Topological Amplitudes in String Theory

TL;DR

The paper demonstrates that genus-$g$ type II string amplitudes in four dimensions coincide with the topological partition function $F_g$ of the twisted internal $N=2$ theory, yielding effective couplings $W^{2g}$ in the $N=2$ supergravity action. The holomorphic anomaly of $F_g$ is interpreted as nonlocality arising from massless propagation and is shown to hold for general Calabi–Yau compactifications, with the relation $F_g = A_g/(g!)^2$ linking string amplitudes $A_g$ to the topological function. The authors embed these results into the $N=2$ supergravity framework via the invariant term $I_g=W^{2g}F_g(X)$ and discuss non-renormalization, duality constraints, and potential non-perturbative insights via the master equation, outlining open questions such as extending to heterotic strings. Overall, the work connects topological recursion to physical string amplitudes, providing a route to extract moduli-dependent couplings and non-perturbative information.

Abstract

We show that certain type II string amplitudes at genus $g$ are given by the topological partition function $F_g$ discussed recently by Bershadsky, Cecotti, Ooguri and Vafa. These amplitudes give rise to a term in the four-dimensional effective action of the form $\sum_g F_g W^{2g}$, where $W$ is the chiral superfield of $N=2$ supergravitational multiplet. The holomorphic anomaly of $F_g$ is related to non-localities of the effective action due to the propagation of massless states. This result generalizes the holomorphic anomaly of the one loop case which is known to lead to non-harmonic gravitational couplings.