Characters of superconformal algebra and mock theta functions

$\mathcal{N}=4$ superconformal algebra

$c=6k$ with $k=1$ case

• non-BPS characters : $h>k/4,\ell=1/2$

$$ \operatorname{ch}^{\tilde R}_{h=1/4+n,\ell=0} $$

• BPS characters : $h=1/4,\ell=0,1/2$

$$ \operatorname{ch}^{\tilde R}_{h=1/4,\ell=0}=\frac{[\theta_{11}(z;\tau)]^2}{\eta^3}\mu(z;\tau)\\ \operatorname{ch}^{\tilde R}_{h=1/4,\ell=1/2} $$ 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

• http://www.google.com/search?hl=en&tbs=tl:1&q=

related items

• 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
• 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.
• 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