Article 4, Section 2.
"No person held to service or labor in one state, under the laws thereof, escaping into another, shall, in consequence of any law or regulation therein, be discharged from such service or labor, but shall be delivered up on claim of the party to whom such service or labor may be due."
The movement in the North was to do away with this. The Underground Railroad was an example of an abuse. Taking labor from one state, and withholding them.
Amendment X
"The powers not delegated to the United States by the Constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people. "
This is more of a rule that allows the States to leave. The North, by inacting war, denied this right.
And upon secession, Amendement III (thought III may be argued prior to as well)
What would YOU call a fort in the middle of your property belonging to a different government? I'd call it quartering of troops in my property.
& XI and Article 4, Section 4 of the Constitution.
"The judicial power of the United States shall not be construed to extend to any suit in law or equity, commenced or prosecuted against one of the United States by citizens of another state, or by citizens or subjects of any foreign state. "
This means that holding that State's laws as "subjet" to other laws not in the Constitution is against the Constitution.
AND
IF: (as has been suggested) The actions of the South were NOT Legally recognizable.
THEN: The US government had NO Constitutional right to enter Southern Territory as an armed force!
4 or 5, depedning on if you view the secession of States as lawful.