Eric Lippert nailed it, I'll use a different way to say what he said. All of the members of an interface are virtual and they all need to be overridden by a class that inherits the interface. You don't explicitly write the virtual keyword in the interface declaration, nor use the override keyword in the class, they are implied.
The virtual keyword is implemented in .NET with methods and a so-called v-table, an array of method pointers. The override keyword fills the v-table slot with a different method pointer, overwriting the one produced by the base class. Properties, events and indexers are implemented as methods under the hood. But fields are not. Interfaces can therefore not contain fields.