STRING THEORY Problem Set # 7 March 13, 1997

Problem 1

Let M be a Riemannian manifold with local coordinates x^{μ}, metric G = G_{μν}dx^{μ}⊗dx^{ν}, two form B = B_{μν}dx^{μ} ∧ dx^{ν}, and scalar field Φ, and let l be a real positive parameter. The generalized non-linear sigma model action on a surface Σ, with metric g, Gaussian curvature R_{g}, volume form dμ_{g}, and anti-symmetric tensor ϵ^{mn}, is given by

S[x; g] = \frac{1}{8πl^{2}} ∫_{Σ} dμ_{g}[g^{mn}G_{μν} + ϵ^{mn}B_{μν}]∂_{m}x^{μ}∂_{n}x^{ν} + \frac{1}{2π} ∫_{Σ} dμ_{g}R_{g}Φ

a) Derive the form of the vector field V^{μ}(x) in T M, such that the infinitesimal Weyl transformation δg = 2δσg, combined with the transformation δx^{μ} = l^{2}δσV^{μ}(x) yields a trace of the stress tensor T_{mn}g^{mn} which is of the same form as the Lagrangian density of S[x; g], i.e. such that

δS[x; g] = −\frac{1}{2π} ∫_{Σ} dμ_{g}[g^{mn}β_{μν}^{G} + ϵ^{mn}β_{μν}^{B}]∂_{m}x^{μ}∂_{n}x^{ν} + \frac{1}{2π} ∫_{Σ} dμ_{g}R_{g}β^{Φ}

b) Derive the resulting expressions for β^{G}, β^{B} and β^{Φ}. (These are the tree level contributions to the β-functions defined for Weyl transformations.)

Problem 2

Let ϕ be a real scalar field on a Riemann surface Σ, with action

S[ϕ; g] = \frac{1}{8π} ∫_{Σ} dμ_{g}[g^{mn}∂_{m}ϕ∂_{n}ϕ + m^{2}ϕ^{2}]

where m is an IR regulator mass, which we let to zero whenever the limit is well-defined.

a) Use dimensional regularization to derive the Weyl transformation properties of the Green function G(ξ, ξ′) = ⟨ϕ(ξ)ϕ(ξ′)⟩ and its first and second derivatives, evaluated on the diagonal (i.e. at coincident points).

b) Repeat the calculation of a), but now with a heat-kernel regularization of the Green function. Compare your result with that of a).