Term 2, Problem Set Four

(1) In the two-dimensional sigma model with target space a sphere S^{n}, for large n, we introduced a Lagrange multiplier field σ in order to quantize the theory.

(a) By using the equations of motion, identify σ as an operator in terms of the original variables.

(b) What is the dimension of the σ field and therefore what would you expect to be the large q behavior (with Euclidean q) of ⟨σ(q)σ(−q)⟩. Compare to the actual behavior.

(2) Consider in two dimensions a U(1) gauge theory with two species of fermion, ψ_{i}, i = 1, 2, in the same representation of U(1), of bare masses m_{i}.

The Lagrangian is thus

L_{ψ}(ψ_{1}, m_{1}) + L_{ψ}(ψ_{2}, m_{2}) + ∫ (\frac{F^{2}}{4e^{2}} + i\frac{F}{2π} )  (1)

where L_{ψ} is the Lagrangian that we considered in class:

L_{ψ}(ψ, m) = ∫ (\overline{ψ}_{+}D_{−}ψ_{+} + \overline{ψ}_{−}D_{+}ψ_{−} + m\overline{ψ}_{−}ψ_{+} + \overline{m}\overline{ψ}_{+}ψ_{−}) d^{2}x.  (2)

(a) Show that the θ dependence disappears if either of the m_{i} vanishes.

(b) Show that in the limit that m_{1} becomes very large keeping m_{2} fixed (or letting m_{2} go to zero while m_{1} becomes large) the theory reduces to the one that we studied in class, but with a possibly shifted effective value of θ.

(c) Describe qualitatively the behavior when both of the m_{i} are large.

(d) Describe the behavior when both of the m_{i} are small.

(e) Show that if the m_{i} are equal, the theory has an SU(2) symmetry. Now, by considering the case that the m_{i} are equal and small, deduce that the two-dimensional sine-Gordon model, that is the model of a scalar field with potential V (ϕ) = μ^{2} · cos(βϕ), has an SU(2) symmetry for a certain value of β which you should identify.

(f) Taking the limit as μ goes to zero, where the sine-Gordon theory mentioned in (e) becomes a massless conformally invariant free field theory, deduce that the two-dimensional theory of a free massless scalar of the correct radius (or equivalently of fixed radius 2π and the correct coefficient k of the kinetic energy in L = k·∫ d^{2}x |dϕ|^{2}) has an SU(2) symmetry. Because of the conformal invariance this theory actually represents the affinization of SU(2) (more exactly, two copies of it, one for left-movers and one for right-movers) and not only the finite-dimensional group SU(2). This is called the vertex construction of the affine Lie group of SU(2).

To obtain it, we didn’t really need to go via the gauge theory. One could, more generally, simply consider several massless free fermions and, by applying the bose-fermi correspondence to them, get the vertex construction for SO(2n) and (with a slight modification) for SU(n).

Hint: a reference for most of Problem 2 (except the last paragraph) is Coleman’s “More On The Massive Schwinger Model.”