Zhu reduction for Jacobi $n$-point functions and applications

Abstract

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...

$$ \mathcal{N} $$ = 2* Schur indices

“…where G k (z, τ ) is the twisted Eisenstein series (A.35). Also the logarithm of the spectral zeta function has an expansion [68] log (2πiνZ 1 (u; ξ; q)) = log(2πiν) + P 0 (ν, τ…”

“…The G k (τ ) defined here are rescaled by (2πi) −k . They the same as the G k (τ ) in[68], the (2πi) −k G k (τ )in[18,80], the E k (τ ) in[19].…”

“…The G k (z, τ ) defined here are the same as the (2πi) −k Ĝk (e 2πiτ , e 2πiz ) = Ĝk (e 2πiτ , e 2πiz ) in[40], the (−1) k+1 J k /k! in[83], the G k (z, τ ) in[68].…”

“…The P k (z, τ ) is the same as the P k (z, τ ) in[68], the (−2πi) −k P k (e 2πiz , q) in[84] and the (2πi) −k E k (z, τ )in[16,20].…”

Reductive cohomology associated with vertex algebras

“…The main example: cohomology associated to Jacobi forms In many cases, n-point correlation functions for vertex algebras generate modular forms for certain groups related to appropriate underlying manifolds. Corresponding spaces of differential operators acting on n-point correlation functions can be also considered [1,9,21]. In this paper we continue to study the reduction cohomology of vertex algebras in the particular case of correlation functions denerating Jacobi forms.…”