E ( m ) K (1 − m ) + E (1 − m ) K ( m ) − K ( m ) K (1 − m ) = π $_{2}$. (A.12) Δ = 2( J$_{3}$ + 6 16 $_{π}$) + ϵ$^{√}$ 3 2 π exp $^{[}$− 8 6 π 3 ( J$_{3}$ + 6 16 $_{π}$) $^{]}$+ · · · , (A.13) Δ = 2 3 3 ( Q$_{3}$ + 3 2 32 $_{π}$) + ϵ$^{√}$ 6 4 π exp $^{[}$− 8 2 π 3 ( Q$_{3}$ + 3 2 32 $_{π}$) $^{]}$+ · · · . (A.14) To get the expansion for E (1 − m ) given above, it is convenient to use the Legendre’s relation, Using the q -series expansions we can eliminate the y (1 /y ) dependence of conserved charges in Table II (III) and find the functions Δ( J$_{3}$ ) , Δ( Q$_{3}$ ). For Case 1 and 3, one can verify and on the other hand for Case 2 In the above, ϵ is 1( − 1) for toroidal(cylindrical) membranes. [1] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, JHEP 10 , 091 (2008), arXiv:0806.1218 [hep-th]. [2] I. R. Klebanov and E. Witten, Nucl. Phys. B536 , 199 (1998), arXiv:hep-th/9807080. [3] Some authors call it M 3 , $^{2}$. [4] D. Fabbri, P. Fre, L. Gualtieri, and P. Termonia, Nucl. Phys. B560 , 617 (1999), arXiv:hep- th/9903036. [5] D. Fabbri et al. , Nucl. Phys. B577 , 547 (2000), arXiv:hep-th/9907219. [6] D. Martelli and J. Sparks, Phys. Rev. D78 , 126005 (2008), arXiv:0808.0912 [hep-th]. [7] A. Hanany and A. Zaffaroni, JHEP 10 , 111 (2008), arXiv:0808.1244 [hep-th]. [8] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Nucl. Phys. B636 , 99 (2002), arXiv:hep- th/0204051. [9] N. Beisert, Phys. Rept. 405 , 1 (2005), arXiv:hep-th/0407277. [10] A. A. Tseytlin(2004), arXiv:hep-th/0409296. [11] J. Plefka, Living Rev. Rel. 8 , 9 (2005), arXiv:hep-th/0507136. [12] M. Alishahiha and M. Ghasemkhani, JHEP 08 , 046 (2002), arXiv:hep-th/0206237. [13] S. A. Hartnoll and C. Nunez, JHEP 02 , 049 (2003), arXiv:hep-th/0210218. [14] P. Bozhilov, JHEP 03 , 001 (2006), arXiv:hep-th/0511253. [15] P. Bozhilov, JHEP 08 , 087 (2005), arXiv:hep-th/0507149. 19