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QCD with chemical potential on S^1 x S^3

In this proceedings we summarize our calculation of the phase diagram of QCD at non-zero temperature and chemical potential on S^1 x S^3 from one-loop perturbation theory [1], which is valid in the limit R << 1/Lambda, where R is the radius of S^3. We calculate several observables including the Polyakov lines and the quark number, for large number of colors N and large number of quark flavors Nf, on S^1 x S^3, and compare with results for the same system with N = 3, and with results for N=2 lattice QCD. For N > 2 the action is complex and the dominant contributions to the path integral occur in the space of complexified gauge field configurations. This results in the expectation values of the eigenvalues of the Polyakov line lying off the unit circle and out in the complex plane. This is an important issue for the lattice, and also for the calculation on S^1 x S^3 in the large N limit where we obtain analytical results using the saddle point approximation. It is thus necessary to adapt available techniques to locate the stationary solutions in the complexified gauge field configuration space.