have and have got to

The structures have to and have got to are used to talk about obligation.
The meaning is similar to must.

Sorry, I have (got) to go. (= I must go.)

Have got to can be used to express certainty.

You have got to be joking. (= You must be joking.)

This structure was once peculiar to American English, but is now becoming common in British English as well.

When have is followed by to, it can be used like an ordinary verb or like an auxiliary verb. When have is used like an ordinary verb, questions and negatives are made with do. When have is used like an auxiliary verb, questions and negatives are made without do.

When do I have to be back? OR When have I got to be back?

To talk about a future obligation we use ‘will have to‘. If the obligation already exists, we may use ‘have got to’.

I will have to find a new job.
She will have to get a new passport.

Will have to is often used to make instructions sound more polite.

You can go out, but you will have to be back before ten.

Gotta
Gotta is the conversational pronunciation of got to. It is common in informal American English.

I gotta go home. (= I got to go home.)