STRING THEORY Problem Set # 8 March 20, 1997

Let G be a semi-simple compact Lie group, with Lie algebra G, and structure constants f_{abc}, a, b, c = 1, ⋯, dim G. Let Σ be a surface with the topology of the plane, with a flat metric. The Wess-Zumino-Witten model is defined in terms of maps g : Σ → G and classical action

S[g] = \frac{k}{16π} ∫_{Σ} trθ ∧ ∗θ + \frac{k}{24π} ∫_{B} trθ̃ ∧ θ̃ ∧ θ̃.

Here, θ denotes the G-valued left-invariant one form θ = g^{−1}dg, B is a 3-dimensional manifold whose boundary is Σ. Also, we have θ̃ = g̃^{−1}dg̃ where g̃ : B → G is any differentiable extension of g to B such that g̃|_{Σ} = g.

Classical Part

a) Show that the action S[g] is of the same form as the generalized non-linear sigma models studied in the lectures, with action (for definitions, see problem set 7)

S_{G,B}[x] = \frac{1}{8πl^{2}} ∫_{Σ} dμ_{δ}[δ^{mn}G_{μν} + ϵ^{mn}B_{μν}]∂_{m}x^{μ}∂_{n}x^{ν}

b) Show that the following multiplication formula holds

S[gh] = S[g] + S[h] + \frac{k}{16π} ∫_{Σ} trg^{−1} ¯∂gh^{−1}∂h

and use it to show that S is well-defined, up to shifts by integer multiples of 2πk.

c) Using the formula of b), show that S[g] is invariant under the infinite dimensional current (Kac-Moody) algebra Ĝ × Ĝ :

g(z, z̄) → Ω(z)g(z, z̄)Ω̄(z̄)^{−1}

where Ω(z) and Ω̄(z̄) are holomorphic and anti-holomorphic maps from Σ into G (actually into the complexification of G) respectively. Derive the G-valued conserved current J = J^{a}t^{a} resulting from the transformations Ω. (t^{a} are generators of G, satsifying the structure relations [t_{a}, t_{b}] = f_{abc}t_{c}.) Derive the transformation law of J under Ω.

d) Derive the stress tensor T, show that it is holomorphic and express T as a quadratic form in terms of J.

Quantum Part

We now quantize the theory defined by S[g], and assume that the quantum current J and stress tensor T continue to be holomorphic. We also assume in part f) that the relation between the quantum current J and the quantum stress tensor T is given by the relation derived in d), except for an unknown multiplicative constant κ :

2κT (z) =: J^{a}(z)J^{a}(z) :

e) Show that the Ward identities for Diff(Σ) and Ĝ yield the following OPE’s on T and J

\begin{aligned}T (z)T (w) & ∼ \frac{c/2}{(z − w)^{4}} + \frac{2}{(z − w)^{2}} T (w) + \frac{1}{z − w} ∂_{w}T (w) \\ T (z)J^{a}(w) & ∼ \frac{1}{(z − w)^{2}} J^{a}(w) + \frac{1}{z − w} ∂_{w}J^{a}(w) \\ J^{a}(z)J^{b}(w) & ∼ \frac{kδ^{ab}}{(z − w)^{2}} + \frac{f^{abc}}{z − w} J^{c}(w) \end{aligned}

f) By considering the current correlation functions

⟨J^{a_{1}}(z_{1}) ⋯ J^{a_{n}}(z_{n})⟩

and limits thereof when different z_{i} approach one another, show that the OPE’s for J and T are consistent only if

c = \frac{k dim G}{k + C_{2}(G)}

where C_{2}(G) is the quadratic Casimir value of the adjoint representation of G, defined by f^{acd}f^{bcd} = C_{2}(G)δ^{ab}.

g) Show that, in the large k approximation, the metric G and the anti-symmetric tensor field B of the Wess-Zumino-Witten model, satisfy the equations for conformal invariance and for the central charge that were derived (in the small l^{2} limit) in the lectures.