I was listened to Bump of chicken's song, Parade. I tried to figure out the grammar they used on each sentence and this phrase catch my attention.
忘れないで 忘れないで 心だけが世界。
Why do they use で here ? Is it work as " です” or て form or what?
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忘れないで here is the casual form of 忘れないでください, which means "don't forget". In general, when it's not the last word in the sentence it means "without forgetting".
As you guessed, it's basically one of two て forms for negative verbs. The other is 忘れなくて, but it's used in different contexts like 忘れなくてもいいです, "it's alright if you forget".