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In this paper we show that both of the Green-Schwarz anomaly factorizationformula for the gauge group $E_8\times E_8$ and the Ho\v{r}ava-Witten anomalyfactorization formula for the gauge group $E_8$ can be derived through modularforms of weight 14. This answers a question of J. H. Schwarz. We also establishgeneralizations of these factorization formulas and obtain a newHo\v{r}ava-Witten type factorization formula.

Anomaly Cancellation and Modularity. II: $E_8\times E_8$ case