SL(2,R) to SL(2,Z) in Type IIB String Theory I heard from Prof. Katrin Becker (in her "SUSY for Strings and Branes - Part 1" lecture) that the classical $SL(2,\mathbb{R})$ symmetry in type IIB String theory becomes $SL(2,\mathbb{Z})$ in Quantum because of charge quantization. However, I cannot see how does it work. Is there any rigorous Mathematical derivation for this? 
Thank you.
 A: It's simple. The dilaton-axion (complexified) field in supergravity (and similar classical theories with a noncompact symmetry) is invariant under $SL(2,R)$ transformations
$$\tau \to \frac{a\tau+b}{c\tau+d}, \quad ad-bc=1$$
However, the same transformation must also transform the charges of objects. For example, one-dimensional branes always carry the charge like $m$ fundamental strings superposed on top of $n$ D1-branes. So the general charge (density) of a D1-brane is given by two numbers $(m,n)$. Under the transformation above, they transform to 
$$(m,n)\to (am+bn, cm+dn)$$
because the $SL(2,R)$ transformation mixes the two types of one-brane charges (and similarly for other dimensions of branes, including the instantons).
In the classical theory, the charge of a black $p$-brane, including the one-branes above, is (a generalization of the charge of a charged black hole) given by any real numbers (charges) $m$ and $n$. However, quantum mechanically, $m$ and $n$ have to be integers in certain units. Consequently, we only get allowed one-branes after the transformation if the final charges, $(am+bn, cm+dn)$, are integers for all integers $(m,n)$. This requirement of the integrality of charges implies that $a,b,c,d$ have to be integers by themselves and only the $SL(2,Z)$ subgroup of $SL(2,R)$ maps allowed states in the Hilbert space (a superselection sector) to other allowed states in the same Hilbert space.
Note that the quantization (integrality) of the charges such as $m,n$ above is required by quantum mechanics. The one-branes are electromagnetic duals of five-branes that carry their charges, too – a combination of D5-brane and NS5-brane charges (completely analogous to the two one-branes). Because all these four types of charges (D1,F1,D5,NS5) are allowed to be nonzero but quantum mechanics enforces the Dirac quantization rule that the spacing of the D1-brane charge is inverse up to a $2\pi$ factor to the spacing of the D5-brane charge, and similarly for F1 and NS5, it follows that all these four charges must belong to a lattice. In other words, there has to exist a linear redefinition or convention in which all these four charges are integers.
