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- by Changhai Lu -

During the time in Columbia, I have been working with Professor Kimyeong Lee on theoretical physics. My research has been focused in two directions:

1. Chern-Simons Theory at Finite Temperature

Along this direction, I (in collaboration with Gerald Dunne and Kimyeong Lee) studied the finite temperature Chern-Simons coefficient problem. It is well known that the coupling between a gauge field and a fermion field in an odd dimensional spacetime will induce a Chern-Simons term in the effective action of the theory. Perturbative finite temperature calculations of the coefficient of such term yield a continuous function of temperature. A serious problem arises since gauge invariance requires the coefficient to be an integer. This problem has been puzzling physicists for more than a decade. In our work On Finite Temperature Chern-Simons Coefficient (Physical Review Letters 78, 3434, 1997), we computed the exact finite temperature effective action in a (0+1)-dimensional field theory that shares all the significant features of the (2+1)-dimensional Chern-Simon theories. This exact result explained the origin and meaning of the temperature dependent coefficients found previously.

2. Topological Objects in Gauge Theories

Along this direction, I have studied three related topics on topological objects in Yang-Mills-Higgs Systems.

In Two Massive and one Massless Sp(4) Monopoles (Physical Review D 57, 5260, 1998), Kimyeong and I explored a BPS state made of two massive and one massless monopole in an Sp(4) theory with the gauge symmetry broken to SU(2)×U(1). This monopole system carries a purely Abelian total magnetic charge and is a highly nontrivial system containing a non-Abelian cloud. One of the major topics in monopole physics is to compute the moduli space metrics which not only determine their low energy dynamics but also play an important role in the study of non-perturbative effects of gauge theories. It turns out that computing moduli space metrics is quite difficult and so far only a few exact results are known. In this work, we computed the exact moduli space metric of this Sp(4) system by employing the Nahm formalism. We also constructed field configurations with axial symmetry and analyzed in detail the properties of various submanifolds of the moduli space.

In SU(2) Calorons and Magnetic Monopoles (Physical Review D 58, 025011, 1998), Kimyeong and I investigated a single SU(2) caloron, or periodic instanton, with non-trivial Higgs expectation value at spatial infinity. This work generalized the previous results by considering gauge symmetry breaking. We constructed the explicit field configuration of the caloron and studied various limits. From this analysis we found a new picture of instantons in which an instanton is composed of two self-dual monopoles of opposite magnetic charge. As an application of the constituent monopole picture, we computed the moduli space metric of the caloron.

In Two-Monopole Systems and the Formation of Non-Abelian Clouds (Physical Review D 58, 125010, 1998), I studied two-monopole systems in SU(3) and Sp(4) theories with non-Abelian unbroken gauge symmetry. I computed the energy densities of both systems and checked several special limits. As one of the major motivations of the work, particular attention was paid on the massless limit of the two theories and some insights were obtained concerning the formation of non-Abelian clouds. Another major result of the work is that the coefficient of the internal part of the moduli space metric of the Sp(4) system was computed from the analytic expression of energy density. This result gives an attractive mechanical interpretation to the moduli space metric.

3. Ph.D Thesis

My Ph.D thesis is entitled Topological Objects in Gauge Theories.