Instanton Moduli and Compactification

1.2. Primer on Connections. Recall from Peng’s talk that a connection on P → X can be thought of in three ways: (1) As a field of horizontal subspaces: TpP = HpP⊕VpP , where V P = ker (π∗). (Note that Vp ∼= g, and Hp ∼= Tπ(p)X). (2) As a g-valued 1-form A ∈ ΩX (g) which is invariant w.r.t. the induced G-action on ΩX (g). (3) Given a representation of G on W , as a vector bundle connection on the associated bundle E := P ×GW → X ∇A : ΩX (E)→ ΩX (E) , where∇A is a linear map satisfying the Leibniz rule: ∇ (f · s) = f ·∇s+df ·s for f ∈ C∞ (X), s ∈ Γ (X,E). The first two of these definitions are seen to be equivalent by setting HpP = kerAp for (2)→(1), or Ap = TpP → VpP (projection) for (1)→(2). For (3), we think of P as being the frame bundle for E, and then describe a horizontal frame of sections. For concreteness, we will mostly use (3) in this talk, i.e., fix a vector bundle E → X, and then think of P as the frame bundle of E. However, everything can still be done for principal bundles, too. From ∇A, we can build a new operator dA : ΩX (E)→ Ωk+1 X (E)