21.1. Boundary States for a bosonic field Consider the two analytic and anti-analytic U(1) currents j(z) =n jnz−n−1 and

¯j( ¯ z) =n

¯jn ¯ z−n−1 related to a massless bosonic field ϕ(z, ¯ z) by the relations j(z) = i∂zϕ

and ¯j( ¯ z) = i∂¯ zϕ. Let us use the conformal map z = exp[τ +iσ] to map the upper halfplane into an infinitely long strip of width π.

a. Show that it holds the relations ∂σ ϕ = ∞

n=−∞
jn e−n(τ+iσ) − ¯jn e−n(τ−iσ)
, i∂τ ϕ = ∞

n=−∞
jn e−n(τ+iσ) + ¯jn e−n(τ−iσ)
.

b. The Neumann boundary condition at σ = 0 is implemented by the condition ∂σ ϕ|σ=0 = 0, while the Dirichlet boundary condition by ∂τ ϕ|σ=0 = 0. Show that these imply the following relations between the modes jn and ¯jn jn − ¯jn = 0 Neumann boundary condition.
jn + ¯jn = 0 Dirichlet boundary condition.

c. Now swap the role of σ and τ and consider the boundary states associated to both Neumann and Dirichlet boundary conditions ∂τ ϕ|τ=0|BN = 0 Neumann boundary state.
∂σ ϕ|τ=0|BD = 0 Dirichlet boundary state.
Show that their exact expression (up to normalization) is given by |BN = exp−
∞

k=1 1 k j−k ¯j−k  |0, |BD = exp+
∞

k=1 1 k j−k ¯j−k  |0.