Term 2, Problem Set 3

(1) Let G(k, n) be the Grassmannian of complex k planes in C^{n}. Consider the twodimensional sigma model with G(k, n) as target space. Determine the large n behavior for fixed k. Be sure to include the “theta angle” related to H^{2}(G(k, n), Z) = Z. Answer the usual questions: is there a mass gap? Are symmetries broken? What are the quantum numbers of the lightest particles?

(2) Now do the same for G_{R}(k, n), the Grassmannian of real k planes in R^{n}. Note that the “theta angle” is now Z/2Z-valued as H^{2}(G_{R}(k, n), Z) ≅ Z/2Z.

In this case, to understand the case that the theta angle is non-zero, you might find it helpful to look at my paper “θ Vacua In Two-Dimensional Quantum Chromodynamics,” Il Nuovo Cimento 51 A (1979) 325. (A few copies will be provided.)