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  What is the advantage of AdS/CFT in studying strongly coupled systems compared with lattice methods

+ 5 like - 0 dislike
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I often heard that AdS/CFT correspondence provides a powerful framework to study strongly coupled systems, to which perturbation is not applicable. However, lattice methods still work in the non-perturbative domain. My question is, what is the advantage of AdS/CFT? Is there any example impossible to access by lattice method (I don't mind lattice get numerical than analytic results)?


This post imported from StackExchange Physics at 2014-07-21 09:31 (UCT), posted by SE-user user26143

asked Jul 21, 2014 in Theoretical Physics by user26143 (405 points) [ revision history ]
edited Jul 21, 2014 by Arnold Neumaier
Lattice methods are computationally very intensive and offer _only_ numbers as results, no insight. AdS/CFT provides more of the latter. Apart from that I can't tell when which approach would be better. Let them compete until time tells...
Forgive my ignorance, I thought AdS/CFT is a technical advance to map a strongly coupled field theory to weakly coupled gravity theory, which is easier to solve. What kind of physical insight it has been provided?

3 Answers

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Since computations at strong coupling are hard, if not impossible, to do, any method that can provide a handle on strong coupling computations is great.  In that sense, both lattice and AdS-CFT are good. To be honest, I don't even think one should be comparing the two methods. I shall however try. Fermion doubling is an issue on the lattice (I am no expert and am sure that there are ways to get around it) but that is not an issue in the AdS-CFT correspondence. The AdS-CFT (more generally, the gauge-gravity) correspondence gets weaker as the supersymmetry is reduced  and things can get dicey, I believe, when there is no supersymmetry. Lattice methods might work better in such situations (think of pure QCD).

answered Jul 21, 2014 by suresh (1,545 points) [ revision history ]
edited Jul 21, 2014 by Arnold Neumaier
+ 2 like - 0 dislike

AdS/CFT is not a direct way of studying a strongly coupled system. You have to first find a dual gravitational theory, use the AdS/CFT dictionary and then calculate the dual of the quantity you actually want to calculate on the gravity side and then interpret the results on the gauge theory side. In principle, the opposite should also work (i.e calculate things in weakly coupled gauge systems and predict strongly coupled gravity), but this has been very less explored. 

In lattice computations, one calculates the observables directly in the strongly coupled system (given that they can be formulated consistently and continuum limit meaningful). So lattice and AdS/CFT have different advantages.

However, if there exists a situation where one solve a specific problem via both AdS/CFT and lattice,  it would be a great check (or proof) of the conjecture [Note that AdS/CFT is just a conjecture without any rigorous proof]. 

Also, lattice (mostly) is used to study non-supersymmetric systems like QCD and AdS/CFT applies to gauge theories with sufficient number of supersymmetries.

answered Jul 22 by aperyconstant (30 points) [ revision history ]
edited Jul 22 by aperyconstant
+ 0 like - 2 dislike

You mention dual systems.  In a cellular automata similar to the game of Life, except in an explosive rule where the cells appear to flicker randomly, since the diagonal of a cell is longer than the edges, any chain of cells "moving" diagonally, in going from A to B diagonally, must travel a further distance than going from A to D vertical in the same number of cells, the there is the appearance and the math of diagonal motion travelling faster.  What happens in this dual system is that a perfectly straight line shoots almost instantly across the flicker of cells, and a "double circle" or closed path appears somewhere on the screen in the random flicker.  Now if the vertical cells could turn on faster so that groupings of cells in the vertical direction now move at the same speed or near-abouts the diagonal, then closed paths still have to occur, but they may not be circles.  Since the diagonal is an irrational number, it's uncertain whether you can "square the circle" an ancient problem, and you may have to investigate lunes, or instead analyze what sort of instantaneous motion across all flickering cells, what path it takes instead of a perfectly straight horizontal line that resolves to a double closed circle, it may be a curved line, and the closer you get to the irrational number approximation, perhaps the more highly curved lines that shoot instantly through the flicker, so that it may be an automata that approximates gravity.  You can look on an arduino screen that has a slower update on a cheap arduino screen like those small lcd screen in the game of life, and all have a delay, but with the delay you can see how the diagonal increase in speed I mentioned causes the small patterns to make rotations in a circle.  I'm talking about varying the update speed of how fast a cell turns on to match the diagonal speed so there is no preferred direction (the diagonals in a square lattice have been criticised for that preferred direction).

It may be that the lunes in the ancient problem of squaring the circle would be the most obvious presence on screen, since you can have exact "squaring of the circle" with certain lunes.  If this seems like total nonsense, I did write a post titled "quantum computer" on Conway's Game of Life in the forum -other cellular automata- that shows the pattern, in particular the larger scale placement of the X's, pasted in a hexagonal super pattern in a square underlying grid.

update, There doesn't need to be any change to the rule in the pattern I made in Golly.  However, how you present them does need to be changed.  You would use the same rules, but as the cells turn on or off, you'd present the product of the computation (where a cell turns on), you would delay everything, but have faster "turn on a cell" if a cell is directly above a cell that is born (not above and to the left, a diagonal), or if a dell is directly to the right (or left) that is born  ---- no change in the rule just presenting vertical "toppers" and horizontal "toppers" as they're produced within the sequence of delay, to present those on screen before the diagonal ones even though "NOTHING HAS CHANGED" but the effect is different-----.  That's what I learned from the delay in all cells in the arduino screen and also in a lossy form of video compression, they're similar, but I'll have to program a delay just in the -presentation-.  I believe that the hexagonal lattice I made in that form (a slightly rotated hexagon), there is a function in Golly, --FLIP PATTERN VERTICAL-- AND --FLIP PATTERN HORIZONTAL--.   When you take the original hexagonal pattern, and the paste a flipped one on top in vertical or horizontal, the X's start to line up in rows and columns, a necessary ingredient to have strings flowing through them.  I'm going to start working on it, and if you're interested I'll show the results on that Conway forum.  The point is to match the diagonal speed and to remove as much circular closed paths as possible.

Edit: If you think about it, having just -in the chain of delay- the vertical cells presented before diagonal born cells, since the diagonals are already "fast", it will make large collections of cells move in a roughly larger circular shape (moving in a rotated square relative to a facing square or a diamond shape, a larger collection of cells moving along the diamond path).  Now if you overlay that over another automata (overlay meaning no interaction but you can see both cells as they update) the -another automata- has both vertical and horizontal updating faster than the diagonal within the chain of delay as they are presented, a new presentation of the order of the cells with the same rule.  If you think about the -another automata- it's tuned not for rotation since there is no preferred "faster" direction, so you'll have just points or dots or fixed points appear.  I'm guessing that the -another automata- are the hidden variables of the large 2d clouds of rotating pixels.  Before, with no alteration of anything just the rule B3/S12456 in Golly, upon playing a video in a lossy format, it only produced very small groupings of pixels rotating everywhere, and that's because the X-shaped pattern has a repeating "circular" small oscillator at the center of each X-shape pattern.  And instead of just overlaying, you could genetically mutate the two just described -the larger rotation and the fixed points- by overlaying both presentations and having a genetic algorithm mix the two --> in this case the fixed points would get mixed in the larger 2d rotations.

If you just wrap the top and bottom that increases the speed of the vertical, since objects can teleport from top to bottom.  That makes a larger rotation with the void area covered by the vertical motion making a continuous area to rotate.  Contrast that with a different rotation by wrapping the horizontal.  By switching between those two you have a gate (cylinder wrap in golly either vertical or horizontal) and you have a gate ---if you place the X's (minus the diagonals of the X's if you want to have the oscillators interact closer) in a certain pattern, you can switch the gate between vertical and horizontal.  By torus wrapping, any fixed points "drop out" since there is no preferred "fast" direction anywhere.

answered Jul 26 by anonymous [ revision history ] 2 flags
edited Jul 31

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