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Communications in Mathematical Physics - Volume 254 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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1. Notation and main results. Throughout this paper, we consider the trivial G-bundle over a 4-manifold X. Here G is a compact Lie group with Lie algebra g. We denote the Lie bracket on g by [·, ·], and we equip g with a G-invariant inner product ·, · . A connection on the trivial bundle G × X is a g-valued 1-form A ∈ 1 (X; g). We denote the space of smooth connections by A(X) := 1 (X; g). Associated to a connection A ∈ A(X) one has the exterior derivative dA on g-valued differential forms given by dA η = dη + [A ∧ η] ∀η ∈ k (X; g).

An n-tuple of Lagrangian submanifolds Li ⊂ A0,p ( i ) as in (1) then constitutes a gauge invariant Lagrangian submanifold L := L1 × . . × Ln of the symplectic Banach space 0,p A0,p ( ) = A0,p ( 1 ) × . . × A0,p ( n ) such that L ⊂ Aflat ( ). In order to linearize the boundary value problem (2) together with the local slice condition, fix a smooth connection A + ds ∈ A(S 1 × Y ) such that As := A(s)|∂Y ∈ L for all s ∈ S 1 . Here ∈ C ∞ (S 1 × Y, g) and A ∈ C ∞ (S 1 × Y, T∗ Y ⊗ g) is an S 1 -fam1,p ily of 1-forms on Y (not a 1-form on X as previously).

N ), then V ≡1. Hence g¯ V is the standard metric g0 , which has a natural S(O(3)×O(3))-symmetry. If {pj± }N j =1 are simultaneously collinear, that is, if they lie on a two-dimensional subspace in R41 , then g¯ V has an (S 1 ×S 1 )-symmetry. Indeed, the extra S 1 -symmetry is given by the rotation around the intersection of the subspace and the totally geodesic sphere S 2 = {ρ = 0}. In particular, if N = 1, then g¯ V always has an (S 1 ×S 1 )-symmetry (cf. Poon [21]). We also introduce another construction of explicit solutions (V , θ ) of (4) satisfying the conditions in Proposition 6, by using the separation method.

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