A major sponsor of the American Ballet Theater has pulled its financial support, and transferred its fiscal loyalties to New York City Ballet, saying that mismanagement at ABT forced the move. The Movado Watch Company has sponsored ABT productions for nearly two decades, to the tune of more than $400,000 per year, but the company’s chairman said in a letter that he was taken aback by the ABT management’s unwillingness to listen to criticism from its board. The sponsorship loss is being seen as a potentially crippling blow to ABT, which has struggled mightily in recent years.