## Vertex Operator Algebras and Related Areas by Gaywalee Yamskulna, and Wenhua Zhao Maarten Bergvelt,

By Gaywalee Yamskulna, and Wenhua Zhao Maarten Bergvelt, Maarten Bergvelt, Gaywalee Yamskulna, Wenhua Zhao

Vertex operator algebras have been brought to arithmetic within the paintings of Richard Borcherds, Igor Frenkel, James Lepowsky and Arne Meurman as a mathematically rigorous formula of chiral algebras of two-dimensional conformal box conception. the purpose was once to exploit vertex operator algebras to give an explanation for and turn out the outstanding huge Moonshine conjectures in team conception. the speculation of vertex operator algebras has now grown right into a significant examine zone in arithmetic. those court cases comprise expository lectures and examine papers offered in the course of the overseas convention on Vertex Operator Algebras and comparable components, held at Illinois country college in general, IL, from July 7 to July eleven, 2008. the most features of this convention have been connections and interactions of vertex operator algebras with the next components: conformal box theories, quantum box theories, Hopf algebra, limitless dimensional Lie algebras, and modular types. This ebook could be important for researchers in addition to for graduate scholars in arithmetic and physics. Its goal isn't just to provide an updated evaluation of the fields coated via the convention but in addition to stimulate new instructions and discoveries through specialists within the components

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Reine Angew. , 608 (2007), 35-64. E. 3359. -C. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302. edu This page intentionally left blank Contemporary Mathematics Volume 497, 2009 Vector-valued Modular Forms P. Bantay Abstract. We give a short overview of the approach to the theory of vectorvalued modular functions and forms developed in the past few years in collaboration with T. Gannon. In particular, we present the notions of fundamental and characteristic matrices, discuss the tools available for their computation, and explain the trace and spectral conditions.

Diagonalizable), and satisfy the following Spectral condition [5]: the possible eigenvalues of A are 0 or 1, while those of B are either 0, 1 or 2. 11) A (A − 1) = B (B − 1) (B − 2) = 0 . 13) 4 Tr(X ) = 4 (62β1 + 124β2 − 123α), and that the matrix A has to satisfy A2 =A , 17 A − 2 AΛ2 + ΛAΛ + Λ2 A 18 44 8 +3 (AΛ + ΛA)−4Λ3 + 8Λ2 − Λ + , 9 9 which is a system of quadratic equations – the monodromy equation – for the matrix elements of A for each given diagonal exponent matrix Λ. 8a). 14) is diﬃcult to solve in general, and for ranks d > 5, it will have families of solutions depending on several continuous parameters.

Then we form the vector space Ts ⊂ I spanned by all such fs (q). By using the invariance of I under the group SL(2, Z), it is not hard to see that fs (q) ∈ Ts satisﬁes fs (− τ1 ) = τ s gs (q), where gs (q) is again contained in Ts . 16). Thus, k − 1 has to be the highest power of τ appearing in all generalized characters. 2, where the K = 2 case was considered. 1. For the triplet vertex algebra the space of ordinary characters is precisely the space of irreducible characters. 4. Modular equation for SW(m) All deﬁnitions and results in [Miy] are extendable to C2 -coﬁnite vertex operator superalgebras.