* Shakespeare DID write "A Midsummer Night's Dream". He was a playwright, and this is one of his famous comedies.
* The Globe Theatre DID exist during Shakespeare's time. It was a famous playhouse in London.
* It's highly likely that "A Midsummer Night's Dream" was performed at the Globe. We don't have definitive proof, but it's a very reasonable assumption given the play's popularity and the Globe's status as the leading playhouse in London.
However, we can't say for sure that Shakespeare personally directed the play at the Globe. Playwrights in Shakespeare's time weren't always the directors of their own works.