General overview for people who aren't familiar with this stuff : Given a particular space-time interval \(ds^{2}\) and a D-n brane you define its DBI action as \(\int d^{n+1}\xi \, \sqrt{-\det(P(G_{ab}))}\) where the \(\xi^{a}\) are the n+1 space-time dimensions it's extended in and \(P(G_{ab})\) is the pull back of the space-time metric onto these n+1 dimensions, defined via \(P(G_{ab}) = \frac{\partial X^{\mu}}{\partial \xi^{a}} \frac{\partial X^{\nu}}{\partial \xi^{b}}G_{\mu\nu}\) where \(ds^{2} = G_{\mu\nu}dx^{\mu}dx^{\nu}\). Here is me doing the explicit calculation for a particular space-time also in the picture. But, as the link says, I don't get what a particular paper says. Given it's all pretty straight forward in terms of putting in the various functions and fields, does anyone whose messed with metric pullbacks or done brane dynamics (Ben, Prom?) know what I'm doing wrong or what step in logic I'm missing? Oh and the \(\epsilon_{3}\) is just an angular function which isn't important but you get from the \(S^{3}\) factor of the brane being extended in the \(AdS_{5} \times S^{3}\) part of the space-time. I mean, well obviously! Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image! /edit And \(r^{2} = \sum_{i} y_{i}^{2}\), which are the 6 dimensions which aren't part of our normal space-time, should anyone be actually interested but unsure about that.
Hi AN, To clarify, the brane is extending in the \( \vec{x}, \, S^3 \) and \(\rho \) directions and you're choosing the world volume coordinates to be the same as the spacetime ones except that \(y_5, \, y_6\) are functions of \(\rho\)? The only derivatives that give a non trivial result are those on \(y_5, \, y_6\) and if I'm not mistaken there is some dependence on these coordinates hidden in your \(r\) variable. Could this be the source of the problem?
OK, that was completely useless. I've had a quick go at working it out and I get the same as you did. I've only ever done this for D3 branes which are considerably simpler.
Having spent the evening messing around with various things, looking through 'D Branes' by Johnson and racking my brains, I seem to have come up with a way of working out roughly how to do it using a method which sort of extends the Polyakov action view-point. The Polyakov action is \(\int d^{2}\xi \, \sqrt{-\gamma}\gamma^{ab}h_{ab}\), where \(\gamma_{ab}\) is the worldsheet metric and \(h_{ab}\) is the space-time metric pullback for the string, ie a D1-brane formalism. This has the particular nicety that it's got more than just the usual symmetries (Lorentz and diffeomorphisms), it also has the Weyl symmetry \(\gamma^{ab} \to \e^{\lambda}\gamma^{ab}\) and from the equations of motion for \(\gamma^{ab}\) you get \(\sqrt{-\det(h)} = \frac{1}{2}\sqrt{-\det(\gamma)}\gamma^{ab}h_{ab}\). Unfortunately that's unique to 2 dimensional constructs but if you make the connection between \(\gamma\) and \(g_{ab}\), the induced metric on the brane* and then compute \(g^{ab}h_{ab} = g^{ab}P[G_{ab}]\), where \(g_{\mu\nu} = G_{\mu\nu}\) for the first 8 coordinates, since they are the D7 directions, it turns out that \(g^{ab}P[G_{ab}] = g^{ab}\partial_{a}X^{c}\partial_{b}X^{d}g_{cd} + g^{ab}\partial_{a}X^{I}\partial_{b}X^{J}G_{IJ}\), where I,J = 9 or 10, the \(y_{5,6}\) directions. The first term is the trace of Euclidean 8d metric, so 8 and the second term reduces to \(\frac{R^{2}}{r^{2}}(1+\dot{y}_{5}^{2}+\dot{y}_{6}^{2})\) where the dot is differentiation WRT \(\rho\). So if you set \(R^{2}=8\) you get the right expression, which I think is okay (but the paper doesn't mention it from my quick look through) because you're just making a choice of coordinates, equivalent to energy or RG scale?But then the various terms in the action's product are not of the right power, ie you need \(\sqrt{g^{ab}h_{ab}}\). I don't think you can construct a 'nice' action from factors of \(\det(g)\), and \(g^{ab}h_{ab}\), because you don't have Weyl symmetry, unless you put in \(\sqrt[8]{\det(g)}\)!! Hence why the DBI action is used. I've never actually done any calculations involving it before, hence why I'm not sure about what precisely I'm doing.... And now that you say it, it seems I've completely forgotten that r is a function of \(\rho\)! Doh! But you've said it doesn't seem to make a difference? Over all though, the important thing, in terms of what I plan to do over the weekend, is that I can see how that particular expression involving the \(g^{ab}\partial_{a}y_{5}\partial_{b}y_{5}\) etc comes about and in terms of equations of motion that's the important thing. Up to unimportant factors of angular components and what-not I should be okay. I probably should have gone into the office today but then I'd have had to sit through a 90 minute talk on MHV and something to do with gluons. *The paper I mention calls g the induced metric while Johnson considers h to be the induced metric via the pullback, while the paper's use of 'pull back' is different.
It just so happens that I'm doing some work with this action in an AdS background at the moment, but it's only a D3 brane that looks like an S3 for large r in global AdS, so there's not so many dimensions to keep track of. After a quick look I thought a derivative out of the pull back would hit the r but then I realised that the derivatives in the pull back always act on the map and not the metric itself so the factors of r should just sit there and behave themselves. It usually takes me a couple of goes to get things right. Please Register or Log in to view the hidden image! I got completely confused by this a couple of days ago. The DBI action contains the pull back of the spacetime metric to the world volume of the brane which you correctly state is \(({}^* g)_{a b} = \partial_a X^\mu \partial_b X^\nu g_{\mu \nu}\). This is quite often called the induced metric - as far as I'm concerned the two names are synonymous. Unfortunately there seems to be a lot of different conventions for the pull back. At least you've made some progress towards what you want. Please Register or Log in to view the hidden image!
I'm hoping you'll know the answer to this Prom, I don't have much book knowledge when it comes to various AdS/CFT results and concepts. When you're constructing the metrics, obviously you want it to have an asymptotically \(AdS_{5} \times S^{5}\) space-time layout (in IIB, if you're working in other theories it's different, says Becker & Schwarz). When you go this by putting in things like D3, D5, D7 branes etc that basically dictates the form of the metric and since you're doing it by explicit construction, all is well. But if you want to work backwards, such as tinker with particular terms in some field or brane coordinate's equation of motion to get a 'nice' system, it is not immediately apparent if the function you put into the EoM can be constructed from a D brane system. Do you know if asymptotically \(AdS_{5} \times S^{5}\) is all you need for the system to be workable within the AdS/CFT correspondence (along with large N obviously). If I can write such a metric down there'll definitely be some QFT which it's dual to? I just get the feeling that assuming that because I can write down a metric which is asymptotically \(AdS_{5} \times S^{5}\) then The Correspondence applies and there's certain to be some configuration of branes which gives that metric is assuming a little too much.
The original way of thinking about the correspondence and the way it was derived was to put a stack of D3 branes in flat space and let them back react to give you some non trivial geometry. That you already know, and you may already know that the holographic picture of the correspondence is that the IIB string on \(AdS_5 \times S^5\) is dual to N = 4 SYM on the boundary, which is \(S^3 \times \mathbb{R}_t\). If you're interested in calculating the meson spectrum or simply having AdS/CFT with flavour then you stick a D7 brane in at an appropriate position to preserve supersymmetry. You'd usually assume that the brane doesn't affect the geometry. I've recently done a project based on hep-th/0502111 and hep-th/0205290 where the crux is basically that you can take \(AdS_5\) and do 2 wick rotations so that time becomes a spatial circle and one of the compact directions becomes time. What happens is that the bulk is called a topological black hole (it's a pretty weird space) and the boundary of the space goes from \(S^3 \times \mathbb{R}_t\) to \(dS_3 \times S^1\) and the field theory is defined on that space. If you start with a Schwarzschild black hole in AdS space the result is a "bubble of nothing" which is even weirder. You can then follow the usual Witten's prescription to compute correlation functions. A partial answer to your question is that, if the metric you write down is related to \(AdS_5 \times S^5\) by some operation, then the duality is still the same - IIB string in the bulk = N = 4 SYM on the boundary. I suspect that if you take an AdS space with a boundary and stick some string theory in it, there will be a dual field theory, but you won't necessarily know what that theory is. The usual way of constructing gravity duals is to do some computation in the string theory so that the field theory falls out. Enough is known now about how these sorts of things work so that properties of the gravity side have a known effect on the field theory side. If you wrap some branes on the \(S^5\) for example, that will give you a field theory with less supersymmetry than N = 4. For some examples see hep-th/0007191 and hep-th/0008001. I'm used to tackling problems on the gravity side so I don't really care how horrible the field theory is - I only want the gravity side to be nice. So my answer to "can you modify some term in the field theory by altering the geometry?" is yes, but typically not individual terms in the Lagrangian. What is it exactly that you're trying to do?
The current gravity model we're looking at primarily involves computing the dynamics of the two directions which the D7 isn't extended in. If there's some dynamic which makes the D7 vary in these directions you can stretch strings between it and the D3 and via usual interpretations etc you have a model of some kind of meson-like object in the field theory. The equations of motion can also be recast into a Schrodinger equation like thing, but analytically the potential isn't a nice one, but approximations by a square well or the \(\frac{1}{x^{2}} + \frac{1}{(x-x_{0})^{2}}\) potentials is accurate to within about 10%. Since its these potentials which ultimately dictate the dynamics and the Schrodinger equation is pretty much the best way of looking at things (rather than some horrific equation with various metric coefficients floating around) my supervisor wants to be able to change that potential by hand to something which is more QCD-like. I then get the job of unwinding that equation to turn it into the original metric which gives that equation of motion. But both of us agree its unlikely any and all potentials are constructable because you're restricted in your choice of metric entries by requiring the asymptotic properties as well as particular transformations used to make it Schrodinger-like also contribute. At the moment we've found that approximating the rather unpleasant numerically-solved potential by an infinite square well is accurate to about 8%, which when it comes to predicting QCD masses isn't actually that bad. /edit I should also point out that I'm aware some of these questions I'm asking would be stuff my supervisor will know but I'm sure you understand my apprehension in asking what might be an absolute clanger of a question to him, thus making myself look as clueless as Reiku.
I have felt your pain, especially when I asked a stupid question that I'd asked before and forgotten the answer to! Please Register or Log in to view the hidden image! To me it looks like what you're doing is finding an approximation to the N = 4 theory using gravity, rather than finding an exact gravity dual to some other theory. I didn't realise the correlation between AdS/CFT and experiment was so good, especially considering it's not a physical theory and there are infinite colours etc.
I think it's some kind of middle ground. I've not specifying my space-time metric exactly, just working out what conditions you'd have if you want a QCD-like theory which has some kind of fairly bland meson spectra but which is as close to QCD as possible. Andrew Tedder, a postgrad whose just finished here at Southampton, got a meson (well, those due to chiral symmetry breaking) spectrum which when you did a minimisation of statistical things like root mean squared quantities to within about 5~10% of QCD using 3 inputs and about 2 dozen outputs.
