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Extra info for Communications in Mathematical Physics - Volume 254
This boundary condition arises naturally from examining the ChernSimons functional on a 3-manifold Y with boundary . Namely, the Lagrangian boundary condition renders the Chern-Simons 1-form on the space of connections closed, see [S]. The resulting gradient flow equation leads to the boundary value problem studied in this paper (for the case X = R × Y ). Besides the regularity and compactness properties on noncompact manifolds we also establish the Fredholm theory for the compact model case X = S 1 × Y .
In the first component of D(A, ) we have used the global space-time splitting of the metric on S 1 × Y to identify the self-dual 2-forms ∗γs − γs ∧ ds with families γs of 1-forms on Y . The vanishing of this component is equivalent to the linearization + dA+ ds (α + ϕds) = 0 of the anti-self-duality equation. Furthermore, the boundary condition α(s)|∂Y ∈ TAs L is the linearization of the Lagrangian boundary condition in the boundary value problem (2). Theorem C (Fredholm properties). Let Y be a compact oriented 3-manifold with boundary ∂Y = and let S 1 × Y be equipped with a product metric ds 2 + gs that is compatible with the embedding τ : S 1 × → S 1 × Y .
Wolf, J. : Spaces of constant curvature. R. Douglas Commun. Math. Phys. 1007/s00220-004-1235-z Communications in Mathematical Physics Anti-Self-Dual Instantons with Lagrangian Boundary Conditions I: Elliptic Theory Katrin Wehrheim Department of Mathematics, Princeton University, Fine Hall, Princeton, NJ 08544-1000, USA. edu Received: 6 February 2003 / Accepted: 20 July 2004 Published online: 11 November 2004 – © Springer-Verlag 2004 Abstract: We study nonlocal Lagrangian boundary conditions for anti-self-dual instantons on 4-manifolds with a space-time splitting of the boundary.