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Small coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. I. (English) Zbl 1346.70035

Summary: In this paper and its sequel [ibid. 53, No. 6, 063707, 39 p. (2012; Zbl 1346.70034)], we analyze the space of solutions to the {\(\varepsilon\)}-Dirichlet problem for the Yang-Mills equations on the four-dimensional disk, for small values of the coupling constant \(\varepsilon\). These are in \(1-1\) correspondence with solutions to the Dirichlet problem for Yang-Mills, for small boundary data \(\varepsilon A_0\). We establish a Morse theory for this non-compact variational problem and prove the existence of multiple solutions, and, also, non minimal ones. Here, we describe the problem, state the main theorems and do the first part of the proof. This consists in making the problem finite dimensional, by seeking solutions approximated by the connected sum of a minimal solution with an instanton, plus a correction term due to the boundary. By introducing an auxiliary equation, we solve the problem orthogonally to the space of the approximate solutions.{
©2012 American Institute of Physics}

MSC:

70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
35G31 Initial-boundary value problems for nonlinear higher-order PDEs
37D15 Morse-Smale systems

Citations:

Zbl 1346.70034
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References:

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