Instantons and scattering in N = 4 SYM in 4

We study classical solutions (ic–instantons) in N = 4 SYM in 4D which, in the strong coupling limit, correspond to complex two–dimensional manifolds. Asymptotically in time the latter have boundaries represented by compact real three–manifolds. Therefore they lend themselves to an interpretation in terms of 3–brane scattering. We suggest that these solutions may represent scattering of D3–branes of type IIB theory in 10D. In particular we show that the world–volume theory on complex two–dimensional manifolds is the correct one for D3–branes. Contents 1. Introduction 1 2. Interaction–carrying classical solutions 2 2.1 Ic–instanton equations as BPS solutions 4 2.2 Ic–instanton equations from self–duality in 8D 4 2.3 Ic–instanton equations: other derivations and covariant form 5 2.4 General properties of ic–instantons 6 3. Ic–instantons 7 3.1 Explicit construction of ic–instantons 9 3.2 Generalized ic–instantons 10 4. Spectral coverings and scattering 11 4.1 Scattering of S 3 –branes 12 4.2 Scattering of T 3 –branes. 13 5. Expansion about a classical solution 14 5.1 Summing over line bundles 18 5.2 Counting zero modes 19 6. Discussion 21 6.1 Connection with Matrix theory 21 6.2 Other questions 23