Non-compact Kogut-Susskind gauge theories: 1D lattice models with exact ${\rm SL}(2,\mathbb{R})$ gauge invariance

In the quest to understand the universe at its most fundamental level, physicists often turn to the language of symmetry. Imagine a world where the laws of physics remain unchanged even if you shift your perspective or rotate your view; this is the essence of a gauge theory, a framework that describes how particles and forces interact. For decades, scientists have successfully mapped these interactions using mathematical structures known as compact groups, which behave like closed loops or spheres. These models have been so successful that researchers are now building physical machines, using trapped atoms and superconducting circuits, to simulate them in the lab. However, a major gap remains in our understanding of gravity. Unlike the forces that hold atoms together, gravity is described by a different kind of symmetry, one that is "non-compact." This means its mathematical shape stretches out infinitely, like an open plane rather than a closed circle. While this infinite nature is crucial for describing the fabric of spacetime and black holes, it has been incredibly difficult to translate into the discrete, step-by-step language of a computer simulation.

A team of researchers has now taken a significant step toward bridging this gap by constructing a new type of model that works directly with these infinite symmetries. They focused on a specific, simplified version of the problem: a one-dimensional chain of points, similar to a string of beads, governed by a non-compact group called SL(2, R). In their work, they demonstrated that it is possible to build a quantum model on this lattice that retains exact symmetry, even with the infinite nature of the group. By carefully choosing how the mathematical pieces fit together, they showed that their model reproduces the exact results of a famous theory of gravity known as Jackiw–Teitelboim gravity, which describes the behavior of black holes near their event horizons. This achievement is not just a theoretical exercise; it provides a concrete blueprint for simulating gravitational physics using the same tools currently used to study simpler quantum systems.

The researchers began by revisiting a standard method for building quantum models, known as the Kogut-Susskind formulation, which is typically used for compact groups. In this approach, the space is divided into a grid of links and points. On the links, they placed fields that represent the force carriers, and on the points, they placed matter. The challenge with non-compact groups is that their mathematical representations are infinite, meaning the fields can take on a continuous range of values rather than a discrete set. To handle this, the team chose a specific mathematical basis where the fields act like a ladder with infinitely many rungs. Instead of jumping between fixed steps, the fields hop to neighboring rungs with shifts of exactly ±1. They found that by restricting their model to a specific subset of these mathematical operations—effectively keeping only the positive values—they could eliminate the infinite complications that usually arise. This restriction allowed them to recover the exact partition function of Jackiw-Teitelboim gravity, a key result that describes the thermodynamics of black holes in two dimensions.

Next, the team introduced matter into their system, treating it as static charges sitting on the points of the lattice. In a breakthrough for understanding how gravity and matter interact, they showed that these charges could be realized using fermions, a type of particle, living in a "synthetic dimension." This is an extra layer of state that the particles can occupy, effectively giving them more room to move without needing more physical space. By dressing these charges with the surrounding force fields, the researchers were able to solve the model exactly. They discovered that the charges are connected by strings of force, and the tension in these strings depends on the continuous values of the fields between them. If the tension is positive, the charges pull toward each other; if negative, they push apart. This creates a dynamic picture where the charges oscillate back and forth, much like a pendulum, but with a frequency that depends on the specific quantum state of the system.

The study also explored what happens when these charges are allowed to move freely rather than staying fixed. In this scenario, the model becomes an interacting system where the charges and the force fields influence each other in real-time. The researchers found that at strong self-interaction, double occupancies are energetically forbidden, forcing the system into a state where the force fields are uniquely determined by the arrangement of the charges. This leads to a fascinating phenomenon where the charges perform what are known as Bloch oscillations, moving back and forth in a regular pattern. However, because the system involves a continuous spectrum of possible states, these oscillations eventually lose their rhythm and dephase, a process that mirrors the way information might be scrambled in a black hole.

Crucially, the authors verified that their lattice model matches the known results of Jackiw-Teitelboim gravity by comparing the density of states—the number of ways the system can arrange its energy—with the theoretical predictions. They found that the distribution of energy levels in their model perfectly matched the exponential growth predicted for black hole entropy. This agreement confirms that their bottom-up construction, which builds the theory from the ground up using a lattice, captures the essential physics of the gravitational theory. The work also highlights a key difference between compact and non-compact theories: in the non-compact case, the presence of matter leads to a continuous distribution of energy states that is weighted by a specific mathematical function, whereas in compact theories, the states are discrete.

This research lays a solid foundation for future studies of interacting non-compact gauge theories. By showing that these complex systems can be formulated on a lattice and solved exactly in certain limits, the team has opened the door to using powerful computational tools, such as tensor networks and quantum simulators, to explore gravitational physics. The ability to simulate these models in the lab could eventually allow scientists to test ideas about quantum gravity that were previously only accessible through abstract mathematics. The paper concludes by emphasizing that while their current work is limited to one spatial dimension and specific types of interactions, the principles they established can be extended to more complex scenarios. This approach offers a promising path toward understanding how the quantum world gives rise to the smooth, curved spacetime we observe in the universe, bringing the elusive goal of quantum gravity simulation one step closer to reality.