Roman Jackiw and Chern–Simons theories

Abstract

Recently, the Center of Mathematical Sciences and Applications at Harvard initiated an excellent series of talks on mathematics. In the inaugural talk on March 13, 2020, S.-T. Yau spoke about S. S. Chern as a great geometer of the 20th century [1]. Particularly in the discussion section after the talk, Prof. Yau emphasized the essential role which Roman Jackiw played in bringing Chern-Simons theories into physics. In this note I wish to share some recollections from those times, and especially my interactions with Roman. I do this because I most strongly agree with Prof. Yau’s assessment of Roman’s contribution. In this as other areas, Roman’s work exhibits both the sheer joy of computation, while continually pushing to understand the greater significance of what is truly new and exciting. As a graduate student of David Gross, with Larry Yaffe we computed the fluctuations to one loop order about a single instanton at nonzero temperature [2]. I then went off to Yale University as a postdoc [3], where I worked with Tom Appelquist on gauge theories in three dimensions. Our motivation was to understand the behavior of gauge theories at high temperature, which for the static mode reduces to a gauge theory in three dimensions. We computed the gluon self energy to one and (in part) to two loop order. The gluon self energy isn’t gauge invariant, so the greater significance of our analysis wasn’t clear. But we also computed in an Abelian theory for a large number of scalars, which is a nice soluble model, and from Tom I learned the most useful craft of power counting diagrams. We concentrated on the gluons, since quarks are fermions, and decouple in the static limit. Sometime in the fall of 1980, I went to MIT to give a talk on this work. It was a joint seminar with Harvard, and I remember it well to this day. Talks were then on transparencies, and the evening before I thought I would be clever, and added a comment that while topological charge in four dimensions is quantized classically, perhaps it isn’t quantum mechanically. Sidney Coleman was sitting near the front, in a purple crushed velvet suit which to me looked very much like that of Superfly. When he saw that slide, however, Coleman pounced, and did not let up until I surrendered abjectly, admitting to the idiocy of my suggestion. During the talk and for the entire afternoon after, Roman grilled me about details of the calculation, how gauge theories in three dimensions work, what about gauge invariance, everything. It was very intense and quite exhilarating. Roman and S. Templeton then wrote a paper on theories in three dimensions [4], which because Roman is a superb calculator, appeared as a preprint a few weeks before ours. When their paper came out, I remember looking at it, and thinking, ah, very good, they were spending their time on something irrelevant, two component fermions in three dimensions. Now if you forget about nonzero temperature and just do dimensional reduction, the natural thing is to go from four component fermions in four dimensions to four component fermions in three dimensions. And thus I missed the really new physics, which Roman grasped. The most interesting part of the paper by Roman and Templeton appears a bit pedestrian, at the beginning of Sec. III, Eqs. (3.3) and (3.4). Under the Lorentz (or Euclidean) group in three dimensions, all one needs are two component fermions, since in three dimensions one can just take the Dirac matrices to be the Pauli matrices. What I did not work out is that under the discrete transformations of parity and charge conjugation, that a mass term for a single two component fermion is parity odd! Roman and Templeton discuss in their Ref. (11) [4]: the dimensional reduction of a four component fermion gives two two-component fermions. The masses for these are of equal in magnitude, but opposite in sign. This is how the mass for four dimensional fermions, which is certainly parity even, remains so after dimensional reduction to three dimensions. Thus while if wasn’t present in the original paper by Roman and Templeton, if one computes the gauge self energy for two-component fermions in three dimensions, then a parity mass term for the gauge field will appear immediately. Yes, it has nothing to do with nonzero temperature, but so what? It is an absolutely beautiful, novel, and gauge invariant mass term, special to three dimensions. This was first proposed in two papers by S. Deser, Roman, and Templeton, in a Physical Review Letter [5], and a long paper in Annals of Physics [6]. The year before, Jonathan Schonfeld has proposed the same theory [7]. What Deser, Roman, and Templeton realized, however, and which Schonfeld did not, is that there is something extraordinary about a non-Abelian Chern-Simons term [8]: the ratio of the Chern-Simons mass term to the gauge coupling is an integer,