Characters of superconformal algebra and mock theta functions

$\mathcal{N}=4$ superconformal algebra

generators and relations

• Virasoro algebra

$$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n}$$

• Affine sl(2)

$$[J_m^i,J_n^j]=\epsilon_{ijk}J_{m+n}^k+\delta_{m+n}\delta^{i,j}\frac{c}{3},\quad i,j,k\in \{1,2,3\},\quad m,n\in \mathbb{Z}$$ $$[L_m,J_n^i]=-nJ_{m+n}^i,\quad m,n\in \mathbb{Z}$$

• 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=

related items

• Supersymmetric minimal models
• Appell-Lerch sums
• Mathieu moonshine

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