it follows from the reconstruction formulae written in the form ( 4.45 ) that the reconstructed currents are anti - hermitian j † ± = −j ± , i.e. j ± ∈ su ( 2 ) . Now with the extra reality conditionθ 0 ∈ R/2πZ = S 1 on the extension of the algebro - geometric data , the reconstruction formula ( 4.24 ) is also easily checked to give an anti - hermitian current so that j ± ∈ su ( 2 ) . It therefore follows that as a result of imposing the reality conditions , the reconstructed field g in ( 4.27 ) becomes SU(2)-valued , and in particular , the original fields X 1 , . . . , X 4 describing the embedding of the string into the S 3 ⊂ R 4 part of the target space are real valued as required .