 is a horizontal subspace. Now to see that 퐻 is principal, note that 휔푝⋅푔(푇푝휎푔(푣)) = Ad푔−1 (휔푝(푣)). Since both 푇푝휎푔 and Ad푔−1 are isomorphisms, we have that 푣 ∈ ker 휔푝 if and only if 푇푝휎푔(푣) ∈ ker 휔푝⋅푔, and thus 푇푝휎푔(퐻푝) = 퐻푝⋅푔. Finally, smoothness follows from the fact that 휔 is a smooth form. ■ From now on, if 휔 is a connection 1-form, we will simply call it a connection.