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On the 2d ∞ 1 24 , η(τ )2 η(2τ )η( τ2 ) = n=1 with q = e2π iτ , where η(τ ) = q 24 other hand, we have 2d η(2τ ) η(τ ) = n = η(τ )2d , (1 − q ) n=1 tr SF(θ ) θ q d L0 + 12 = q 2d ∞ 1 − 48 (1 − q n− 12 ) = n=1 η( τ2 ) η(τ ) 2d . 2) (see [25, Chap. 4]). 3) φ1 (τ )2d ± φ2 (τ )2d . 5) = φ2 (τ ) which follow from the well known modular transformation lows πi η(τ + 1) = e 12 η(τ ), η − 1 τ = (−iτ )1/2 η(τ ). By using the formula we have the following proposition. 6) 790 T. 5 The modular transformations of SSF ± (τ ) and SSF(θ )± (τ ) with respect to the transformations τ → τ + 1 and τ → − τ1 are given by SSF ± (τ + 1) = e SSF ± 1 − τ = 1 2d+1 dπi 6 SSF(θ )+ (τ ) − SSF(θ )− (τ ) ± SSF(θ )± (τ + 1) = ±e− SSF(θ )± − 1 τ = SSF ± (τ ), dπi 12 (−iτ )d (SSF + (τ ) − SSF − (τ )), 2 SSF(θ )± (τ ), 1 SSF(θ )+ (τ ) + SSF(θ )− (τ ) ± 2d−1 SSF + (τ ) + SSF − (τ ) .

4]). 3) φ1 (τ )2d ± φ2 (τ )2d . 5) = φ2 (τ ) which follow from the well known modular transformation lows πi η(τ + 1) = e 12 η(τ ), η − 1 τ = (−iτ )1/2 η(τ ). By using the formula we have the following proposition. 6) 790 T. 5 The modular transformations of SSF ± (τ ) and SSF(θ )± (τ ) with respect to the transformations τ → τ + 1 and τ → − τ1 are given by SSF ± (τ + 1) = e SSF ± 1 − τ = 1 2d+1 dπi 6 SSF(θ )+ (τ ) − SSF(θ )− (τ ) ± SSF(θ )± (τ + 1) = ±e− SSF(θ )± − 1 τ = SSF ± (τ ), dπi 12 (−iτ )d (SSF + (τ ) − SSF − (τ )), 2 SSF(θ )± (τ ), 1 SSF(θ )+ (τ ) + SSF(θ )− (τ ) ± 2d−1 SSF + (τ ) + SSF − (τ ) .

Math. Phys. 202, 169–195 (1999) 8. : Classification of irreducible modules for the vertex operator algebra M(1)+ II. Higher Rank, J. Algebra 240, 389–325 (2001) 9. : Induced modules for vertex operator algebras. Commun. Math. Phys. 179 (1), 157–183 (1996) 10. : On quantum Galois theory. Duke Math. J. 86 (2), 305–321 (1997) 11. : On axiomatic approaches to vertex operator algebras and modules. Mem. Am. Math. Soc. 104, (1993) 12. : Vertex operator algebras and the monster. Pure and Applied Mathematics, 134, Academic Boston (1988) 13.