Case 1 Case 2 Case 3 y 16 κ 2 ω 2 − 1 64 κ 2 3 ν 2 − 3 16 κ 2 ω 2 − 1 Δ √$_{6(}$$_{y}$$_{+1)}$ 8 π K √$_{6(}$$_{y}$$_{+3)}$ 8 π K √$_{6(}$$_{y}$$_{+1)}$ 8 π K R √ 6 16 3 √ 2 16 √ 6 16 Q$_{3}$ 0 3 √ 2 32 $_{π}$(4 K − E ) 3 √ 6 128 Q$_{8}$ 3 √ 2 64 9 √ 2 64 − 3 √ 2 128 J$_{3}$ √ 6 16 $_{π}$(2 K − E ) 0 √ 6 16 $_{π}$(2 K − E ) TABLE II. Conserved charges for cylindrical membranes. K ≡ K ( y ) ,E ≡ E ( y ). results in Table II. The charges are all normalized by λ , we set n = 1, and all the elliptic integrals have an argument y . The analysis for toroidal membranes with a nontrivial potential goes in a similar way. Let us again illustrate it with Case 1. From the periodic boundary condition, we have Then the energy is expressed as The results for other cases and different charges are reported in Table III. One notes that Case 1 and Case 3 give identical results for toroidal membranes. For both cylindrical and toroidal membranes, it is obvious that one can take the large energy limit in y → 1, where K ( y ) , K (1 /y ) develop a logarithmic divergence. For Case 1 and 3, the normalized charges scale as And for Case 2, One can of course do better and expand Δ in terms of J$_{3}$ or Q$_{3}$ , but we relegate the results as well as the details to the appendix. 15 π 2 = $^{√}$3 n $^{2}$e2 128 ∫ π/ 2 0 dθ $^{√}$κ 2 − ω 2 $_{16}$(1 + sin$^{2}$θ ) = √ 6 ne 4 ω √ y K (1 /y ) . (49) Δ √ λ = n $^{√}$6( y + 1) /y 8 π K (1 /y ) . (50) Δ ≈ √ 2 J$_{3}$ ; R, Q$_{3}$, Q$_{8}$ −→ 0 . (51) Δ ≈ 2 √ 3 3 Q$_{3}$ ; R, Q$_{8}$, J$_{3}$ −→ 0 . (52)