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Is there a fundamental equation in string theory and can it be derived from a variational principle?
If there is "a" fundamental equation at all in string theory, I guess you're looking for the Polyakov action,Is there a fundamental equation in string theory and can it be derived from a variational principle?
This is an FAQ: see What are the equations of string theory?Is there a fundamental equation in string theory and can it be derived from a variational principle?