Is the wavefunction a real physical wave or only a mathematical abstraction? Does interaction with physical slits in a double slit experiment indicate that the wavefunction is a real physical wave, as opposed to a mathematical abstraction? This question pertains to the psi ontology debate as described here https://arxiv.org/pdf/1409.1570.pdf. In summary, the debate is about whether the wavefunction is a real physical wave as the psi-ontologists claim, or simply a mathematical abstraction that provides information in the form of probabilities as the epistemologists claim. Why doesn't interaction with physical slits prove the wavefunction is a real physical wave, i.e., an ontic entity?
 A: "Physical" in physics is defined as "what is measurable". "meaurable" has developed to mean not only direct measurements with rulers and voltmeters, but also by using mathematical models to arrive at a set of real numbers that consistently describe direct measurements. Sometime described as "proxy" measurements .
We have a table of elementary particles. Are these particles "physical"? We see a track of  an electron in a bubble chamber. In truth it is a "measurement" by proxy, where we model mathematically the electron ionizing on its track so that we see a macroscopic manifestation of its passing by proxy. We postulate that the elementary particles are "real" because of the predictability of the mathematical models we are using.
In this sense the complex conjugate square of the wave function that gives the probability for an event to happen, is physical. It is a real number and accumulating measurements with the same conditions always gives the same probability distribution, even though there are levels upon levels of modeling.

So how can a non physical entity interact with physical slits in a double slit experiment?

The wavefunction is the mathematical modeling of what happens when "electron scatters off double slit" . It is not the wavefunction that interacts, it is the electron which interacts with the boundary conditions of a double slit that can be fitted with a wavefunction which complex conjugate squared gives the probability distribution for the experiment. 
When you throw a projectile in the air , it traces a parabola which can be predicted by Newtonian mechanics perfectly. It is not the parabola that interacts with the gravitational field, it is the ball.
A: Probability is indeed an abstraction pertaining to the occurrence of events, probability amplitude is not--rather it is a projection onto an eigenstates. That is, a wave function $\psi(x)$ tells us that the projection of that state onto an eigenstate of the $\bf x$ operator, $\delta(x_0)$, is $\psi(x_0)$. Of course, all the projections add in quadrature, so the total 'amount' in $\delta(x_0)$ is $||\psi(x_0)||$.
Now if we let all the eigenstates propagate accordingly (e.g., through slits), they all add up (coherently), to look just like it is $\psi(x)$ that is a wave of probability amplitudes propagating.
A: Yes, it is physical enough. It is real enough. The specific words you use to describe that may be relevant and not the best way to ask a physics question, but your question and point are clear enough, from a physics point of view. 
We get too purist if sometimes, and in truth it is often necessary to be precise in what one states or writes, but sometimes we get a little pedantic. 
Eigenstates or projections or some other description of the state of the particle, such as a wave function, are equivalent for your purposes. And the fact is that they have an amplitude (sqrt of probability) and a phase. Both are real. One is how likely and the other one how to interfere (roughly; you know what phase is). It's been shown experimentally  that both are real, that phase makes a difference and of course so does amplitude, i.e. the full wave function affect the results. 
So, whichever words, the property (and we call it all those like prob. amplitude, projection into eigenstates, wave function, etc) is real, is physical. Not just a math concept. As you said, otherwise they would not interfere. 
A: I think this doubt doesn't really pertain quantum physics but, as already mentioned by others, derives from a bad use of concept of "physical" and "abstract mathematical concept". Well, when you model a physical system through "abstract" mathematics that mathematics is going to interact with your physical objects (which in reality are either abstracted in the model).We don't need quantum mechanics to show this.
Suppose we have a corp (think of it even like a classical newtonian corp) moving toward a slit. We don't know its exact position and velocity, we only have probability distribution (non physical mathematical notion you would say). Now when we know that the corp passed through the slit, this information impact our probability distribution, since we now know that all the position we initially considered as possible that lead to the corp impacting the wall instead of passing the slit must be refused (we get a conditioned probability p(x| it passed through the slit). So we update our probability distribution, but we still have some uncertainty on the position (the slit is not a single point) and about velocity, so our distribution will evolve expanding again like a wave generated by the slit, since the corp could be in the area within the slit, and could have different velocity (both speed and direction).
So well, you can see that our probability distribution interacted with a slit. Is it physical?
