What is matter field? I am a mathematics student, and I recently read a paper which refers to the matter field. I just know about the gravity field. What is a matter field? Why can it be represented by smooth tensor fields? Besides, what is the mean of "slots"?  



 A: Well, in quantum field theory all kinds of matter and force particles are described as excitations of corresponding quantum fields. "Matter" usually means a field of fermions (such as electrons or quarks) while bosonic fields describe forces acting on and between the fermions.
And that is what is meant by matter field, a field whose excitations describe (probably fermionic) matter particles. As your paper says, such fields are tensor valued and transform in various representations of the Lorentz group and internal gauge symmetry groups.
I can't do much more except give you some examples, for a real understanding of this I guess you should read the first few chapters of some QFT book. But for example


*

*In quantum electrodynamics (QED), "matter" is electrons and positrons. This is described by a $\mathbb C^4$-valued field $\psi^A$ (electrons and positrons, spin-up and spin-down makes four degrees of freedom). The field transforms in the Dirac-/spinor representation of the Lorentz group like the Dirac wavefunction in relativistic quantum mechanics.

*In quantum chromodynamics (QCD), the up-, strange- and top quarks e.g. form a triplet under an internal $\mathrm{SU}(3)$ gauge group. That means, they are described all together by a $\mathbb C^4 \otimes \mathbb C^3$-valued field, i.e. a field $\psi^{A,i}$ with two indices. The Lorentz group acts on the first index ("slot") $A$ like in the QED-example above, while the $\mathrm{SU}(3)$ acts in the fundamental representation on the $i$-index.

