# Why does the Weyl vector $\rho = \sum_{\lambda_i\in\Lambda_W} \lambda_i = \dfrac 12 \sum_{\alpha_i \in\Lambda_R^+}\alpha_i$ represents vacuum?

I was reading https://arxiv.org/pdf/1003.2861.pdf, and in p.5, just below equation (3.1), it was written that

The state $$|\rho\rangle$$ associated with the Weyl vector $$\rho$$ corresponds to the vacuum (no Wilson loop inserted).

Here $$\rho$$ is given by $$\sum_{\lambda_i\in\Lambda_W} \lambda_i = \dfrac 12 \sum_{\alpha_i \in\Lambda_R^+}\alpha_i$$. I was puzzled why this is true. I would think like, why is the vacuum not represented by $$|0\rangle$$?

I read from math.stackexchange that somehow this $$\rho$$ is called Weyl vector and is some kind of $$0$$ in certain kind of cohomology, and appears in the equation of the (affine) action $$w\cdot \lambda = w(\lambda + \rho)-\rho$$ of the Weyl group.

Can you help me put the bits and pieces together and tell me a complete picture why the quote is true? Thank you.