Modular Forms and $SL(2, {\mathbb Z})$-covariance of type IIB superstring theory

Understand

The local higher-derivative interactions that enter into the low-energy expansion of the effective action of type IIB superstring theory with constant complex modulus generally violate the $U(1)$ R-symmetry of IIB supergravity by $q_U$ units.

• These interactions have coefficients that transform as non-holomorphic modular forms under $SL(2, {\mathbb Z})$ transformations with holomorphic and anti-holomorphic weights $(w,-w)$, where $q_U=-2w$.
• In this paper $SL(2, {\mathbb Z})$-covariance and supersymmetry are used to determine first-order differential equations on moduli space that relate the modular form coefficients of classes of BPS-protected maximal $U(1)$-violating interactions that arise at low orders in the low-energy expansion.
• These are the moduli-dependent coefficients of BPS interactions of the form $d^{2p} \mathcal{P}_n$ in linearised approximation, where $\mathcal{P}_n$ is the product of $n$ fields that has dimension $=8$ with $q_U=8-2n$, and $p=0$, $2$ or $3$.