complex geometry

# Contents

## Idea

In type IIB supergravity the dilaton/string coupling constant $g_s$ and the degree-1 RR-field $C_0$ of the D(-1)-brane combine into a complex function

$\tau \coloneqq C_0 + i/ g_s$

which transforms under the modular group $SL(2,\mathbb{Z})$ S-duality $\subset$ U-duality as

$\left( \array{ a & b \\ c & d } \right) : \tau \mapsto \frac{a \tau + b}{c \tau + d} \,.$

This combined field is also called the axio-dilaton.