  We give the first examples of Fano manifolds with multiple optimal tori,
i.e.~we construct monotone Lagrangian tori $L$, such that the weighted number
of holomorphic Maslov index two discs with boundary on $L$ equals the upper
bound given by the symplectic invariant $\limsup_n ([m_0(L)^n]_{x^0})^{1/n}$,
where $m_0(L)$ is the Floer potential.
  To every trivalent graph $\gamma$ of genus $g$ we associate an optimal torus
$L_\gamma$ on the celebrated symplectic Fano manifold $\mathcal{N}_g$ (of
complex dimension $3g-3$) with $\mathrm{T}_{\mathcal{N}_g} = 8g-8$), given by
the character variety of rank 2 on a genus $g$ surface with prescribed odd
monodromy at a puncture, We moreover show that all pairs
$(\mathcal{N}_g,L_\gamma)$ are pairwise non-isotopic. In particular, we confirm
a form of mirror symmetry between the A-model of the pairs
$(\mathcal{N}_g,L_\gamma)$ (and also spaces $\mathcal{N}_g$ standalone) and
B-model of graph potentials, a family of Laurent polynomials we introduced in
earlier work.
  A crucial input from outside of symplectic geometry is an analysis of Manon's
toric degenerations of algebro-geometric models $\mathrm{M}_C(2,\mathcal{L})$
for the spaces $\mathcal{N}_g$, as moduli spaces of stable rank $2$ bundles on
an algebraic curve with a fixed determinant, constructed using conformal field
theory.
