Non-unitary c(2,2k+1) minimal models

central charge and conformal dimensions

• central charge\(c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}\)
• primary fields have conformal dimensions\(h_j=-\frac{j(2k-1-j)}{2(2k+1)}\), \(j\in \{0,1,\cdots,k-1\}\) or by setting i=j+1\(h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}\) \(i\in \{1,2, \cdots,k\}\) (this is An's notation in his paper)
• effective central charge\(c_{eff}=c-24h_{min}\)\(c_{eff}=\frac{2k-2}{2k+1}\)

character formula and Andrew-Gordon identity

• Andrews-Gordon identity
• character functions\(\chi_i(\tau)=q^{h_i-c/24}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)
• to understand the factor \(q^{h-c/24}\), look at the finite size effect page also
• bosonic characters of Virasoro minimal models(Rocha-Caridi formula)\(\chi_{r,s}^{(p,p')}=\frac{q^{\Delta_{r,s}^{(p,p')}}}{(q)_{\infty}}\sum_{n=-\infty}^{\infty}(q^{pp'n^2+(rp'-sp)n}-q^{(pn+r)(p'n+s)})\)\(\Delta_{r,s}^{(p,p')}=h_{r,s}^{(p,p')}-\frac{c}{24}\)

Let's specify p=2, p'=2k+1, r=1, s=i

\(\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-q^{(2n+1)((2k+1)n+i)})=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2n+1)((2k+1)n+i)})\)

\(=\sum_{n=-\infty}^{\infty}(q^{2(2k+1)n^2+(2k+1-2i)n}-\sum_{n=-\infty}^{\infty}q^{(2(-n)+1)(-(2k+1)(-n)+i)})\)

\(=\sum_{n=-\infty}^{\infty}(q^{2n\left[(2k+1)(2n)+(2k+1-2i)\right]/2}-\sum_{n=-\infty}^{\infty}q^{(2n-1)\left[(2k+1)(2n-1)+2k-2i+1\right]/2}\)

\(=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}\)

\(=\sum_{n=-\infty}^{\infty}(-1)^n(q^{\frac{(2k-2i+1)}{2}})^{n}(q^{\frac{(2k+1)}{2}})^{n^2}=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})(1-q^{-\frac{2k-2i+1}{2}}q^{\frac{2k+1}{2}(2m-1)})\)

\(=\prod_{m=1}^{\infty}(1-q^{(2k+1)m})(1-q^{(2k+1)m-i})(1-q^{(2k+1)m-(2k-i+1)})\)

Thus,

\(\chi_{r,s}^{(p,p')}=q^{\Delta_{r,s}^{(p,p')}}\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)

Using a result from 베일리 격자(Bailey lattice)

\(\sum_{n_1\geq\cdots\geq n_{k-1}\geq0}\frac{q^{n_1^2+\cdots+n_{k-1}^2+n_i+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots (q)_{n_{k-2}-n_{k-1}}(q)_{n_{k-1}}}=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{\frac{(2k+1)n^2}{2}}q^{\frac{n(2k-2i+1)}{2}}=\prod_{n\neq 0,\pm i\pmod {2k+1}}(1-q^n)^{-1}\)

asymptotic analysis of Andrews-Gordon identity

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

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

• non-unitary \(c(2,2k+1)\) minimal models

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

different expressions for central charge

• from above\(h_i-c(2,2k+1)/24\)\(c(2,2k+1)=1-\frac{3(2k-1)^2}{2k+1}\)\(h_i=-\frac{(i-1)(2k-i)}{2(2k+1)}\), \(i\in \{1,2, \cdots,k\}\)
• L-values\(\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}\)

Dirichlet L-function

• 디리클레 L-함수

\(L(-1, \chi) = \frac{1}{2f}\sum_{n=1}^{\infty}{\chi(n)}{n}\)

\(n\geq 1\) 이라 하자. 일반적으로 \(\chi\neq 1\)인 primitive 준동형사상 \(\chi \colon(\mathbb{Z}/f\mathbb{Z})^\times \to \mathbb C^{*}\)에 대하여 \(L(1-n,\chi)\)의 값은 다음과 같이 주어진다 \[L(1-n,\chi)=-\frac{f^{n-1}}{n}\sum_{(a,f)=1}\chi(a)B_n(\frac{a}{f})\] \[L(-1,\chi)=L(1-2,\chi)=-\frac{f}{2}\sum_{(a,f)=1}\chi(a)B_2(\frac{a}{f})\]

여기서 \(B_n(x)\) 는 베르누이 다항식(\(B_0(x)=1\), \(B_1(x)=x-1/2\), \(B_2(x)=x^2-x+1/6\), \(\cdots\))

Let N=2k+1

\(\omega=\exp \frac{2\pi i}{2k+1}\)

G: group of Dirichlet characters of conductor N which maps -1 to 1

G has order k and cyclic generated by \(\chi\)

\(c_i=\frac{1}{2k}\sum_{s=1}^{k}\omega^{is}L(-1,\chi^s)\)

Then,

\(c_i=-\frac{2k+1}{12}+\frac{j(N-j)}{2N}\)

where j satisfies \(\chi(j)=\omega^{k-i}\)

Vacuum energy is given by

\(d_i=\frac{1}{2}L(-1,\chi^{k})-c_i\)

Since

\(L(-1,\chi^{k})=\frac{N-1}{12}=\frac{k}{6}\), the vacuum energy

\(d_i=\frac{1}{2}L(-1,\chi^{k})-c_i=\frac{k}{12}+\frac{2k+1}{12}-\frac{j(N-j)}{2N}=\frac{j(2k+1-j)}{2(2k+1)}-\frac{k+1}{12}\).

These are equal to \({h_i-c/24}\)

1. k := 5 f[k_, j_] := (2 k)/ 24 + ((2 k + 1)/12 - (j (2 k + 1 - j))/(2 (2 k + 1))) Table[{j, f[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*f[k, j]}, {j, 1, 2 k}] // TableForm d[k_, j_] := (2 (k - j) + 1)^2/(8 (2 k + 1)) - 1/24 Table[{j, d[k, j]}, {j, 1, 2 k}] // TableForm Table[{j, -24*d[k, j]}, {j, 1, 2 k}] // TableForm cef[k_, j_] := -((j (2 k - 1 - j))/(2 (2 k + 1))) - (1 - (3 (2 k - 1)^2)/(2 k + 1))/24 Table[{j, cef[k, j]}, {j, 0, 2 k - 1}] // TableForm Table[{j, -24*cef[k, j]}, {j, 0, 2 k - 1}] // TableForm

1. w := Exp[2 Pi*I*1/k] L[j_] := -(2 k + 1)/2* Sum[DirichletCharacter[2 k + 1, j, a]* BernoulliB[2, a/(2 k + 1)], {a, 1, 2 k}] c[k_, i_] := 1/(2 k) Sum[w^(i*s)*L[Mod[3*s, 2 k]], {s, 1, k}] Table[DiscretePlot[{Re[DirichletCharacter[2 k + 1, j, a]], Im[DirichletCharacter[2 k + 1, j, a]]}, {a, 0, 2 k + 1}, PlotLabel -> j], {j, 1, EulerPhi[2 k + 1]}] Table[c[i], {i, 1, 2 k}]