# How does holographic duality work?

In holographic theories about the universe, the one being in the boundary of the space, takes information of the higher dimensional universe, so, two different theories can describe the two universes. If the lower dimensional universe gathers information about the higher one, couldn't exist also the possibility that this universe would be described, not by a different but equivalent theory than that describing the higher one, but by the same exact theories of the higher one?

Can this "holographic duality" between theories (as it is the case for AdS/CFT), be true for whatever two theories? If not, then what does establish what theories can be dual?

https://en.wikipedia.org/wiki/AdS/CFT_correspondence

https://en.wikipedia.org/wiki/Holographic_principle

## 1 Answer

In holographic duality, the two mathematical theories (e.g. a string theory in AdS space, and a conformal field theory in flat space of one less dimension) are believed to be alternative descriptions of exactly the same thing.

You might say, but in the most famous example, the AdS space has ten dimensions, and the flat space has four dimensions, how can they be the same thing? The answer is that the boundary-at-infinity of the AdS space only has four dimensions. The activity of the "fields in the flat space" describe the entry and exit of strings from the boundary-at-infinity, into the "center" (the bulk) of the AdS space.

From the perspective of the four-dimensional theory, the radial direction of the AdS space corresponds to the size of objects in the field theory. The larger an object is in the four dimensions, the deeper it is in a fifth dimension of AdS space. So the four-dimensional view is (metaphorically) like many slices of five dimensions, that have then been superimposed on each other.

So that's five out of the ten dimensions. The other five dimensions are closed on themselves, and correspond to field strengths in the four dimensions, in a way that I am a little unsure about.

The actual way that four dimensions corresponds to ten dimensions is still a little mysterious - not all the details are worked out. The duality itself was guessed by Maldacena (in this paper) after he studied four-dimensional black holes in ten-dimensional string theory. It seemed like the inside of the black hole should be described in two ways, one four-dimensional, the other ten-dimensional. Mathematical equivalence is not absolutely proven, but it has passed every test (when you calculate two things in the different frameworks, that should be equal, they come out equal).

• I won't answer those questions directly because I hope I can make this clearer and then they will answer themselves... When we say these two theories are equivalent, it is very similar to saying that 10 cubed + 9 cubed = 1 cubed + 12 cubed. That's really saying 10 cubed + 9 cubed = some number, and 1 cube + 12 cubed = the same number... – Mitchell Porter Feb 2 '18 at 23:00
• The two "theories" in an AdS/CFT duality each describe some kind of "quantum universe". In a quantum theory, you have physical properties, and then formulas for probabilities. The duality is saying that although the AdS theory and the CFT theory look very different, that they are actually describing the same possible world. People have tested this by making parallel calculations that come out the same, but they haven't proven that this will happen for all possible calculations. – Mitchell Porter Feb 2 '18 at 23:04
• The duality isn't due to one universe gathering information from a different universe. Instead, there is a mathematical mapping between activity of the fields in the d=4 theory, and possible states of strings in the d=10 theory (possible states as they leave or approach the edge of AdS space). This mathematical mapping is part of the reason why the two apparently different theories can actually be talking about the same thing. – Mitchell Porter Feb 3 '18 at 12:02
• In that sentence, "captures information about..." just means "contains information about...". Anyway, the idea of the correspondence is that the two mathematical frameworks are describing the same thing. – Mitchell Porter Feb 3 '18 at 22:41
• Holography in quantum gravity means that a quantum theory with gravity (like string theory) behaves like a non-gravitational theory with fewer dimensions. It was predicted because the entropy of a quantum black hole is proportional to its surface area rather than its volume. AdS/CFT provided the first concrete examples. en.wikipedia.org/wiki/Holographic_principle – Mitchell Porter Feb 4 '18 at 2:31