# Modular invariance Modular transformations are [[0028 Conformal symmetry|conformal]] transformations that are not continuously connected to the identity, aka *large* conformal transformations. The partition function is not generally modular invariant due to the existence of conformal anomaly. On a torus though, because the metric is flat, the partition function *is* modular invariant. The modular group on a torus is $SL(2,\mathbb{Z})$ and is generated by the so-called $S$ and $T$ transformations. In higher dimensions, the situation can be more or less interesting depending on your perspective. If we place a higher-dimensional CFT on $\mathbb{T}^{d-1}\times \mathbb{R}$, then the modular group is $SL(d,\mathbb{Z})$. But with a more general topology, there may not be any nontrivial modular invariance. The key ingredient is the existence of another (spatial) circle besides the thermal circle so that they can be "swapped" with each other. ## Implications The most well-known consequence of modular invariance is the [[0406 Cardy formula|Cardy formula]], which gives the high-energy density of states in a 2d CFT. Others include the determination of universal formulas for the OPE coefficients using 4-point crossing on the sphere, 1-point modular covariance on the torus, and modular invariance on the genus-2 surface: - [[2016#Kraus, Maloney]]: light-heavy-heavy OPE from modular covariance of one-point function on the torus - [[2017#Das, Datta, Pal (Jun)]]: with additional $u(1)$ symmetry - [[2017#Cardy, Maloney, Maxfield]]: heavy OPE from modular invariance of genus-two partition function - [[2017#Cho, Collier, Yin]] - [[2017#Das, Datta, Pal (Dec)]]: one of the operators is averaged over heavy primaries - [[2019#Collier, Maloney, Maxfield, Tsiares]]: unified all previous formulas and now works for HHH, HHL and HLL (averaging over heavy ones) ## Related - [[0028 Conformal symmetry]]