Making manifest the symmetry enhancement for coincident BPS branes

We consider g coincident M-5-branes on top of each other, in the KK monopole background Q of multiplicity N . The worldvolume of each M-5-brane is supposed to be given by the local product of the four-dimensional spacetime R and an elliptic curve. In the coincidence limit, all these curves yield a single (Seiberg-Witten) hyperelliptic curve Σg, while the gauge symmetry is enhanced to U(N). We make this gauge symmetry enhancement manifest by considering the hypermultiplet LEEA which is given by the spacetime N=2 non-linear sigma-model (NLSM) having Q as the target space. The hyper-Kähler manifold Q is given by the multicentre Taub-NUT space, which in the coincidence limit amounts to the multiple Eguchi-Hanson (ALE) space QmEH. The NLSM is most naturally described in terms of the hyper-Kähler coset construction on SU(N,N)/U(N) in harmonic superspace, by using the auxiliary (in classical theory) N=2 vector superfields as Lagrange multipliers, with FI terms resolving the AN singularity. The Maldacena limit, in which the hypermultiplet LEEA becomes extended to the N=4 SYM with the gauge group U(N), arises in quantum field theory due to a dynamical generation of the N=2 vector and hypermultipet superfields, when sending the FI terms to zero. 1 Brane technology and KK monopoles The (Seiberg-Witten-type) exact solution to the N=2 super-QCD can be identified with the LEEA of the effective (called N=2 MQCD) field theory defined in the worldvolume of the single M-5-brane, given by the local product of the uncompacftified four-dimensional spacetime R and the hyperelliptic curve Σg of genus g = N − 1: Witten (1997) (see, e.g. Karch, Lüst and Smith (1998), Ketov (1997), Ketov (1998) for a review or an introduction). The hyperelliptic curve Σg is supposed to be holomorphically embedded into the hyper-Kähler four-dimensional multiple Taub-NUT space QmTN associated with the multiple KK-monopole. The identification of the low-energy effective actions (LEEA) in these two, apparently very different, field theories (the N=2 super-QCD in the Coulomb branch, on the one side, and the N=2 MQCD defined in the M-5brane worldvolume, on the other side) is highly non-trivial, since the former is defined as the leading contribution to the quantum LEEA in a gauge field theory, whereas the latter is determined by the classical M-5-brane dynamics or the D = 11 supergravity equations of motion, whose BPS solutions preserving some part of supersymmetry are the M-theory branes under consideration.