From your query, it is not clear whether it is an anonymous mail or not. Irrespective, Deadsoul has given an apt example of why it should be investigated. If the manager is higher in rank than you, then you need to get your boss involved. Let me give some examples of how two situations (where complaints were anonymous) were dealt with.
Examples of Handling Anonymous Complaints
In the first case, an academic was accused of improper conduct under the influence of alcohol while on an away course with a mature female student. The matter was investigated by the departmental boss, and the academic was given a warning. In the second case, a student had pasted posters accusing two tutors of racism at a few points across the campus. The dean of the faculty, who saw that, got it all removed, called the concerned tutors, asked a few questions, and wanted to know how they would deal with it. The tutors did not want anything done as they did not know who the accuser was, and there the matter ended.
If such complaints are not investigated and are just brushed under the carpet, the organization will have more problems. Hence, kindly follow the sound advice of (Cite Contribution).
Have a nice day.
Regards,
Simhan
A retired academic in the UK