First of all I do not recommend using FastQuery, because it can be really slow and many people have had problems with it: https://blog.coates.dk/2014/11/19/sitecore-fast-query-syntax-can-kill-your-sql-server-or-website/
But if you still want to use here is the trick:
The field in Sitecore is not Display Name, it is "Display Name".
All the standard fields
are prefixed by "" 2 underlines char.

Your fast query should look like this:
fast:/sitecore/content//*[@#__Display Name#='homedisplayname']
The Display Name
for my home item is: homedisplayname
I added #field name# because multi word names should be encoded with "#"
I tried fast query in Developer Center and here are the results.
You need to change a bit of your code to have same pattern like my query from Developer Center.
