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on a string
Under someone's control, influence, or manipulation. She kept teasing him with the possibility of a relationship in order to keep him on a string, convincing him buy her clothes and pay for her rent. The boss really has the new intern on a string. If he so much as coughs, he'll come running into his office to see what he can do for him.
on the string
Awaiting a decision. They told me they would call last week! How long am I going to be on the string?
on a/the string
Under one's complete control or influence.
See also: on