Lecture 6: Principal bundles
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1 Lecture 6: Principal bundles Jonathan Evans 6th October 2010 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12
2 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 We are in the process of defining principal G-bundles and connections on them. The last thing we said was: Definition A connection is a G-equivariant choice of horizontal space H p in each T p P, i.e. a subspace which projects via π to T π(p) M and such that g (H p ) = H g(p) for all g G. We write X for the unique horizontal vector field which projects along π to the vector field X on M. We see that this is the same as a g-valued 1-form α on P, which projects tangent vectors onto their vertical part (i.e. ker α = H) and takes vertical vectors v g to themselves. G-equivariance of H translates into the equivariance α(g v) = g α(v) of α.
3 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 In the case of U(1)-bundles we saw that the difference of two connections H andh is a 1-form on M. This has an analogue for G-bundles. Since both H p and H p project bijectively to T π(p) M we can write H p as the graph of a g-valued matrix A : H p g. In other words we can think of A : π TM P g. Equivariance of H now implies A(g v) = g A(v) for all g G We want to think of A as a 1-form a living on M, so we want to divide π TM and P g by G.
4 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 Recall Remember that on a Lie group, the tangent space at the identity e (the Lie algebra g) is naturally identified with the space of left-invariant vector fields ξ g such that ξ g = (L g ) ξ e where L g denotes left-multiplication by g. The fibre of our bundle P g TP over a point p P is precisely the space of left-invariant vertical vector fields on the fibre P π(p). There is a bundle over M whose fibre at q is the bundle of left-invariant vertical vector fields on the fibre P q. This is clearly the bundle in which this putative 1-form a takes its values since a(v) = A(ṽ) is a G-equivariant vertical vector field! Left-invariant is the same as G-equivariant: (L g A)(ṽ) = A(g ṽ) = g A(ṽ) = (L g ) A(ṽ) We want a better description of this bundle so we define the notion of associated vector bundle.
5 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 Associated bundles You may be more used to/comfortable with vector bundles than principal bundles. In this case you ll be happy to know that there s a nice way to translate between the two. Given a representation V of G (if you like just bear in mind the case G = U(n) acting on C n ) and a principal G-bundle P you automatically get a vector bundle P G V where the notation means that the total space of the bundle is P V divided by the group action of G by (p, v) (pg, g 1 v) This inherits a projection π : P G V M.
6 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 Lemma The fibre π 1 (m) can be identified with the vector space V by picking p P m and sending v [(p, v)] Proof. Surjectivity: If [(q, w)] π 1 (m) then p = qg for some g G so (q, w) (p, g 1 w). Injectivity: if (p, v) (p, w) then there is a g G such that p = pg, v = g 1 w, however the G-action on P is free so g = 1.
7 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 Lemma Sections of the bundle associated to P via a representation ρ correspond 1-1 with maps f : P V satisfying f (pg) = ρ(g 1 )f (p) Proof. Given such a map, form the map F = id f : P P V. This is G-equivariant with respect to the usual G-action P and the diagonal G-action on P V and hence it descends to a section M P G V. Conversely a section σ lifts to a G-equivariant map F : P P V and since π σ = id, the map F has the form id f.
8 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 One can recover P as the bundle of frames in the vector bundle P G V. A frame is just a G-equivariant identification of the fibre with V (i.e. a choice of basis! The basis is orthonormal, unitary or whatever else G may be) which is precisely what we picked to see that the fibre of P G V over m was isomorphic to V. Lemma The bundle over M we described earlier, whose fibre over q M is the space of left-invariant vertical vector fields on P q, is associated to P via the adjoint action of G on g Recall that the adjoint action of G on itself is Ad g (h) = ghg 1 and the adjoint action on the Lie algebra is ad g = dad g = dr g 1 dl g where R g and L g are the right and left actions of G on itself.
9 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 Proof. To each ξ g associate the vector field V (ξ) p = d dt t=0 (pe tξ ). If we write R g for the right action of G on itself then (R g ) V (ξ) p = d ) dt (pe tξ g t=0 = d ( )) dt (pg g 1 e tξ g t=0 = V (ad g 1ξ) pg. Note that a section of the bundle associated to P via the representation ρ : G Aut(V ) lifts to a map f : P V satisfying R g f = ρ(g 1 )f.
10 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 Outcome The difference of two connections on P is an ad(p)-valued 1-form on M, where ad(p) is the vector bundle associated to P via the representation ad : G Aut(g). If we fix a connection then for any ad(p)-valued 1-form A we write A = + A and we observe that the space of connections A is an affine space modelled on the space of ad(p)-valued 1-forms. In particular it is contractible!
11 Jonathan Evans () Lecture 6: Principal bundles 6th October / 12 Differentiating sections of associated bundles A connection on P induces a connection on the vector bundle E associated to some representation ρ : G Aut(V ). Recall that a connection on a vector bundle is just a differential operator on sections σ which obeys the Leibniz rule with respect to functions f : X (f σ) = df (X )σ + f X σ Sections of the associated bundle are in 1-1 correspondence with G-equivariant maps σ : P V and we can define X σ = dσ( X ) = Lie X σ (The last equality holds by Cartan s formula, considering σ as a V -tuple of functions).
12 A step back What I have written is actually extremely confusing (for a change...). Let s deconstruct it in the case of the trivial U(1)-bundle and the associated complex vector bundle E (with the usual action of U(1) on C). Here P = M U(1), E = M C and a section is the same as a map s : M C. The corresponding map σ : P C is NOT just s π, rather it is σ(m, e iθ ) e iθ s(m) Now if = d is the trivial connection, X = (X, 0) so X σ = e iθ ds(x ) If = d + ia then X = (X, A(X )) so X σ = e iθ (ds(x ) + ia(x )s(m)) The reason things are complicated is because we are remembering more information than we need (a whole G s worth of frame data) in order to make it easier to write down. Jonathan Evans () Lecture 6: Principal bundles 6th October / 12
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