(3,4) Ising minimal model CFT

introduction

• Ising model
  

• first review the minimal models page
• Weber functions and conformal field theory
  
• rank 1 case
  

 

 

Ising model as a minimal model

• Ising model is a unitary minimal model and thus can be understood by the representation of Viraroso algebra
  
• the representation is given by following data
  \(m= 3\)
  central charge \(c = 1-{6\over m(m+1)} = \frac{1}{2}\)
  \(h_{p,q}(c) = {(4p-3q)^2-1 \over 48}\)
  \(p= 1,2\), \(q = 1,\cdots p\)
  \((p,q)=(1,1), (2,1), (2,2)\)
  (1,1)=(2,3), (1,2)=(2,2), (1,3)=(2,1)
  
• possible values of \(h\)
  \(0, 1/2, 1/16\)
  

 

 

graded dimensions

• associated chiral algebra has three irreducible modules with the following graded dimensions
  h=0, \(\chi_0=\chi _{1,1}^{(3,4)}=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
  h=1/2, \(\chi_{\epsilon}=\chi_{1,3}^{(3,4)}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
  h=1/16, \(\chi_{\sigma}=\chi _{1,2}^{(3,4)}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)
  
• Rocha-Caridi character[RC84] bosonic characters of Virasoro minimal models(Rocha-Caridi formula)
  

\(\chi_{\sigma}=\chi _{1,2}^{(3,4)}=\chi _{2,2}^{(3,4)}=\frac{\eta (2\tau )}{\eta (\tau )}=q^{1/24}\sum _{m=-\infty }^{\infty } (-1)^mq^{3 m^2- m }=q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)

\(\mathfrak{f}_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})=\sqrt{2}q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)

 

 

modularity of graded dimensions

\(\chi_M(-1/\tau)=\sum_{N} S_{M,N}\chi_N(\tau)\)

\(\chi_M(\tau+1)=\sum_{N} T_{M,N}\chi_N(\tau)\)

 

\(T=\left(\begin{array}{ccc}e^{-\pi i/24} & 0 & 0 \\ 0 & e^{23\pi i/24} & 0 \\ 0 & 0 & e^{\pi i/12}\end{array} \right)\)

\(2S=\left(\begin{array}{ccc}1 & 1& \sqrt{2} \\ 1 & 1 & -\sqrt{2} \\ \sqrt{2} & -\sqrt{2} & 0\end{array} \right)\)

 

 

matching two sets of funtions

• http://www.research.att.com/~njas/sequences/A000009
  

 

1

\(f(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)

\(\chi_0+\chi_{\epsilon}=\chi_{1,1}+\chi_{1,3}\)

 

{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}

http://www.research.att.com/~njas/sequences/A027349

2

\(f_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)

\(\chi_0-\chi_{\epsilon}=\chi_{1,1}-\chi_{1,3}\)

{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}

http://www.research.att.com/~njas/sequences/A081362

 

 

3

\(\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)

\(\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)

 

{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}

{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}

sum http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)%2B(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)

{2,-1,0,0,1,0,1,0,3,...} ->  \[q^{-1/48}(1-1/2q^{1/2}+q^{4/2}+q^{6/2}+3q^{8/2}+\cdots)\]

difference http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)-(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)

{0,1,0,2,-1,2,-1,2,-1,...} ->  \[\frac{1}{2}q^{-1/48}(q^{1/2}+2q^{3/2}-q^{4/2}+2q^{5/2}-q^{6/2} +2q^{7/2} -q^{8/2}\cdots)\]

 

 

4

\(f_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)

\(\chi_{\sigma}=\chi_{1,2}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)

{1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296}

http://www.research.att.com/~njas/sequences/A000009

 

related items

• Integrable perturbations of Ising model‎
• determinantal identities and Airy kernel
• Weber functions and conformal field theory

 

 

 

articles

• http://dx.doi.org/