The short answer to your question is that the verb that follows did should be used in its base form. Read below for more information.
The verb did, the past tense form of do, commonly appears before another verb in these two sentence types:
1. Questions about the past
2. Negative statements about the past
In both of these sentence types, did is an auxiliary verb (or “helping verb”) that is followed by a main verb, which carries the real verb meaning.
The auxiliary verb (did) is marked for past tense, but the main verb is not. It appears in its base form. A helpful way to remember this is that when there is an auxiliary verb, the main verb does not need to be marked for tense, because the tense is shown in the auxiliary. However, in a sentence about the past without an auxiliary verb, the main verb does need to be in the past tense form, as in this sentence:
I hope this helps.