Is anti-gravity (i.e. repulsive gravity) possible in theoretical physics?

As it happens we do observe anti-gravity (i.e. repulsive gravity) in the form of the dark energy. There are speculative ideas to describe this from string theory, though these are far from widely accepted.

In general relativity the curvature of space time is controlled by the stress-energy tensor. To get anti-gravity, or more precisely gravitational repulsion, you just need an appropriate stress-energy tensor. The problem is that normal matter cannot create a stress-energy tensor of the required form: instead you require exotic matter and we have never observed this in a lab.

However you have probably heard of dark energy, which is causing the expansion of the universe to accelerate. This does indeed create a gravitational repulsion. The trouble is that we have no idea what dark energy is.

You specifically mentioned string theory in your question, and one of the ideas from string theory is the string landscape. One possible scenario from this is that our universe is in a metastable state, and if so this would cause a cosmological constant that behaves like the dark energy we have observed.