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In the AdS/CFT correspondence I know that the mapping of global symmetries involves also the S duality that in the field theory side is $SL(2,Z)$. In Type IIB supegravity this duality is $SL(2,R)$. I don't understand how to matching these symmetries.

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The AdS/CFT correspondence (or this particular, most famous example of it) is the equivalence between the non-gravitational ${\mathcal N}=4$ supersymmetric Yang-Mills theory on one side ("the boundary CFT"); and the type IIB superstring theory in $AdS_5\times S^5$ ("the bulk theory") on the other side. The equivalence is exact and the gauge theory is well-defined, so the dual bulk AdS description has to be consistent, too. String/M-theory is the only consistent theory of quantum gravity so the bulk theory has to be a string theory and not just "supergravity", too.

Supergravity is just a (nonrenormalizable, and for this reason and others, inconsistent) low-energy approximation (effective field theory) of the full theory, string theory. This particular one has the noncompact symmetry $SL(2,{\mathbb R})$. However, quantum effects break it to the discrete subgroup $SL(2,{\mathbb Z})$, the same symmetry that we see on the boundary CFT side. It's not hard to see how it happens.

For example, the supergravity contains black 1-branes carrying one of the two string-like charges under the 2-form B-fields. One of them is the "fundamental (F1) string" charge and the other one is the "D1-brane" charge. These two charges transform as a doublet under the $SL(2)$ group.

In supergravity used as a classical description, these 1-branes are seen as black-hole-like extended solutions – the black 1-branes – but the amount of charge these BPS objects carry is arbitrary, a continuous, real number. The same holds for the two types of 5-branes, those with the NS5-brane charge and the D5-brane charge, another doublet.

However, in the quantum theory, the charges cannot be continuous. One way to prove this assertion is to appreciate the Dirac quantization rule (which results from the requirement of single-valuedness of the wave function of one object when it orbits around the electromagnetically dual object's Dirac string). F1-strings are electromagnetically dual to the NS5-branes and D1-branes are electromagnetically dual to D5-branes. For the charges of mutually dual objects (just like for the electric charges and magnetic monopoles in 4D), the Dirac quantization rule has to hold: $$ Q_E Q_M \in 2\pi {\mathbb Z} $$ It follows that the allowed F1,NS5; D1, D5-charges are not continuous. They actually belong to a lattice that is "self-dual" in the sense of the Dirac quantization condition above.

Such a lattice isn't quite unique. When the allowed minimum charge $Q_E$ is increased $k$ times and the minimum $Q_M$ is decreased $k$ times, the Dirac quantization rule continues to hold. Also, one may mix the electric and magnetic charges (in a way that fully matches the QCD-like theta-angle in the gauge theory).

When the conditions are taken into account, one sees that there is a 2-real-dimensional family of possible charge lattices and it may be written as the usual moduli space $$ SL(2,{\mathbb Z}) \backslash SL(2,{\mathbb R}) / SO(2, {\mathbb R}) $$ It may be represented as the "fundamental domain" of the modular group. It's the same as the space parameterized by $g_{YM}$ and $\theta_{QCD}$ in the gauge theory, of course. The moduli space is a quotient on both sides. The left-multiplication identifies the points with respect to the actual "U-duality group" $SL(2,{\mathbb Z})$ because elements of this group act in such a way that they map the lattice of allowed charges onto the same lattice.

The approximation of the moduli space by "one point" is basically equivalent to assuming that the charges of all the brane-like objects are continuous, not quantized, and ignoring to the Dirac quantization rules and all similar quantum effects. This approximation is good enough at "low energies". When we consider the low-energy limit, the only objects that carry such charges are "large branes" that look like macroscopically large extended black holes. And those carry so huge charges, relatively to the minimum units, that these charges may be considered continuous.

I mentioned the quantization of the brane charges as a reason why the dual theory is the full string theory and not just the supergravity. There exist other differences between string theory and supergravity. All of these discriminating features may be checked in AdS/CFT and in all of them, one may see that the gauge theory is dual to the full string theory and not just "supergravity". For example, one may see all the excited string states in terms of traces of long products in the gauge theory (complex composite operators), verify that they interact just like in string theory, see all the wrapped branes, check that the string theory cures all the problems of SUGRA with its nonrenormalizability, and many other things.

On the gauge theory side, the supergravity limit is obtained by focusing on the planar limit of simple enough single-trace operators etc. But if one sees beyond this subset of observables and beyond the approximations with which it is being computed to compare the gauge theory with SUGRA, one may see all of string theory, too.

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  • $\begingroup$ Clear, Thanks! One more question: in summary, if we take the classical limit of supegravity theory, that is $g_s\to 0$ , the group SL(2,Z) enlarges to SL(2,R) in gravity side. The same limit corresponds, in field theory, (thanks to the relations between the parameteters on both sides) to some other limit that has the same result of enlarging the the group SL(2,Z) to SL(2,R). It's true? What is this limit in field theory ? $\endgroup$ – Andrea89 Aug 21 '15 at 11:27
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    $\begingroup$ Good question. I think that the limit is a bit harder to see on the gauge theory side. Try to see e.g. arxiv.org/abs/hep-th/9811047 $\endgroup$ – Luboš Motl Aug 21 '15 at 12:03

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