Elliptic genera and E8 Bundles in odd dimensions

TL;DR

This work develops a modular-genus framework that couples $SL(2,\mathbf{Z})$-modular forms with $E_{8}$ and $E_{8}\times E_{8}$ bundles on odd- and spin$^{c}$-structured manifolds. By constructing twisted Witten-type genera using theta-function technology and odd Chern-character forms, the authors show that associated twisted forms are modular of specific weights across dimensions $7$–$15$, enabling explicit anomaly-cancellation identities. They extend the construction to two-bundle (i.e., $E_{8}\times E_{8}$) cases and to spin$^{c}$ settings, producing a family of modular forms with weights $2k+4$ or $2k+8$ and deriving coefficient relations by comparing $q$-expansions against standard modular bases like $E_{4}$ and $E_{6}$. The results provide new odd-dimensional anomaly cancellation formulas and generalize known modular-invariance methods to encompass $E_{8}$-bundle twists, with potential implications for higher-dimensional gauge/gravity anomaly analyses. The approach offers a systematic procedure to generate elliptic genera-linked cancellations in diverse dimensions and bundle configurations.

Abstract

This paper aims to derive new anomaly cancellation formulas by combining modular forms with E8 and E8*E8 bundles. To this end, we systematically twist and generalize known SL(2,Z) modular forms to define new modular forms associated with these bundles on odd-dimensional spin and spin^c manifolds, leading to a new series of anomaly cancellation formulas.

Theorems & Definitions (30)

• Definition 2.1
• Lemma 3.1
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof