STRING THEORY Problem Set # 4 February 13, 1997

Problem 1

In this problem, we propose to prove a certain number of intermediate steps in the calculation of the Weyl anomaly of the determinant combinations

Z_{(n)}^{−}(g) = \frac{Det′Δ_{(n)}^{−}}{det(ϕ_{j}, ϕ_{k})_{g}det(ψ_{a}, ψ_{b})_{g}}

Here, we use the following expressions for covariant derivatives and Laplacian operators on sections f of K^{n}, (K is the canonical bundle of Σ and z, z̄ are local complex coordinates such that g = 2g_{zz̄}|dz|^{2})

\begin{matrix}∇_{(n)}^{z}f = & (g_{zz̄})^{−1}∂z̄f  Δ_{(n)}^{+}f = −2∇_{(n+1)}^{z}∇_{z}^{(n)}f \\ ∇_{z}^{(n)}f = & (g_{zz̄})^{n}∂_{z}(g_{zz̄})^{−n}f  Δ_{(n)}^{−}f = −2∇_{z}^{(n−1)}∇_{(n)}^{z}f \end{matrix}

Also, ϕ_{j} ∈ Ker∇_{z}^{(n−1)}, ψ_{a} ∈ Ker∇_{(n)}^{z} and these kernels have dimensions N_{(n−1)}^{+} and N_{(n)}^{−} respectively. Show that under infinitesimal Weyl transformations of the metric δ_{σ}g = 2δσg, we have

a) N_{(n)}^{±} are invariant,

b) and the trace of the heat-kernel changes by

δ_{σ}Tre^{−tΔ_{(n)}^{−}} = −t\frac{∂}{∂t}Tr[2nδσe^{−tΔ_{(n−1)}^{+}} − (2n − 2)δσe^{−tΔ_{(n)}^{−}}].

c) Establish the following short time (t) expansion of the heat-kernels

Trδσe^{−tΔ_{(n)}^{±}} = \frac{1}{4πt} ∫_{Σ} dμ_{g}δσ + \frac{1 ± 3n}{12π} ∫_{Σ} dμ_{g}R_{g}δσ + O(t),

d) as well as their long time limits

Trδσe^{−tΔ_{(n)}^{−}} = −\frac{1}{2n − 2}δ_{σ} ln det(ϕ_{j}, ϕ_{k})_{g} + O(e^{−λ_{1}t})
Trδσe^{−tΔ_{(n)}^{+}} = \frac{1}{2n}δ_{σ} ln det(ψ_{a}, ψ_{b})_{g} + O(e^{−λ_{1}t}),

where λ_{1} > 0.

e) Check that the result in c) reproduces the Riemann-Roch-Atiyah-Singer index theorem for the operators ∇_{(n)}^{z} and ∇_{z}^{(n+1)}.

Problem 2

a) Construct the vertex operators for all physical states with mass squared M^{2} = 2, in closed (oriented) string theory in flat space-time M = R^{26}, by requiring that vertex operators are integrals of primary fields of conformal dimension (1, 1).

b) Verify that the vertex operators you find are in one to one correspondence with the physical states |ϕ⟩, obtained by imposing the Virasoro conditions L_{n} − δ_{n,0} and L̃_{n} − δ_{n,0} on |ϕ⟩, (as was carried out in Lecture 3).