By Yi-Zhi Huang

"The exposition is apparent and available. the required history material...is defined intimately in 3 appendices [and] one other appendix includes solutions to a couple workouts formulated within the text... Self-contained to a excessive degree... hugely recommended."

**--ZAA**

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**Additional info for Two-Dimensional Conformal Geometry and Vertex Operator Algebras**

**Example text**

In [FHLJ, in order to prove that the graded dual space of a module for a vertex operator algebra is still a module, an identity in representations of s[(2, q is proved. Later we need a generalization of this identity to representations of the Lie sub algebra of the Virasoro algebra generated by Lj, j E N - 1. One of the two identities gives this generalization in the special case that the representation is given by _x j +1 d~' j E N - 1. These three identities are the sewing identities. 1. FORMAL POWER SERlES AND EXPONENTIALS OF DERIVATIONS 37 unless a different meaning is given.

Fn). 3. Two canonical spheres with tubes (Zl •... 3. 11) of type (1,n) (n E Z+) are conformally equivalent if and only if = 1, .. ,n -1 and fi = i = 0, .. ,n (as power series). it, i Zi = Zi, Proof. 11). The conclusion of the proposition is equivalent to the assertion that F must be the identity map of C. By definition F is a complex analytic automorphism of C, that is, a projective transformation. 13) F 0 -1 IBmin(ro,ro). 15) for some nonzero complex number a. 15) we have lim wfo w-+O Since fo is of the form w+ (W) = 1.

N, aj , l = , ... , n, J E 1U+, WIt Zi = , -1- J. ; J. -- 1 , ... z -- 1 , ... , n, an d aj(i) -- 00, z• -- 0 , ... , n, J. " h nl ·bl 1U+, as teo y POSSI e poIes. 5. 19). Thus a function on the moduli space of spheres with tubes of type (1,0) can be viewed as a function of aj, j E Z + + 1. A meromorphic function on the moduli space of spheres with tubes of type (1,0) is a polynomial in aj, j E Z+ + 1 when viewed as a function of aj, j E Z+ + 1. We denote the space of meromorphic functions on the moduli space of spheres with tubes of type (1, n), n E N, by Dn.