EMERGING SPACE-TIME: PLAYING GOD

We have been living in what we can consider modern civilizations for just over two centuries. In this very short period of time, humanity has gone from building the first steam engine to presenting viable theoretical models about the ultimate nature of space and time. This last question probably represents the most fundamental and transcendent question in physics. It is difficult to imagine how a species of hominids whose brain was originally "designed" by evolution

To survive, we can even dream of coming close to answering this question. Although it may seem incredible, we already have some models that allow us to glimpse the answer to the "final frontier of science."

Although it may sound a bit pretentious, in this article we're going to "play God." Using a model based on string theory, we're going to see what happens to the geometry of space-time when we modify the value of a certain conserved physical quantity. Using the metaphor of a "god" playing with the Universe by modifying a simple dial, we'll begin to glimpse the possible ultimate nature of space-time itself.

The AdS/CFT correspondence

In 1997, theoretical physicist Juan Maldacena published what would become the most cited work in the history of theoretical physics. In this work, he presented what is probably the most important advance in quantum gravity in recent decades. The conclusion of this work is simple but incredibly powerful: a (conformal) quantum field theory defined in D dimensions is physically indistinguishable from a theory of

(quantum) gravity defined in an AdS spacetime in D+1 dimensions. This is a great example of the power of Mathematical Physics: two completely different systems (with and without gravity and defined in different dimensions) are physically equivalent. This is possible basically because the symmetries and quantum operators that define both systems are the same. The great power of this correspondence is that for the first time we can explore features of quantum gravity (the AdS side) simply by analyzing the features of ordinary quantum systems (CFT side). Metaphorically speaking, it is as if by analyzing the properties of an electronic circuit you could obtain information about the

space-time that contains it.

One of the most important problems in the study of this correspondence is that to study situations where gravity is weak (similar to the environment we live in) on the AdS side we must compute quantum systems where the strength of the interactions is very strong on the CFT side and we do not know how to do this. On the contrary, if we study weakly coupled systems on the CFT side we are describing systems with enormous gravity on the AdS side (black holes) and we do not fully understand their characteristics. Fortunately there is a solution to this problem: use the so-called "1/2 BPS sector" that offers us the

supersymmetric properties of both systems.

The 1/2-BPS sector

Both the AdS part of the duality and the CFT part use a special symmetry called Supersymmetry. This symmetry establishes an equivalence between the particles that make up matter (spin 1/2 particles or fermions) and those that constitute the interactions (spin 1 particles or bosons). The fundamental characteristic of the 1/2-BPS sector is that when we increase the coupling (the strength of the interactions) this sector of the theory that contains half of the states "decouples" and maintains its supersymmetry for any energy range, that is, it remains unchanged for all energy ranges. This allows us to study the

characteristics of quantum gravity on the AdS side despite having a strong coupling on the CFT side! This sector is defined by the value of a conserved quantity called R-charge , which is related to the angular velocity J. This quantity will be our "dial". We will see what happens to it below

to gravity and therefore to the geometry of space-time on the AdS side as we modify the R-charge "dial" on the CFT side.

The geometry for R<

We start with the dial at its lowest point. This means that the value of the R-charge is much smaller than the number of fundamental constituents of the quantum system N. Duality implies that for every quantum operator on the AdS side, there exists a dual operator on the CFT side. When R is small, the energy of the system is low, and calculations can be performed using the partition function expressed as a path integral. Calculations show that the dual operators (on the AdS side) to operators with R< gravitons! Therefore, if we perturb a CFT operator with R<

The geometry for R=√N

The next position on our dial is where R is of the order of √N. In this case, the energy of the system is higher than in the previous case, and calculations indicate that operators with R=√N on the CFT side correspond to 1-dimensional objects on the AdS side. These objects are strings! In addition, the calculations indicate something fascinating: the 1/2-BPS sector can be described as a series of complex matrices.

which represent the dynamics of N free fermions. This system of fermions has a mean value or expectation value that represents the average value of the "up" spins and "down" spins of the system. The most striking result is that the calculations show that this mean value represents the position of the string in AdS spacetime!

In principle, our "god" could control the position of the string by manipulating the mean value of the spin network!

Schematic image of "god" modifying the R-charge on the CFT side and producing strings on the AdS side

The geometry for R=N

Next, we turn the dial to the R=N position. The energy of the system is now so large that our previous calculation methods are invalid, and we must resort to a technique based on the so-called "Schur polynomials." This technique allows us to perform the calculations under certain reasonable assumptions.

As we mentioned earlier, the R-charge on the CFT side corresponds to the angular velocity of the dual object in AdS spacetime. Upon reaching an R-charge of the order of N, the angular velocity of the dual object is so large that it exceeds the tension on the object, and it begins to expand. An extended object in string theory is called a brane, so the dual object in this case is a D-brane! The D indicates the dimension of the extended object, which in superstring theory is a number between 0 and 10.

Schematic image of "god" modifying the R-charge on the CFT side and producing branes on the AdS side

New geometries: space-time for R=N 2

This is when our "god" shows his full power: the change in the geometry of space-time. When the R-charge is of the order of N2, something truly impressive begins to happen: the dual objects on the AdS side are so heavy that

They begin to interact with space-time itself! The consequence of this is that a change in the geometry and even the topology of space-time itself occurs. The key question at this point is: Can we determine the geometry of the new space-time? To try to determine the shape of the new space-time, we can introduce a "probe" into our AdS space-time and study its geodesics to determine the metric of the new space-time. Like explorers of new worlds, theoretical physicists set out to explore a Universe with a new space-time! To achieve this, it is enough to perturb the operator

corresponding in the CFT and check how this perturbation (graviton, string or membrane) evolves in the dual AdS space using the so-called "correlation functions".

If we perturb an operator corresponding to a graviton we find that when the R-charge is of the order of N2 the coupling constant undergoes a sudden change, this is a sign that a change in the geometry of spacetime has occurred. If we use a string as a probe instead of a graviton we find that from a certain value of R the ends of the string behave as if they were not attached to any object, as if they had a new degree of freedom. This can be interpreted as the motion of closed strings that are associated with

New geometries of space-time. Unfortunately, our "probes" cannot explore the entire dual space-time; they can only move within a limited environment near a plane. Despite this, we can make a series of general considerations to try to glimpse the overall geometry of the new space-time.

In the original formulation of the AdS-CFT duality proposed by Maldacena, the AdS part has 5 dimensions (AdS5) and the CFT part has 4 dimensions. Since the AdS part is described by superstrings, it must have 10 dimensions in total. This is achieved by rolling the remaining 5 dimensions into an S5 sphere. Therefore, the overall geometry on the AdS side is AdS5xS5. The new geometry must respect the global RxSO(4)xSO(4) symmetry since the symmetry and therefore its associated charges must be preserved.

The so-called LLM geometries meet this and other requirements and are general solutions to the equations. One possibility offered by these geometries is that the S5 sphere on the AdS side can change its topology from a 5-dimensional sphere to a 5-dimensional object with a non-trivial topology. This can be seen in the following drawing:

In geometry 1 and geometry 2, a 5-dimensional space-time is achieved by "merging" (fibering) a 3-dimensional sphere with the two-dimensional surface that appears shaded in the figures.

Conclusions

Although we still have a long way to go to explain the true nature of space-time in our 4-dimensional Universe, there are models based on superstring theory that indicate that certain extended objects (strings and branes) could be the fundamental constituents we are looking for.

Furthermore, these models support a hypothesis that is increasingly resonating in the world of fundamental physics: the space-time we detect at macroscopic scales is an emergent property, arising from the interaction of fundamental components. Theoretical physics, armed with the power of mathematics

It's reaching beyond what was imaginable a century ago. The latest work is exploring exciting new possibilities, and future work will undoubtedly bring us even closer to the true nature of the incredible Universe we inhabit.

Sources: Emergent spacetime, Bubling AdS spaces and 1/2 BPS geometries