On the space of elliptic genera

TL;DR

The paper addresses how modular invariance and spectral flow constrain the polar part of SCFT spectra by studying the elliptic genus. It develops a concrete framework: decompose the elliptic genus into a vector of holomorphic pieces $h_\mu(\tau)$ and theta functions, identify the polar terms, and count constraints via the dimension of vector-valued cusp forms using a Selberg trace approach. The key results are explicit dimension formulas: $\dim \tilde{J}_{0,m} = p(m) - \dim S_{2\frac{1}{2},\mathbf{M}}(\Gamma(4m)^*)$, with modular-corrected expressions that grow linearly with central charge, and analogous constructions for $\mathcal{N}=(4,0)$ spectra, including a CP$^2$ example illustrating the presence or absence of weakly holomorphic solutions. The work provides a practical tool to rule out incompatible polar spectra, connects to the AdS$_3$/CFT$_2$ correspondence, and suggests extensions to other Jacobi form spaces and potential links to holomorphic anomalies and Euler numbers in gauge/string theory contexts.

Abstract

Invariance under modular transformations and spectral flow restrict the possible spectra of superconformal field theories (SCFT). This paper presents a technique to calculate the number of constraints on the polar spectra of N=(2,2) and N=(4,0) SCFT's by analyzing the elliptic genus. The polar spectrum corresponds to the principal part of a Laurent expansion derived from the elliptic genus. From the point of view of the AdS_3/CFT_2 correspondence, these are the states which lie below the cosmic censorship bound in classical gravity. The dimension of the space of elliptic genera is obtained as the number of coefficients of the principal part minus the number of constraints. As an additional illustration of the technique, the constraints on the spectrum of N=4 topologically twisted Yang-Mills on CP^2 are discussed.