Vertex operators for closed strings

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Abstract
A systematic construction of vertex operators for bosonic on-shell physical particles is presented for the closed bosonic, type-II, and heterotic strings in their critical dimensions. Anomalous Weyl dependences arising from contractions are explicitly calculated. Space-time gauge invariances are traced back to the presence of total derivative operators. For fermionic strings the use of superfields leads to both conceptual and computational simplifications.