Coherent U(N) intertwiners in LQG and a measure on the Grassmanian
This is a detailed question about $U(N)$ intertwiners in LQG, and it comes from the the paper by <PERSON> and <PERSON> (2011 - archive). It is very specific but related to finding a measure on a quotient space so I hope it will get more general mathematical physics interest.
We start by casting LQG into the spinor language. 346
Most of the details are not necessary, but we have a set of $N$ spinors
$$|z_i\rangle =\left(\begin{array}{c} z^0_i\ z^1_i\end{array}\right).$$ In addition to the usual spinor relations, they satisfy the closure relation, which is
$$\sum_i |z_i\rangle\langle z_i| =A(z)\mathbb{1},\qquad A(z):=\frac{1}{2}\sum_i\langle z_i|z_i\rangle.$$
Now the states in the coherent intertwiner framework are holomorphic functions of $\mathbb{C}^{2N}$ that are homogenous functions under $GL(2,\mathbb{C})$ transformations. Thus, they are elements of a Grassmanian
$$Gr_{2,N}=\mathbb{C}^{2N}/GL(2,\mathbb{C})$$
Now, we want a measure on this Hilbert space (and a resolution of unity) so we start by relating the measures on the three spaces above (Eq (93) in that paper):
$$\int_{\mathbb{C}^{2N}}\left( \Pi_id^4z_i \right)e^{-\sum_i\langle z_i|z_i\rangle}F(z_i)=\int_{GL(2,\mathbb{C})}d\lambda e^{-Tr(\lambda^\dagger\lambda)}|det(\lambda)|^{2(N-2)}\int_{Gr_{2,N}}\Omega(z_i)F(z_i) $$
Here is my first question; Why does the measure on $\mathbb{C}^{2N}$ have the exponential? Is it there because we are looking at coherent states? I am also a bit confused about the measure on $GL(2,\mathbb{C})$, but for the moment I'm ok just thinking it probably comes from the <PERSON> measure and not the main point of my question.
The next thing they do in that paper is to say the measure on $Gr_{2,N}$can be represented as
$$\Omega=(\Pi_i d^4z_i) \delta^{(4)}\left(\sum_i|z_i\rangle \langle z_i|\right)$$
Then they go on to find the volume of $Gr_{2,N}$ as $\pi^{2(N-2)}/((N-1)!(N-2)!)$ which gives a normalized measure on the Grassmanian.
So the next question is how does this measure jive with the closure constraints? If we perform the integration over the Grassmanian we enforce the constraint
$$\sum_{i}|z_i\rangle\langle z_i|=0\rightarrow A(z)=0.$$
While that doesn't directly contradict the closure constraints, it does seem like it is adding an extra condition which is not implied by anything else. Can someone help me understand this measure? (and extra points for anything about that $GL(2,\mathbb{C})$ part) 669