# Communications in Mathematical Physics - Volume 188 by A. Jaffe (Chief Editor)

By A. Jaffe (Chief Editor)

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**Extra resources for Communications in Mathematical Physics - Volume 188**

**Example text**

1. Here we consider the genus 1 case. e. a compact Riemann surface of genus 1, realized as the quotient X ∼ = L\C, where L is a rank 2 lattice in C, generated by the translations α(z) = z + 1 and β(z) = z + τ , where Im τ > 0. e. a ||µ||∞ < 1 function on C satisfying µ◦γ =µ for all γ ∈ L, and let f = f µ be the normalized (fixing 0, 1, ∞) solution of the Beltrami equation on C fz¯ = µfz . ˜ It is easy to see that f ◦ L = L ◦ f , where L˜ is the rank 2 lattice in C generated by 1 and τ˜ = f (τ ).

Here we pass to the dual computations in cohomology. Let fzz µz d z ∧ d z¯ , ω[f ] = fz be the density of Polyakov’s action functional in the genus zero case, where µ = fz¯ /fz . Obviously, ω[f ] can be considered as an element in C2,0 , that is a two-form valued zero cochain on . Then there exist elements θ[f ] ∈ C1,1 and Θ[f ] ∈ C0,2 such that δω[f ] = d θ[f ] so that the f -dependent cochain cocycle in Tot C, that is f and δθ[f ] = d Θ[f ] , def = ω[f ] − θ[f ] − Θ[f ] of total degree two is a D(ω[f ] − θ[f ] − Θ[f ]) = 0 .

G. [22, 31]), Eq. 2) is equivalent to the following Cauchy-Riemann equation (∂¯ − µ∂) c δW − T (z) /(fz )2 δµ(z) 12π =0 with respect to the complex structure on C defined by the coordinates ζ = f (z, z), ¯ ζ¯ = f (z, z). ¯ Using the regularity of the stress-energy tensor at ∞ one gets that δW = T(z) δµ(z) µ = c T (z) . 3) This variational equation for determining W was explicitly solved by Haba [18]. Specifically, let f tµ be the family of self-mappings of C associated to the Beltrami coefficients tµ, 0 ≤ t ≤ 1.