AdS/CFT with no humor…

well, maybe not that humor-less… however, let’s start from the very beginning… well, starting from the very beginning will mean I will jump over several aspects but I hope not over the important ones. So, you know that in quantum mechanics you have to sum over all possible wavefunctions and square the result in absolute value to get the final probability density. Well, in the case of collisions between particles that may have some structure and may interact through specific interactions you have to sum (or integrate) over all inequivalent configurations (emphasis on inequivalent). The situation can be represented in the form of some diagrams called Feynman diagrams after the smart guy called in the same way that invented them. There you represent some lines for particles but also use some other lines for whatever happens “in between” and you don’t really observe… you also have to sum (integrate) over whatever is inside and you don’t see in order to get the right amplitude so in the end a feynman diagram represents some sort of integral (in momentum or position space). Now, the propagators (internal lines) you find by solving some form of field equations and in this way you get some expressions you put in the integral. Now, these equations are equations of motion. They are derived from Lagrangians (or, if you are unlucky and work in condensed matter from Hamiltonians…) Now, you observe that a change in the phase of your field (yeah, that damn thing) should not affect the results… this is a relatively fundamental principle (although in some cases the phase difference becomes relevant, see Berry phases, Bohm Aharonov experiment, etc. due to topological features…). The name of this principle is called “gauge invariance” and the global gauge invariance (you change the phase in the same way everywhere) is a symmetry that is associated to the “conserved electric charge” (see Noether theorem). Now, the situation becomes slightly more complicated if you allow local phase changes. The results should be invariant to that too but in order to have this condition imposed you need a “connection” in your space that allows you to define the change of a quantity from one point to the other. Surely, this won’t change your results but you still need the mathematical structure to impose this and in order to have that mathematical structure you need to have a new field (well, new… you knew it as the Maxwell field). Now, of course you will need some equations for this field too, and it will appear in the inner part of your Feynman diagram especially if you describe e.m. phenomena. Well, the thing is that this field also has some freedom (gauge freedom) and when you integrate over it you practically integrate over lots of equivalent situations. This can be easily solved by fixing “the gauge”… ok, thing done, you obtain a consistent Feynman diagram. However, when you put a loop into your diagram you start obtaining divergencies. These are simple divergencies, you remove them by renormalization (there are several types of renormalizations and methods to achieve them, Minimal subtraction, multiplicative, counter-term, etc.) This renormalization appears due to a bad behavior of your theory at very high momenta or very small distances… remember, it is a perturbative approach (oh, damn it, I forgot to tell you this, it is of course a damn perturbative approach all this…) Yeah, and there are of course IR divergencies but these are easily removed by either introducing some regularization or some massive terms. Ok, so we have a theory that is UV and IR renormalized and represented by a series of graphs… this is the situation when the gauge algebra (the algebra associated to the gauge group transformations of the interaction fields) is abelian. If it is non-abelian the situation becomes lightly more complicated. The commutator must be considered. Now, when we have a gauge group like SU(3) we may have fermionic fields in its fundamental representation (having an internal index) and some gauge fields living in its dual representation (adjoint) and having 2 indexes. This is important… Now, Gerard ‘t Hooft ‘s idea was to represent the feynman diagrams such that the indices i and j are represented by different lines according to how they are i>j, i=j or i<j as lines with arrows. Now, these diagrams can be organized such that the internal lines cover a surface. Now, you obtain a very ugly figure over the plane with some lines and some doubled lines with directions. Well, at this you can look in a direct way but also in a dual way, where the distance between the lines becomes “your object”… now, as SU(3) becomes SU(N) and N->infinity. In principle you have a surface with holes in it that can be classified according to N and the coupling at each vertex, call it g. Euler’s theorem in topology will give you some exponents for you factors and here we go, a topological classification of your surface. Now, when N is large you can make an expansion in 1/N and this gives you a series expansion of the kind you want (string sheet expansion). Now wait a bit, this is just a representation of a formal power series. You don’t have a SU(Infinity) group and the “dual” representation of the fillings instead of the lines doesn’t mean nature is made out of strings and branes… Actually ‘t Hooft said that pretty clear in his paper but … lokomotiv string theory doesn’t stop… however, I said I am serious around now… so, let’s assume you make this connection then AdS/CFT says in principle that what you have is a conformal field theory on the string world-sheet surface (don’t forget, you speak about a 2 dimensional surface, CFT is the preferate choice but you did not start with that in the first place, so the relation is “jumpy”). You obviously don’t have a good 1/N approximation because in QCD N=3 which is not known to be terribly large… well, 4 is certainly larger than 3… and by all means electrons in a solid are NOT QCD… well, never mind, the idea is that you look at this and you see that your CFT is in fact a theory on a boundary … what boundary? well, if you desperately search for something you can expand in a series you will need a weak coupling and that one you find if you consider the CFT on the boundary of an AdS_{5}xS^{5} space. In that case the intuition may suggest (but it is NOT clearly proved, only “verified” and I don’t agree with the verifications either) that your strongly coupled theory on the boundary (hard to compute as you know only series expansions in small couplings) is “dual” (whatever that may mean) to a gravity (supergravity) theory in the “bulk” space of AdS_{5}xS^{5}. Of course the quantum case of the bulk supergravity must in principle also be a quantized theory of gravity so you may end up having to integrate over non-equivalent geometries… whatever “nonequivalent” may mean in this case… Well, this is the place where all the craziness starts… you can start playing with your AdS space, transforming it into a general manifold M and see what happens, how you do quantization on that one, what else can you put instead of S^{5} and so on, and so on… you see however, that the whole thing is based on a CONJECTURE… and some alternative (dual space) representations… 


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