"the is the deep groove that separates the brain into two halves."

The biggest part of our brain, the cerebrum is separated into two halves, the left and the right hemisphere by a deep groove that is called the longitudinal fissure or cerebral fissure. 
The hemispheres are connected by a bundle of nerve fibers called the corpus callosum in order to enable communication between the two of them, as their communication is necessary for various cognitive functions of the brain.