Tensionless Strings and the Weak Gravity Conjecture (1808.05958v3)

Abstract: We test various conjectures about quantum gravity for six-dimensional string compactifications in the framework of F-theory. Starting with a gauge theory coupled to gravity, we analyze the limit in K\"ahler moduli space where the gauge coupling tends to zero while gravity is kept dynamical. We show that such a limit must be located at infinite distance in the moduli space. As expected, the low-energy effective theory breaks down in this limit due to a tower of charged particles becoming massless. These are the excitations of an asymptotically tensionless string, which is shown to coincide with a critical heterotic string compactified to six dimensions. For a more quantitative analysis, we focus on a $U(1)$ gauge symmetry and use a chain of dualities and mirror symmetry to determine the elliptic genus of the nearly tensionless string, which is given in terms of certain meromorphic weak Jacobi forms. Their modular properties in turn allow us to determine the mass-to-charge ratios of certain string excitations near the tensionless limit. We then provide evidence that the tower of asymptotically massless charged states satisfies the (sub-)Lattice Weak Gravity Conjecture, the Completeness Conjecture, and the Swampland Distance Conjecture. Quite remarkably, we find that the number theoretic properties of the elliptic genus conspire with the balance of gravitational and scalar forces of extremal black holes, such as to produce a narrowly tuned charge spectrum of superextremal states. As a byproduct, we show how to compute elliptic genera of both critical and non-critical strings, when refined by Mordell-Weil $U(1)$ symmetries in F-theory.

• The paper reveals that in six-dimensional F-theory a nearly tensionless heterotic string emerges as the U(1) gauge symmetry approaches a global limit, fulfilling quantum gravity conjectures.
• It employs an analysis of Kähler moduli space and elliptic genera to show that diminishing gauge couplings lead to massless towers of charged states and adherence to the SWGC bounds.
• The study connects modular invariance in weak Jacobi forms with arithmetic properties to refine the elliptic genus, emphasizing implications for the Completeness and Swampland Distance Conjectures.

Analyzing Tensionless Strings and the Weak Gravity Conjecture

The paper offers a comprehensive paper of the interplay between gauge and gravitational effects in the context of six-dimensional F-theory compactifications. It focuses particularly on the emergence of a nearly tensionless heterotic string as an $U(1)$ gauge symmetry approaches the limit where it becomes a global symmetry, in light of various conjectures in quantum gravity.

At the core of the analysis is the exploration of Kähler moduli space and its implications for the moduli-dependent features of such theories. The authors consider a scenario where a gauge coupling tends to zero while maintaining the dynamical nature of gravity. They examine the resulting scenario within the framework of F-theory and demonstrate that this gauge limit is located at an infinite distance point in Kähler moduli space, effectively adhering to the Swampland Distance Conjecture. This leads to a breakdown of the low-energy effective theory through the emergence of a tower of charged particle states becoming massless, bound to the excitations of a tensionless string.

A strong focus of the paper is evaluating the modularity and duality structures surrounding these tensionless heterotic strings. By examining these strings in the context of elliptic genera, it provides insights into the charge-to-mass ratios of string excitations, illustrating how these are determined by the modular properties of meromorphic weak Jacobi forms. These modular forms, in conjunction with arithmetic properties, participate in the tuneful spectrum of superextremal states that intriguingly meet the Sublattice Weak Gravity Conjecture (SWGC) bounds.

The authors further utilize the theory of weak Jacobi forms to refine the elliptic genus, forming a critique of the SWGC by evidencing how the series of nearly tensionless heterotic strings can indeed satisfy, at a significant level of detail, the expectations of the SWGC. This research also studies the Completeness Conjecture and the general understanding that the charge lattice should be densely populated by string states, hinting at potential cosmological and phenomenological impacts.

The geometric configuration of the base of the F-theory compactification and its intricate relationship with other algebraic and enumerative-geometric properties is further explored. Particularly, the existence of a $K3$ fibration, common to heterotic dual pairs, is used to explore the relationship these have with the entropy bounds of black holes, amplifying understanding in diverse quantum gravity conjectures. Additionally, the paper addresses how Modular Invariance in algebraic geometry converges with physical expectations when relating charge-mass configurations.

Overall, the paper presents an advanced inquiry into the theoretical frameworks that bind string excitations and massive spectrums in many supernatural physics systems. It underscores the importance of thoroughly examining the algebraic, number-theoretic, and geometric properties of string configurations within F-theory to measure the implications on SWGC and global symmetries in quantum gravity models. The research insinuates at future directions, including the reflections of these findings in phenomenological realities and further refinements or investigations into string compactifications that break away from typical constraints.