What is Kappa deformation in quantum gravity? I know this is a very broad question with few references but what is kappa deformation in the context of quantum gravity and non-commutative QFT (https://doi.org/10.1016/j.physletb.2019.01.063)?, I know that the reference does not have much to do with what I am trying to understand but I am quite lost about it.
 A: Firstly, $\kappa$-deformations is a really specific example of non-commutative geometry and non-commutative field theory.
As far as I'm aware kappa-deformations ($\kappa$-deformations) usually refers to one of two things, the $\kappa$-deformation of $n$-dimensional Minkowski space or the $\kappa$-Poincare quantum group.
A really good reference for this area is Anna Pachol's thesis:
https://arxiv.org/pdf/1112.5366.pdf
These are intimately related, for instance, $\kappa$-Minkowski spacetime possesses $\kappa$-Poincare symmetries. Anna Pachol's work goes through this in detail.
$\kappa$-Minkowski
Let $x^0$ be the timelike coordinate and $x^j$ the spatial coordinates of $n$-dimensional Minkowski spacetime, $\mathbb{R}^{(1,n-1)}$.
Usually, in ordinary geometry, we assume that these coordinates commute with each other. $[x^\mu, x^\nu] = 0$ for $\mu, \nu \in \{0,1,\dots ,n-1\}$.
This would mean that their quantum mechanical equivalent operators would be simultaneously diagonalisable, and so we would be able to find all coordinates of an event infinitely precisely.
However, we know that we can't know the position and momentum of an object simultaneously in quantum mechanics. This is realised in the variables/operators no longer commuting.
The idea of $\kappa$-deformation is to impose a special and specific non-commutativity between the timelike coordinate and the spatial coordinates.
It is usually written as: $[x^0, x^k] = \frac{i}{\kappa} x^j$ and $[x^j, x^k] = 0$, where $\kappa>0$.
This change causes deep reverberating changes throughout the physics defined up such a space. But it has been claimed that such a space appears in doubly-special relativity and spin-foams, [1][2].
$\kappa$-Poincare
This is a quantum group - which is a difficult topic for me to describe.
It is a specific modification to the relationships that the ordinary Poincare group satsfies. The modification is done in such a way so that various properties still hold.
The symmetries of a non-commutative space can be expressed as quantum groups, and the $\kappa$-Poincare is just a specific example of a procedure to create a quantum group from an ordinary Lie group.
There are quantum groups of $SU(2)$ etc.
Relation to Quantum Gravity and Noncommutative QFT
The idea behind a lot of this area of research is that if you drop the assumption that the key variables/observables of spacetime commute, then maybe you will get a quantised version of spacetime.
$\kappa$-Minkowski/$\kappa$-Poincare is just one strand in this tapestry of ideas.
Noncommutative Field Theory takes the idea of $\kappa$-Minkowski and expands upon it. Exploring other modifications to the commutation relations between coordinate variables, and the effect these will have on any quantum field theory defined in terms of these variables.
Quantum Groups have a lot of applications to difficult areas of study. And I'm not really capable of giving much more detail about them, but there are plenty of books about quantum groups.
There are whole areas of non-commutative geometry which are not based on this idea of subtly changing the commutation relations of spacetime variables.
References
[1]: Amelino-Camelia, Giovanni. "Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale." International Journal of Modern Physics D 11.01 (2002): 35-59.
https://www.worldscientific.com/doi/abs/10.1142/S0218271802001330
[2]: Freidel, Laurent, and Etera R. Livine. "3D quantum gravity and effective noncommutative quantum field theory." Physical review letters 96.22 (2006): 221301.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.96.221301
