We describe a type of n-point function associated to strongly regular vertex operator algebras V and their irreducible modules. Transformation laws with respect to the Jacobi group are developed for 1-point functions. For certain elements in V , the finite-dimensional space spanned by the 1-point functions for the irreducible modules is shown to be a vector-valued weak Jacobi form. A decomposition of 1-point functions for general elements is proved, and shows that such functions are typically quasi-Jacobi forms. Zhu-type recursion formulas are provided; they show how an n-point function can be written as a linear combination of (n − 1)-point functions with coefficients that are quasi-Jacobi forms.
We prove an SL 2 (Z)-invariance property of multivariable trace functions on modules for a regular VOA. Applying this result, we provide a proof of the inversion transformation formula for Siegel theta series. As another application, we show that if V is a simple regular VOA containing a simple regular subVOA U whose commutant U c is simple, regular, and satisfies (U c ) c = U , then all simple U -modules appear in some
Let V be a strongly regular vertex operator algebra. For a state h ∈ V 1 satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr M q L(0)−c/24 ζ h(0) (M a V -module) is a vector-valued weak Jacobi form of weight 0 and a certain index h, h /2. We discuss refinements and applications of this result when V is holomorphic, in particular we prove that if g = e h(0) is a finite order automorphism then Tr V q L(0)−c/24 g is a modular function of weight 0 on a congruence subgroup of SL 2 (Z). MSC(2010): 17B69. IntroductionThe theory of n-point functions at genus g = 1 for regular vertex operator algebras (VOA) and their orbifolds was established in [DLM2] and [Z]. In particular, the modular-invariance (in the sense of vectorvalued modular forms) of the space of partition functions of V -modules was proved. The purpose of the present paper is to show how the portion of this theory concerned with partition functions can be extended to a setting in which elliptic modular forms are replaced by weak Jacobi forms. In particular, we prove that appropriately defined trace * Supported by an NSA summer graduate fellowship † Supported by the NSA and NSF
One-point theta functions for modules of vertex operator algebras (VOAs) are defined and studied. These functions are a generalization of the character theta functions studied by Miyamoto and are deviations of the classical one-point functions for modules of a VOA. Transformation laws with respect to the group SL 2 (Z) are established.
We establish precise Zhu reduction formulas for Jacobi n-point functions which show the absence of any possible poles arising in these formulas. We then exploit this to produce results concerning the structure of strongly regular vertex operator algebras, and also to motivate new differential operators acting on Jacobi forms. Finally, we apply the reduction formulas to the Fermion model in order to create polynomials of quasi-Jacobi forms which are Jacobi forms. 1 approach using the shifted theories for VOAs (see [5,21] for discussions on such theories). As a corollary to Propositions 3.4-3.7 below, we obtain the following theorem.Theorem 1.1. A Jacobi n-point function for a VOA V does not contain poles in C × H if the (n − 1)-point functions do not contain poles for any n − 1 many vectors in V .The reasons why poles exist in the reduction formula coefficients and yet not in the associated n-point functions are interesting in their own right, and quite exploitable. Indeed, the bulk of this paper examines this process in more detail. At its core, the possible poles must either never exist (i.e., there are no elements in the VOA which produce the quasi-Jacobi forms giving rise to the poles), or the poles must correspond to zeros of the partition function. Gaberdiel and Keller [10] used the Zhu reduction formulas for Jacobi n-point functions (or elliptic genus) and the fact that no poles arise in the N = 2 superconformal field theories to create new differential operators of Jacobi forms of different (higher) degrees. They also highlighted the use of this for investigating extremal N = 2 superconformal field theories.In Section 4 we study a family of differential operators M k,α defined for k ∈ N and α ∈ Z \ {0}. For certain k and α these operators collapse to those studied in [10] and for other values to those considered in [28]. However, some subtle additional cases are included here. Along with showing certain coefficients of functions under the image of this form are nonzero (see Lemma 4.2), we also establish Lemma 4.1 which, in a simplified version, can be paraphrased as follows.Lemma 1.2. Suppose α ∈ Z \ {0}, k, m ∈ N 0 , and φ is a weak Jacobi form of weight k and index m (with a possible multiplier system). Then M k,α (φ) transforms like a Jacobi form of weight k + 2 and index m (with the same multiplier system). Additionally, if α = ±1, ±2, then M k,α (φ) is holomorphic for either k even or odd if certain conditions on the multiplier system are satisfied.The operators studied in Section 4 are motivated by applying the Zhu reduction formulas to strongly regular VOAs. This analysis is performed in Section 5. We highlight the fact that an additional Lie algebra structure contained in the strongly regular VOAs is what gives rise to the new differential operators. This is described further in Section 5. One could also consider higher degree differential operators here, much as in the same way as Gaberdiel and Keller do in [10] but where one does not have the Lie algebra structure. However, while interesting, thi...
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