Tensionless Strings and the Weak Gravity Conjecture

TL;DR

The article investigates how quantum gravity constraints, notably the Weak Gravity Conjecture, Completeness, and the Swampland Distance Conjecture, constrain 6d F-theory vacua as a gauge coupling vanishes while gravity stays dynamical. It identifies a geometric limit in Kähler moduli space that forces a tensionless string, shown to be the heterotic string in 6d, and develops a robust framework using the elliptic genus and weak Jacobi forms to extract the charge spectrum of string excitations. By relating the elliptic genus to topological string data on the Calabi–Yau threefold, the authors derive bounds on the maximal charge per excitation level and demonstrate the existence of a sublattice of superextremal states, consistent with the Sublattice Weak Gravity Conjecture and Completeness. Across explicit F-theory models with a U(1) symmetry, the results illustrate both perturbative and non-perturbative heterotic realizations and reveal how number-theoretic properties of modular forms interplay with gravitational physics to censor global symmetries in quantum gravity.

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 Kähler 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.

Figures (5)

• Figure 1: Maximal charge $\frak{q}_{\rm max}(n)$ per excitation level $n$ for a 6d F-theory compactification on $Y_3$ with base $B_2 = \mathbb F_1$. The model refers to the values $({\rm x},{\rm y}) = (4,4)$ in the notation of Section \ref{['sec_dP1']}, and has charge index $m=\frac{1}{2} C\!\cdot\! C_0=2$. The solid blue curve is given by $\frak{q}(n)=\sqrt{4m(n-1)}$, which corresponds to the modified, scalar weak gravity bound derived in ref. toappear for the relevant extremal black holes. Observe that the charges for some excitations lie just barely above this curve, as a consequence of the plateau pattern in conjunction with the offset of the vacuum energy by $-1$. The maximally superextremal states, marked in red, lie on the dashed curve given by $\frak{q}_{\rm max}(n)=\sqrt{4mn}$ and populate a charge sublattice with spacing given by $\Delta\frak{q}=2m=4$. Together with the additional superextremal states lying between the red and the blue curves they populate the full charge lattice. This is a feature of this particular example.
• Figure 2: Shown are left- and right-moving spectra relevant for the elliptic genus of a perturbative heterotic string with $(1,0)$ supersymmetry in $d=6$. In the right-moving periodic Ramond sector, the massive towers cancel due to world-sheet supersymmetry, but they can also be mapped into towers that cancel between bosons and fermions due to space-time supersymmetry. In this way we see that the left-moving, $U(1)$-charged excitations can be level-matched to excitations in the vacuum modules of the right-moving sector, to form physical supermultiplets with fixed masses.
• Figure 3: Maximal charge $\frak{q}_{\rm max}(n)$ for a ${\rm Bl}_1\mathbb P^2_{112}[4]$ fibration over $\mathbb F_1$ with $(\rm x, \rm y)=(4,6)$. Again we oberve that maximally superextremal excitations exist whose charges form a sublattice of index $2m=4$. However, in contrast to the previous example, the set of all superextremal states does not populate the full charge lattice.
• Figure 4: Maximal charge $\frak{q}_{\rm max}(n)$ for a ${\rm Bl}_1 \mathbb P^2_{112}[4]$ fibration over $dP_2$, with $(\rm x, \rm y_1, \rm y_2)=(6,2,2)$. The general features of this strongly coupled theory are completely in line with the findings of the previous examples, which correspond to weakly coupled strings.
• Figure 5: Maximal charge $\frak{q}_{\rm max}(n)$ per excitation level for ${\rm Bl}_1\mathbb P^2_{112}[4]$ fibration over $\mathbb F_2$ with $(\rm x, \rm y)=(2,4)$. The maximally superextremal states form a sublattice of spacing $2m=8$.