No the files are made by SQL Express…You have a master MDF file that is created by SQL Express… that mdf file contains references to all databases… including any for other applications… It is not a single file database created by the app. When you deleted it… SQL Express cannot start any databases or create them. You need to reinstall localdb and repair the mdf.
Give me a minute I will google some stuff for you.
this has some interesting information.
Open Server Explorer, right-click on Data Connections, select Create New SQL Server Database. For "Server Name" use "(LocalDB)\v11.0".
There is another option, as described here but it requires installation of SQL Server Data Tools. A version of the instructions for Visual Studio 2012 is also available.
Since you also mention SQL Server Management Studio, you can simply connect to the LocalDb instance and right-click on Databases, then Create, the standard way. It is more-or-less a regular SQL Server instance and all standard operations will function as usual.
Creating the database can also, obviously, be done from the application code as well, but that requires setting up appropriate database permissions. Depending on your environment that may or may not be a good idea.