"(3,4) Ising minimal model CFT"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
Ising model as a minimal model==
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
||
| 1번째 줄: | 1번째 줄: | ||
| − | ==introduction | + | ==introduction== |
* [[Ising models|Ising model]]<br> | * [[Ising models|Ising model]]<br> | ||
| 11번째 줄: | 11번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">Ising model as a minimal model | + | <h5 style="margin: 0px; line-height: 2em;">Ising model as a minimal model== |
* Ising model is a unitary minimal model and thus can be understood by the representation of [[Virasoro algebra|Viraroso algebra]]<br> | * Ising model is a unitary minimal model and thus can be understood by the representation of [[Virasoro algebra|Viraroso algebra]]<br> | ||
| 21번째 줄: | 21번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">graded dimensions | + | <h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">graded dimensions== |
* associated chiral algebra has three irreducible modules with the following graded dimensions<br> h=0, <math>\chi_0=\chi _{1,1}^{(3,4)}=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br> h=1/2, <math>\chi_{\epsilon}=\chi_{1,3}^{(3,4)}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br> h=1/16, <math>\chi_{\sigma}=\chi _{1,2}^{(3,4)}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)</math><br> | * associated chiral algebra has three irreducible modules with the following graded dimensions<br> h=0, <math>\chi_0=\chi _{1,1}^{(3,4)}=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br> h=1/2, <math>\chi_{\epsilon}=\chi_{1,3}^{(3,4)}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br> h=1/16, <math>\chi_{\sigma}=\chi _{1,2}^{(3,4)}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)</math><br> | ||
| 34번째 줄: | 34번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">modularity of graded dimensions | + | <h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">modularity of graded dimensions== |
<math>\chi_M(-1/\tau)=\sum_{N} S_{M,N}\chi_N(\tau)</math> | <math>\chi_M(-1/\tau)=\sum_{N} S_{M,N}\chi_N(\tau)</math> | ||
| 50번째 줄: | 50번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">matching two sets of funtions | + | <h5 style="margin: 0px; line-height: 2em;">matching two sets of funtions== |
* [http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009]<br> | * [http://www.research.att.com/%7Enjas/sequences/A000009 http://www.research.att.com/~njas/sequences/A000009]<br> | ||
| 114번째 줄: | 114번째 줄: | ||
| − | ==history | + | ==history== |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| 122번째 줄: | 122번째 줄: | ||
| − | ==related items | + | ==related items== |
* [[determinantal identities and Airy kernel]] | * [[determinantal identities and Airy kernel]] | ||
| 133번째 줄: | 133번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia== |
* http://en.wikipedia.org/wiki/ | * http://en.wikipedia.org/wiki/ | ||
| 145번째 줄: | 145번째 줄: | ||
| − | ==books | + | ==books== |
| 157번째 줄: | 157번째 줄: | ||
| − | ==expositions | + | ==expositions== |
| 163번째 줄: | 163번째 줄: | ||
| − | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles | + | <h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles== |
| 179번째 줄: | 179번째 줄: | ||
| − | ==question and answers(Math Overflow) | + | ==question and answers(Math Overflow)== |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
| 191번째 줄: | 191번째 줄: | ||
| − | ==blogs | + | ==blogs== |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
| 202번째 줄: | 202번째 줄: | ||
| − | ==experts on the field | + | ==experts on the field== |
* http://arxiv.org/ | * http://arxiv.org/ | ||
| 210번째 줄: | 210번째 줄: | ||
| − | ==links | + | ==links== |
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | * [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | ||
* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내] | * [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내] | ||
2012년 10월 28일 (일) 14:12 판
introduction
- 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==
\(f(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)
\(\chi_0+\chi_{\epsilon}=\chi_{1,1}+\chi_{1,3}\)
{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
\(f_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)
\(\chi_0-\chi_{\epsilon}=\chi_{1,1}-\chi_{1,3}\)
{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
\(\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
\(\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
{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,...} -> \(\frac{1}{2}q^{-1/48}}(q^{1/2}+2q^{3/2}-q^{4/2}+2q^{5/2}-q^{6/2} +2q^{7/2} -q^{8/2}\cdots)\)
\(f_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)
\(\chi_{\sigma}=\chi_{1,2}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)
{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
history
encyclopedia==
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
articles==
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
- http://mathoverflow.net/search?q=
- http://math.stackexchange.com/search?q=
- http://physics.stackexchange.com/search?q=
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field
links
\(m= 3\)
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)
\(0, 1/2, 1/16\)
- 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)
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions