Turn the carpet over so it rests face-down on a clean table or floor. Examine the carpet to locate the torn spot or spots. Mark torn spots with small pieces of duct tape so you don't lose their exact locations.
Push loose fibers through the tear with a putty knife. Avoid cutting the fibers with scissors, as you will make a hole in the carpet. Do not use your fingers, either, because this can increase the size of the tear.
Pull the edges of the tear together so it is not visible anymore. Cut a length of duct tape 2 to 3 inches longer than the tear.
Press the tape over the tear with one hand as you hold the edges of the tear together with the other hand. Flip the carpet on its back and inspect the repair. The tape should not be visible from this side, because you forced the edges of the tear together before taping them. Flip the carpet again and repeat the process to repair any other tears, if necessary.
Thread an upholstery needle with cotton, wool or other heavy-duty thread in the color that matches the Persian carpet background. Knot the thread at the end.
Snip loose carpet fibers with sharp scissors to the height of the rest of the carpet. Insert the needle through the back of the carpet to the front, 1/4-inch away from the edge of the fray. Extend the thread over the frayed section and insert it back through the carpet. Continue wrapping thread over the frayed section this way until you cover the entire frayed area.
Vacuum the repaired section of the carpet to pull up fibers that were matted.