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.


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