Non-unitary c(2,k+2) minimal models

non-unitary \(c(2,k+2)\) minimal models

central charge\(c(2,k+2)=1-\frac{3k^2}{k+2}\)\(k \geq 3\), odd
primary fields have conformal dimensions\(h_j=-\frac{j(k-j)}{2(k+2)}\), \(j\in \{0,1,\cdots,[k/2]\}\)
effective central charge\(c_{eff}=\frac{k-1}{k+2}\)
dilogarithm identity

\[\sum_{i=1}^{k-1}L(\frac{\sin^2\frac{\pi}{k+2}}{\sin^2\frac{(i+1)\pi}{k+2}})=\frac{2(k-1)}{k+2}\cdot \frac{\pi^2}{6}\]

character functions\(\chi_j(\tau)=q^{h_j-c/24}\prod_{n\neq 0,\pm(j+1)}(1-q^n)^{-1}\)
to understand the factor \(q^{h-c/24}\), look at the finite size effect page also
quantum dimension and there recurrence relation\(d_i=\frac{\sin \frac{(i+1)\pi}{k+2}}{\sin \frac{\pi}{k+2}}\) satisfies\(d_i^2=1+d_{i-1}d_{i+1}\) where \(d_0=1\), \(d_k=1\)

(*choose k for c (2,k+2) minimal model*)k := 11 (*define Rogers dilogarithm*) L[x_] := PolyLog[2, x] + 1/2 Log[x] Log[1 - x] (*quantum dimension for minimal models*) f[k_, i_] := (Sin[Pi/(k + 2)]/Sin[(i + 1) Pi/(k + 2)])^2 (*effective central charge*) g[k_] := (k*Pi^2)/(2 (k + 2)) (*compare the results*) N[Sum[L[f[k, i]], {i, 1, k - 1}] + Pi^2/6, 10] N[g[k], 10] d[k_, i_] := Sin[(i + 1) Pi/(k + 2)]/Sin[Pi/(k + 2)] Table[{i, d[k, i]}, {i, 1, k}] // TableForm Table[{i, N[(d[k, i])^2 - (1 + d[k, i - 1]*d[k, i + 1]), 10]}, {i, 1, k}] // TableForm