Abstract
The covariant (in the σ-model sense) background field method for calculating anomalies of the heterotic σ-model is developed in a curved superspace of two-dimensional (1, 0) supergravity. The results of the explicit calculation for the superconformal anomaly up to five loops are found to be in agreement with the known facts about the heterotic string effective action, though our five-loop calculation is only partial for the gauge field dependent terms. This supports Tseytlin's ansatz for the effective action to the case of the heterotic string theory and provides a straightforward way for calculating the action itself.
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