New insights into the problem with a singular drift term in the complex Langevin method

The complex Langevin method aims at performing path integral with a complex action numerically based on complexification of the original real dynamical variables. One of the poorly understood issues concerns occasional failure in the presence of logarithmic singularities in the action, which appear, for instance, from the fermion determinant in finite density QCD. We point out that the failure should be attributed to the breakdown of the relation between the complex weight that satisfies the Fokker-Planck equation and the probability distribution associated with the stochastic process. In fact, this problem can occur in general when the stochastic process involves a singular drift term. We show, however, in a simple example that there exists a parameter region in which the method works although the standard reweighting method is hardly applicable.

BPS solutions in ABJM theory and Maximal Super Yang-Mills on RxS^2

We investigate BPS solutions in ABJM theory on RxS^2. We find new BPS solutions, which have nonzero angular momentum as well as nontrivial configurations of fluxes. Applying the Higgsing procedure of arxiv:0803.3218 around a 1/2-BPS solution of ABJM theory, one obtains N=8 super Yang-Mills (SYM) on RxS^2. We also show that other BPS solutions of the SYM can be obtained from BPS solutions of ABJM theory by this higgsing procedure.