In this sentence the verb, or main verb, is "forgot". The infinitive form "to bring" is the object of the verb "forgot". It answers the question "what". What did you forget? You forgot "to bring your book". We can consider the phrase "to bring my book" the object of "forgot".
There is no auxiliary in this sentence. Auxiliaries are words such as do, did, does, have, has, had, is, are, was, were, can, have to, should, could, will, would, might, may, ought to, must.
Here are a few examples. The underlined words are auxiliaries.
I've forgotten my book again. I should go back to get it. It's won't take long. Can you wait for me here?
I'll go with you.
Okay, it's just a short walk.
If you have any follow-up questions, please, post them. I'd be glad to reply.