"Characters of superconformal algebra and mock theta functions"의 두 판 사이의 차이
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| − | + | ==introduction== | |
| − | |||
| − | + | ==<math>\mathcal{N}=4</math> superconformal algebra== | |
| + | ===generators and relations=== | ||
| + | * [[Virasoro algebra]] | ||
| + | :<math>[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}</math> | ||
| + | * [[Affine sl(2)]] | ||
| + | :<math>[J_m^i,J_n^j]=\epsilon_{ijk}J_{m+n}^k+\delta_{m+n}\delta^{i,j}\frac{c}{3},\quad i,j,k\in \{1,2,3\},\quad m,n\in \mathbb{Z}</math> | ||
| + | :<math>[L_m,J_n^i]=-nJ_{m+n}^i,\quad m,n\in \mathbb{Z}</math> | ||
| + | * fermionic operators | ||
| + | :<math> | ||
| + | G_r^a,\overline{G}_s^b,\quad a,b\in \{1,2\} | ||
| + | </math> | ||
| − | + | ===<math>c=6k</math> with <math>k=1</math> case=== | |
| + | * non-BPS characters : <math>h>k/4,\ell=1/2</math> | ||
| + | :<math> | ||
| + | \operatorname{ch}^{\tilde R}_{h=1/4+n,\ell=0}=q^{h-3/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}=q^{n-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3} | ||
| + | </math> | ||
| + | * BPS characters : <math>h=1/4,\ell=0,1/2</math> | ||
| + | :<math> | ||
| + | \operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}\mu(z;\tau)\\ | ||
| + | \operatorname{ch}^{\tilde R}_{h=1/4,\ell=1/2}+2\operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=q^{-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3} | ||
| + | </math> | ||
| + | where <math>\mu(z;\tau)</math> is the [[Appell-Lerch sums]] which is a holomorphic part of a mock modular form | ||
| + | * this is related to [[Mathieu moonshine]] and the [[elliptic genus]] of K3 surface | ||
| − | |||
| − | < | + | ===<math>k\geq 2</math> case=== |
| + | * this is related to [[Umbral moonshine]] and elliptic genus of hyperKahler manifolds of complex dimension <math>2k</math> | ||
| − | |||
| − | + | ||
| + | |||
| + | |||
| − | + | ==history== | |
| + | * 1986 Eguchi-Taoimina <math>\mathcal{N}=4</math> superconformal algebra | ||
| + | * 1990 Odake, <math>\mathcal{N}=2</math> superconformal algebra | ||
| + | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| − | + | ||
| − | + | ||
| − | + | ==related items== | |
| + | * [[Supersymmetric minimal models]] | ||
| + | * [[Appell-Lerch sums]] | ||
| + | * [[Mathieu moonshine]] | ||
| + | |||
| − | + | ==encyclopedia== | |
| − | * http://en.wikipedia.org/wiki/ | + | * http://en.wikipedia.org/wiki/N_%3D_2_superconformal_algebra |
| + | * http://en.wikipedia.org/wiki/Super_Virasoro_algebra | ||
* http://www.scholarpedia.org/ | * http://www.scholarpedia.org/ | ||
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| − | + | ==articles== | |
| + | * Tohru Eguchi and Kazuhiro Hikami [http://dx.doi.org/10.1088/1751-8113/42/30/304010 Superconformal Algebras and Mock Theta Functions], 2009 | ||
| + | * Kawai, Toshiya, Yasuhiko Yamada, and Sung-Kil Yang. 1994. “Elliptic Genera and N = 2 Superconformal Field Theory.” Nuclear Physics B 414 (1–2) (February 14): 191–212. doi:[http://dx.doi.org/10.1016/0550-3213(94)90428-6 10.1016/0550-3213(94)90428-6]. | ||
| + | * Odake, Satoru. 1990. “c=3d conformal algebra with extended supersymmetry.” Modern Physics Letters A 05 (08) (March 30): 561–580. doi:http://dx.doi.org/10.1142/S0217732390000640. | ||
| + | * Odake, Satoru. 1990. “Character formulas of an extended superconformal algebra relevant to string compactification” International Journal of Modern Physics A 05 (05) (March 10): 897–914. doi:http://dx.doi.org/10.1142/S0217751X90000428. | ||
| + | * Eguchi, Tohru, and Anne Taormina. 1987. “Unitary Representations of the N=4 Superconformal Algebra.” Physics Letters B 196 (1) (September 24): 75–81. doi:[http://dx.doi.org/10.1016/0370-2693(87)91679-0 10.1016/0370-2693(87)91679-0]. | ||
| + | * Eguchi, Tohru, Hirosi Ooguri, Anne Taormina, and Sung-Kil Yang. 1989. “Superconformal Algebras and String Compactification on Manifolds with SU(n) Holonomy.” Nuclear Physics B 315 (1) (March 13): 193–221. doi:[http://dx.doi.org/10.1016/0550-3213(89)90454-9 10.1016/0550-3213(89)90454-9]. | ||
| + | * Yutaka Matsuo [http://ptp.ipap.jp/link?PTP/77/793/ Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra] , Prog. Theor. Phys. Vol. 77 No. 4 (1987) pp. 793-797 | ||
| + | [[분류:migrate]] | ||
| − | * [ | + | ==메타데이터== |
| − | * [ | + | ===위키데이터=== |
| − | + | * ID : [https://www.wikidata.org/wiki/Q6956294 Q6956294] | |
| − | + | ===Spacy 패턴 목록=== | |
| − | + | * [{'LOWER': 'n'}, {'LOWER': '='}, {'LOWER': '2'}, {'LOWER': 'superconformal'}, {'LEMMA': 'algebra'}] | |
2021년 2월 17일 (수) 02:14 기준 최신판
introduction
\(\mathcal{N}=4\) superconformal algebra
generators and relations
\[[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}\]
\[[J_m^i,J_n^j]=\epsilon_{ijk}J_{m+n}^k+\delta_{m+n}\delta^{i,j}\frac{c}{3},\quad i,j,k\in \{1,2,3\},\quad m,n\in \mathbb{Z}\] \[[L_m,J_n^i]=-nJ_{m+n}^i,\quad m,n\in \mathbb{Z}\]
- fermionic operators
\[ G_r^a,\overline{G}_s^b,\quad a,b\in \{1,2\} \]
\(c=6k\) with \(k=1\) case
- non-BPS characters \[h>k/4,\ell=1/2\]
\[ \operatorname{ch}^{\tilde R}_{h=1/4+n,\ell=0}=q^{h-3/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}=q^{n-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3} \]
- BPS characters \[h=1/4,\ell=0,1/2\]
\[ \operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}\mu(z;\tau)\\ \operatorname{ch}^{\tilde R}_{h=1/4,\ell=1/2}+2\operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=q^{-1/8}\frac{[\theta_{11}(z;\tau)]^2}{\eta^3} \] where \(\mu(z;\tau)\) is the Appell-Lerch sums which is a holomorphic part of a mock modular form
- this is related to Mathieu moonshine and the elliptic genus of K3 surface
\(k\geq 2\) case
- this is related to Umbral moonshine and elliptic genus of hyperKahler manifolds of complex dimension \(2k\)
history
- 1986 Eguchi-Taoimina \(\mathcal{N}=4\) superconformal algebra
- 1990 Odake, \(\mathcal{N}=2\) superconformal algebra
- http://www.google.com/search?hl=en&tbs=tl:1&q=
encyclopedia
- http://en.wikipedia.org/wiki/N_%3D_2_superconformal_algebra
- http://en.wikipedia.org/wiki/Super_Virasoro_algebra
- http://www.scholarpedia.org/
articles
- Tohru Eguchi and Kazuhiro Hikami Superconformal Algebras and Mock Theta Functions, 2009
- Kawai, Toshiya, Yasuhiko Yamada, and Sung-Kil Yang. 1994. “Elliptic Genera and N = 2 Superconformal Field Theory.” Nuclear Physics B 414 (1–2) (February 14): 191–212. doi:10.1016/0550-3213(94)90428-6.
- Odake, Satoru. 1990. “c=3d conformal algebra with extended supersymmetry.” Modern Physics Letters A 05 (08) (March 30): 561–580. doi:http://dx.doi.org/10.1142/S0217732390000640.
- Odake, Satoru. 1990. “Character formulas of an extended superconformal algebra relevant to string compactification” International Journal of Modern Physics A 05 (05) (March 10): 897–914. doi:http://dx.doi.org/10.1142/S0217751X90000428.
- Eguchi, Tohru, and Anne Taormina. 1987. “Unitary Representations of the N=4 Superconformal Algebra.” Physics Letters B 196 (1) (September 24): 75–81. doi:10.1016/0370-2693(87)91679-0.
- Eguchi, Tohru, Hirosi Ooguri, Anne Taormina, and Sung-Kil Yang. 1989. “Superconformal Algebras and String Compactification on Manifolds with SU(n) Holonomy.” Nuclear Physics B 315 (1) (March 13): 193–221. doi:10.1016/0550-3213(89)90454-9.
- Yutaka Matsuo Character Formula of C<1 Unitary representation of N=2 Superconformal Algebra , Prog. Theor. Phys. Vol. 77 No. 4 (1987) pp. 793-797
메타데이터
위키데이터
- ID : Q6956294
Spacy 패턴 목록
- [{'LOWER': 'n'}, {'LOWER': '='}, {'LOWER': '2'}, {'LOWER': 'superconformal'}, {'LEMMA': 'algebra'}]