GHY term in loop quantum gravity Does the GHY (Gibbons-Hawking-York) boundary term play any role in loop quantum gravity? If it does where and how does it show up? (I am more interested in 4D)
 A: I'll take a stab at answering this.
Short answer: it does not play a major role in neither the canonical or covariant LQG.
A bit about boundary terms
Terms like that aren't at all mysterious, in fact, they are ubiquitous in all first-order systems. Take for example the action from Hamiltonian mechanics
$$ S[x, p] = \int dt \left( p \dot{x} - H(x, p) \right). $$
This gives rise to the symplectic form
$$ \omega = dx \wedge dp. $$
In fact the same action can be written in a more symmetric form as
$$ S[z] = \int \left( \xi - H(z) dt \right), $$
where $z=\{x,p\}$; $\xi = p dx$ is the form that gives $\omega = d \xi$ after differentiation.
But $\xi$ isn't the only such form, in fact, any function $f(z)$ gives rise to a form $\xi' = \xi + df$ that also has the same property $\omega = d\xi'$ due to nilpotence of the exterior derivative. This means we can add to the action an extra term
$\Delta S = \int df = f(b) - f(a)$.
In field theory, this term becomes a boundary integral through Stoke's theorem. GHY is just a specific form of such a term for gravity.
Canonical LQG
Canonical LQG is based on the Holst action:
$$ S[e, \omega] = \frac{1}{2 \kappa} \int e^I \wedge e^J \wedge \left( \star + \frac{1}{\gamma} \right) F_{IJ}. $$
Here $\gamma$ doesn't change the classical e.o.m. (at least in the absence of spinor degrees of freedom). But it is essential for LQG that $\gamma$ is finite to make sense as a quantum theory, for instance, the spectrum of geometric operators on the kinematical Hilbert space turns out to depend on $\gamma$.
On the classical level, however, we can send $\gamma \rightarrow \infty$ which leaves us with the frame-connection aka tetradic Palatini action
$$ S[e, \omega] = \frac{1}{2 \kappa} \int e^I \wedge e^J \wedge \star F_{IJ} = \frac{1}{4 \kappa} \int \varepsilon_{IJKL} e^I \wedge e^J \wedge F^{KL}. $$
Now if we choose the boundary and fix the values of fields $e^I$ and $\omega^{IJ}$ on the boundary, we can solve the e.o.m. and recover the fields $e^I$ and $\omega^{IJ}$ in the bulk. We can then later plug those fields into the action to obtain the Hamilton's function (which is useful for some practical applications as well as comparing the quantum transition amplitude to it in the classical limit).
Except that we've forgotten that our formalism is 1-st order rather than 2-nd order. That means choosing the values of both $e$ and $\omega$ overspecifies the system. Instead, we should only choose $\omega$, or only $e$.
In LQG, one chooses $\omega$ and this choice is fixed from there on. LQG is a very specific quantization procedure that destroys the symplectic symmetry of the classical phase space. In other words, after quantizing with LQG, you no longer have the option to choose $e$ instead of $\omega$ to be your boundary variable.
However, in the classical theory (which is not directly related to LQG, hence the short answer above), if you chose to use $e$ as your boundary variable, you'd discover that the Hamilton function would differ by a fixed term. That term doesn't impact classical physics (read, e.o.m.) because it is a boundary term, and it is exactly equal to GHY.
Covariant LQG
Covariant LQG is based on Plebanski gravity, which is a reformulation of the Holst gravity into a constrained topological theory:
$$S[B, \omega] = \frac{1}{2 \kappa} \int B_{IJ} \wedge F^{IJ}, $$
with an extra "simplicity constraint" that has to guarantee that the only form $B$ is allowed to take is
$$ B_{IJ} = \left( \star + \frac{1}{\gamma} \right) e_I \wedge e_J $$
for some $e$. The starting point of covariant LQG is the observation that this constraint can be encoded by a linear function over $B$ in the time gauge (this breaks manifest $SO(3,1)$ to its little group $SO(3) \sim SU(2)$, but the full Lorentz symmetry is restored in the quantum theory), and only when $\gamma$ is finite.
One then chooses a specific method of quantizing BF theory, that has to follow "quantize then constrain" rather than "constrain then quantize", because we intend to modify some of the constraints in the quantum theory to account for the simplicity constraint. The perfect fit is the Ooguri spinfoam model. After implementing the simplicity constraint on top of Ooguri model one gets covariant LQG.
The Ooguri model uses a discretization that assigns holonomies of $\omega$ to edges and integrated fluxes of $B$ to faces of spinfoams. This means that on the boundary of the spinfoam, which is a spin network composed of edges, we use the (integrated-exponentiated version of) $\omega$ and not $e$, so we avoid the GHY term again.
As a short conclusion, I think it is fair to say that GHY is a reasonable form of a counterterm to expect in a quantum gravity theory, but LQG is simply too different from conventional means of quantization, so the need for such counterterms simply doesn't arise.
