I will discuss a string theory way to study G_2 instanton moduli space and explain how to compute the instanton partition function for a certain G_2 manifold. An important insight comes from the twisted M-theory on the G_2 manifold. Building on the example, I will explain a possibility to extend the story to a large set of conjectural G_2 manifolds and a possible connection to 4d N=1 SCFT via geometric engineering. This talk is based on https://arxiv.org/abs/2109.01110 and a series of works in progress with Michele Del Zotto and Yehao Zhou.

Let X be an algebraic variety over an infinite field k. In arithmetic geometry we are interested in the set X(k) of k-rational points on X. For example, is X(k) empty or not? And if it is not empty, is X(k) dense in X with respect to the Zariski topology?

Del Pezzo surfaces are surfaces classified by their degree d, which is an integer between 1 and 9 (for d >= 3, these are the smooth surfaces of degree d in P^d). For del Pezzo surfaces of degree at least 2 over a field k, we know that the set of k-rational points is Zariski dense provided that the surface has one k-rational point to start with (that lies outside a specific subset of the surface for degree 2). However, for del Pezzo surfaces of degree 1 over a field k, even though we know that they always contain at least one k-rational point, we do not know if the set of k-rational points is Zariski dense in general.

I will talk about density of rational points on del Pezzo surfaces, state what is known so far, and show a result that is joint work with Julie Desjardins, in which we give sufficient and necessary conditions for the set of k-rational points on a specific family of del Pezzo surfaces of degree 1 to be Zariski dense, where k is finitely generated over Q.