If you've been following our instagram page this week, you'll know that we've made a big fuss about some paper written by some dude named "Juan Maldacena" crossing the 20,000 citations threshold. That many citations for any scientific paper is remarkable, and for a theoretical-high-energy-physics (THEP) paper, it's unheard of.

In fact, there are exactly 4 THEP papers that have over 10,000 citations, as evidenced by this screenshot (the papers are filtered for THEP and sorted by citations)

Maldacena's is first with just over 20k, then there's "A Model Of Leptons" which has had almost 60 years to accrue almost 15k citations (this paper is awesome by the way — in 2.5 pages, it lays down the entire conceptual framework of what is now called the "standard model of particle physics," which itself describes all dynamics in the universe except, famously, gravity). What about the 3rd and 4th-place papers? These are two papers that are also about AdS/CFT and they both cite Maldacena's paper!

The importance of Maldacena's single-author paper — and the "AdS/CFT Correspondence" that it lays out — is tough to put into words. But we're going to try. It hopefully goes without saying that doing so with any amount of detail or rigor will be impossible here, but hopefully we can convey the roughest sketch of the idea(s) involved.

In order to understand why this paper is such a big deal, we have to understand what the paper even does. In short, it sets up what's called the "AdS/CFT Correspondence," so let's get started just with "AdS".

**The "AdS" of "AdS/CFT Correspondence"**

Einstein's theory of General Relativity (GR) says that spacetime bends when mass is present. Therefore, one can ask about the geometry **of spacetime itself**. There are 2 choices for spacetime: it's either flat, or it's not. If space-time is not flat, then its curvature is either constant, or not. Let's assume it is. We then must choose a constant, and the most important thing about this constant is whether it's positive or negative (if it's zero, then we just have flat spacetime again).

These two choices — positive or negative constant curvature — correspond to spacetime geometries that physicists call "de Sitter space" and (creatively) "anti-de Sitter space," and "anti-de Sitter space" is often abbreviated as "AdS." Nice, we've at least been able to write down "AdS".

It turns out that there's something really special about AdS — something that not even flat-spacetime has. Namely, the **boundary **of AdS "looks like" our normal, flat spacetime (at least locally), albeit in one dimension lower (the boundary of a space typically has one dimension fewer than the space itself, like the surface of a sphere vs. the entire, 3-d sphere). In particular, there is a locally well-defined time direction on the boundary of AdS.

Why is this special feature about the boundary of AdS so important? Because we could, in principle, define some well-behaved laws of physics on that boundary (we obviously need "time" in order to do physics). But we're getting ahead of ourself, let's first move on to...

**The "CFT" of "Ads/CFT Correspondence"**

"CFT" in this case stands for "Conformal Field Theory". Let's break this down one step at a time. A "Field Theory" in this case refers to a "Quantum Field Theory", and it is the mathematical framework in which virtually all of fundamental physics — and certainly the entire Standard Model of Particle Physics — is written. For now, we can simply view the phrase "Field Theory" as equivalent to "Laws Of Physics".

Now what does "Conformal" mean? Conformal in this case refers to an extra set of symmetries. For example, our normal, every-day "laws of physics" are symmetric under certain actions. Translations are one such action: the laws of physics are the same "over here" as they are "over there". Rotation is similar: if we rotated the entire universe (and everything in it) by 90 degrees, no one would know.

A "Conformal Field Theory" is, therefore, a field theory (remember, just "laws of physics") that must also respect "conformal symmetry". Precisely how to define conformal symmetry is a little complex, but intuitively it allows for any transformation that keeps **angles between things **constant, even if it changes the lengths of things. Our physical universe is certainly not conformally symmetric: I would die instantly if you stretched every distance in my body to two-times its current distance. But physicists can have fun thinking about such symmetries.

(As a fun side-note: our universe actually **would be** conformally symmetric if we removed all the mass from it. This would not lead to an empty universe, since we could still have massless particles like photons and gluons flying around.)

To summarize, a CFT is simply a set of "laws of physics" that are extra-symmetric — conformally symmetric. Now let's talk about...

**The "Correspondence" of "AdS/CFT Correspondence"**

What Maldacena found in his seminal paper is the following (in reality, he found **really convincing hints** to the following): If you put a CFT on the boundary of AdS — which we can do because the boundary of AdS has a time direction on it — then it "looks" exactly like a completely different theory on the AdS itself (i.e., the not-boundary part — the part that has one dimension more than the boundary).

How can laws of physics with one fewer dimensions — the CFT on the boundary of AdS — "look like" laws of physics on the interior of the space, in one dimension higher? Well, they of course don't "look" the same, but there's a well-defined **dictionary** between everything that happens in one, and everything that happens on the other. To be slightly more technical: the degrees of freedom on each are in one-to-one correspondence. Moreover, there is not just some abstract mathematical "existence" argument for this one-to-one correspondence — one can (often, at least) actually construct the dictionary.

So, the "Ads/CFT Correspondence" is "a correspondence between the degrees of freedom of a CFT placed on the boundary of an AdS, and the degrees of freedom of a different theory that resides on the interior of the same AdS". Sweet. But why is this such a big deal?

**Big Deal #1: Quantum Gravity**

Recall that when we introduced "AdS," we referenced Einstein's theory of General Relativity (GR), and recall that when we introduced "CFT," we referenced Quantum Field Theory (QFT). Therefore, in a precise sense, the "AdS/CFT Correspondence" gives us a correspondence between QFT and GR.

Well, as you may or may not know, finding a unified theory of physics that incorporates both General Relativity and Quantum Field Theory has been quite possibly **the single biggest, and hardest problem** that fundamental theoretical physics has ever faced. Thousands of physicists have been working on this problem for 50-100 years, depending on how you count.

The most common approach to this problem has been, in some sense, to look for a single "umbrella" theory that "reduces to" QFT or GR in separate limits. In other words, much like how GR "reduces to" Newtonian gravity in certain limits, we hope to find some Grand Unified Theory that has GR and QFT (and probably lots of other things) "sitting inside it" somehow.

The AdS/CFT correspondence gives us a slightly different — but just as fascinating — approach. Namely, it shows that perhaps these two things (GR and QFT) are not as different as we initially thought. AdS/CFT says that a CFT on the boundary of an AdS space is "equivalent to" a certain theory of gravity on the inside of that same space. Therefore, maybe we should be viewing GR and QFT as different sides of the same coin, as opposed to (or perhaps in addition to) looking for a **larger** coin that "contains" two separate coins.

That's a pretty cool perspective, and thousands of physicists seem to agree.

**Big Deal #2: Black Holes**

Black holes are both super cool, and super problematic. Their coolness hopefully doesn't need much elaboration, but their annoyingness might be more surprising.

They're annoying because they give rise to what's called the "black hole information paradox." To arrive at this paradox, we must also accept the existence of the (not yet measured) "Hawking Radiation," which is radiation emitted by black holes, and whose existence was predicted by Stephen Hawking.

The problem occurs from the following (simplified) thought-experiment. Imagine some information falling into a black hole. That black hole then radiates and eventually evaporates. Hawking's predicted radiation has the peculiar property that it is independent of the details of the initial state of the black hole, and thus this information that fell into the black hole is truly gone forever once the black hole has evaporated, impossible to reconstruct.

This violates one of the most fundamental principles in physics, known as "unitarity." The relevant part of "unitarity" for this discussion is that it effectively states that information **cannot be lost** in the sense just described above. Hence the paradox.

This was a huge problem for a very long time, with some of the world's best physicists getting involved. AdS/CFT shows, however, that information is **not lost** in a black hole. Namely, the black hole exists on the interior of the AdS, and maps (via the correspondence) to a perfectly good set of objects in the CFT on the boundary. Therefore, we can track the dynamics of the black hole by simply tracking the dynamics of the CFT, which is perfectly unitary.

Therefore, at least to a degree, the AdS/CFT correspondence has put to rest one of the most (technically and philosophically) difficult problems in physics in the last century.

**Big Deal #3: Strong and Weak Coupling**

For a long time, most of the QFT work that physicists could do with any success was in the "weak coupling regime" of QFT. Roughly speaking, this is a regime where there exists some small parameter that we can expand our answers in, much like a Taylor expansion. So, instead of solving problems exactly, we find approximate answers by calculating higher and higher order terms in this "small parameter". This small parameter is called a "coupling constant," and the assumption that it is small is what gives this regime the name "weak coupling."

However, there's plenty of very interesting physics that one would want to study that does **not** involve a small coupling constant, and therefore requires methods for solving things **exactly**. For a long time, these types of questions had to remain unanswered. Until Maldacena came along.

Namely, one of the most amazing parts of the AdS/CFT correspondence is that the strength of the coupling constants on the boundary scales inversely to that of the coupling constants in the interior. In particular, a strongly-coupled CFT on the boundary gives rise to a weakly-coupled theory of gravity in the interior. Since we know how to study weakly coupled systems, we can translate strongly-coupled questions on the boundary to more tractable weakly-coupled questions in the interior.

This blew the door open on a whole host of problems that people wanted answers to, particularly in the field of condensed matter physics, which we turn to next.

**Big Deal #4: Condensed Matter Physics**

Studying one or two particles is "easy" — it's called fundamental physics — and studying hundreds of thousands of particles is also "easy" — it's called statistical mechanics. What's **really** hard, though, is studying, like, 50 particles. Roughly speaking, this is the realm of Condensed Matter Physics (CMP).

One of the many things that makes CMP hard is that many of the interesting phenomena occur at strong coupling. Therefore, in comes AdS/CFT to shed some light.

In fact, not only has the tools from AdS/CFT started making their appearance in CMP, many other string-theoretic ideas and tools have started being picked up by the CMP folks. This is a gigantic and ongoing development, so we'll leave it here for now, but the takeaway is that AdS/CFT is able to find uses even in some pretty "real world" situations.

#### 20k Citations

It's now hopefully clear why this single paper has amassed so many citations, and continues to do so daily. Every one of the "big deals" just mentioned — and surely several more that we are just not thinking about right now — are areas of physics that hundreds, if not thousands, of physicists have built entire careers on. And AdS/CFT gives useful insight into all of them.

I don't know when the next time will be that we'll see a THEP paper reach 20k citations — more specifically, a THEP paper that does **not** have to do with AdS/CFT. It might not be in my lifetime (and I'm old, but I'm not **that** old). So I want to enjoy this one while I can, and now hopefully you can too!