Based on arXiv:2012.15677 [hep-th]

Figure 1: a torus is an example of a geometry that has T-duality

Physicists have been searching for ways to describe the interplay between gravity and quantum mechanics – quantum gravity – for the last century. The problem of finding a consistent theory of quantum gravity still looms physicists to this day. Fortunately, string theory is the most promising candidate for such a task.

One of the strengths of string theory is that at low energies, the equations arising from string theory are shown to be precisely Einstein’s theory of general relativity. Let’s break down what this means. First, we must make sure we know the definition of a coupling constant. Theories of physics are typically described by some parameter that signifies the strength of the interaction. This parameter is called the coupling constant of that theory. According to quantum field theory, the value of the coupling constant depends on the energy. We often plot the logarithm of the energy and the coupling constant to understand how the theory behaves at a certain energy scale. The slope of this plot is called the beta function and when this function is zero, that point is called a fixed point. These fixed points are interesting since they imply that the quantum theory does not have any notion of scale.

Back to string theory, its coupling constant is called α′ (said as alpha-prime). At weak coupling, when α′ is small, we can similarly find the beta function for string theory. At the quantum level, string theory must have a vanishing beta function. At the corresponding fixed point, we find that the Einstein’s equations of motion emerge. This is quite remarkable!

We can go even further. Due to the smallness of α′, we can expand the beta function perturbatively. All the subleading terms in α′, which are infinite in number, are considered to be corrections to general relativity. Therefore, we can understand how general relativity is modified via string theory. It becomes technically challenging to compute these corrections and little is known about what the full expansion looks like.

Fortunately for physicists, string theories are interesting in other ways that could help figure us out these corrections to gravity. Particularly, the string energy spectrum that has radii R and radii α′/R look exactly the same. This relation is called T-duality. An example of a geometry that has this duality is the torus, see Figure 1. Because we know that certain dualities for strings must hold, we can use this to guess what the higher order correction must look like. Codina, Hohm and Marques took advantage of this idea to find corrections to the third power of α′. Using a simple scenario where the graviton is the only field in the theory, they were able to predict what the corrections must be.

This method can be applied at higher orders in α′ as well as a theory with more fields than the graviton, although technical challenges still arise. Due to the structure of how T-duality was used, the authors can also use their results to study cosmological models. Finally, the theory result confirms that string theory should be T-duality at all orders of α′.