How to prove $\Lambda_\mathrm{weight}(\mathfrak{g})/ \Lambda_\mathrm{root}(\mathfrak{g}) = \mathbb{Z}_N$ for $\mathrm {SU}(N)$? I have a question about $\mathrm {SU}(N)$ Lie group from David Tong's Gauge Theory notes (p. 92).
He considers $\Lambda_w(\mathfrak{g})$ and $\Lambda_\mathrm{root}(\mathfrak{g})$ as the weight lattice and the root lattice for $\mathfrak{g}=\mathfrak{su}(N)$ Lie algebra.
And he shows $\Lambda_w(\mathfrak{g})/ \Lambda_\mathrm{root}(\mathfrak{g}) = \mathbb{Z}_N$, $\mathbb{Z}_N$ is the center of $\mathrm {SU}(N)$.
I would like to know how to prove this and how to interpret the $\mathbb{Z}_N$ values. He explains they are labeled by integers and related to possible Wilson lines and 't Hooft lines. Do they correspond to the electric and magnetic charge?
 A: I will merely sketch the derivation for the standard proof that the quotient of the weight lattice by the root lattice produces the centre of an (arbitrary semisimple Lie) group:

*

*prove that the centre of the group is the intersection of its maximal tori $\{T_i\}$

*use the restriction of the exponential map $\exp: \mathfrak t\to T$ to define another abelian group homomorphism $\exp'\ \cdot=\exp 2\pi i\ \cdot$, whose kernel is the dual lattice of the weight lattice

*the preimage of the group centre under the homomorphism $\exp'$ is the dual of the root lattice - this sets up the isomorphism $Z(G)\cong\Lambda_\mathrm{root}^*/\Lambda_\mathrm{weight}^*$ via the first isomorphism theorem

*Use that $\Lambda_1^*/\Lambda_2^*=\Lambda_2/\Lambda_1$
For an explicit computation for $\mathrm{SU}(3)$, see this Maths.SE answer (and links therein for a rigorous proof).
In a non-abelian gauge theory, electric charges take values in a sublattice of $\Lambda_\mathrm{weight}(\mathfrak g)$, while magnetic charges via GNO quantisation are known to take values in a sublattice of $\Lambda_\mathrm{co-weight}(\mathfrak g)$. This is equivalently the weight lattice of the Lie algebra of the Langlands dual group of $G$, leading to the famous Montonen-Olive duality.
In the context of $\mathfrak g=\mathfrak{su}(N)$ which is mapped to itself under the Langlands dual, this means that Wilson lines, which are classified by tensor products of the fundamental representation, i.e. all representations, are far more abundant than 't Hooft line operators that are restricted only to tensor powers of the adjoint. Indeed, analysis is simplified a fair bit since there is a canonical identification between the weight/co-weight and root/co-root lattices.




Operator
$\mathrm{SU}(N)$
$\mathrm{SU}(N)$$/\mathbb Z_n$




Wilson Line
$\Lambda_\mathrm{weight}(\mathfrak g)$
$\Lambda_\mathrm{root}(\mathfrak g)$


't Hooft Operator
$\Lambda_\mathrm{root}(\mathfrak g)$
$\Lambda_\mathrm{weight}(\mathfrak g)$




This works because you can prove that 't Hooft operators are labelled by $\mathrm{Hom}(\mathrm U(1), T)$, with $T$ the maximal torus of $G$ and this is isomorphic to $\Lambda_\mathrm{co-weight}(\mathfrak g)$. Additionally, Wilson lines are labelled by elements of $\mathrm{Hom}(T, \mathrm U(1))\cong\Lambda_\mathrm{weight}(\mathfrak g)/\mathcal W$ (where $\mathcal W$ is the Weyl group) and so this is a state-operator correspondence of sorts - electric charge $\leftrightarrow$ magnetic charge and Wilson lines $\leftrightarrow$ 't Hooft operators under the duality.
Since $\Lambda_\mathrm{weight}/\Lambda_\mathrm{root}\cong\Lambda_\mathrm{co-weight}/\Lambda_\mathrm{co-root}\cong\mathbb Z_N$, you can interpret the $\mathbb Z_N$ as labels for the transformation properties of the Wilson and 't Hooft operators under the electric and magnetic centres of the group: think of the quotient as creating "sectors" labelled by $n_1\in0\dots N$ for transformations under the root lattice and $n_2\in0\dots N$ for transformations under the co-root lattice (i.e. the root lattice while considering the dual theory): the zero sectors transform trivially under the corresponding lattice vectors.
Furthermore, the $\mathbb Z_n$ also describes a discrete generalised one-form symmetry (which can also be gauged) of the corresponding Yang-Mills theory - electric in the case of $\mathrm{SU}(N)$, magnetic in the case of $\mathrm{SU}(N)/\mathbb Z_N$, which when acting on the corresponding operators, cause them to pick up a phase $\phi$ with $\phi^N=1$.
