# Exact results for an STU-model

## TL;DR

This paper derives exact results for the holomorphic function encoding the effective action of the STU-model, utilizing duality symmetries, and explores the topological string partition function with implications for understanding the model's couplings.

## Contribution

It provides the first exact determination of the holomorphic function and topological string partition function for the STU-model, including higher-order ambiguities and their resolution.

## Key findings

- Holomorphic function determined to fifth order with some ambiguities.
- Topological string partition function expanded in invariants of S, T, U.
- Ambiguities in solutions are resolved by comparing effective action and topological string results.

## Abstract

The duality symmetries of the STU-model of Sen and Vafa are very restrictive. This is utilized to determine the holomorphic function that encodes its two-derivative Wilsonian effective action and its couplings to the square of the Weyl tensor to fifth order in perturbation theory. At fifth order some ambiguities remain which are expected to resolve themselves when proceeding to the next order. Subsequently, a corresponding topological string partition function is studied in an expansion in terms of independent invariants of $S$, $T$ and $U$, with coefficient functions that depend on an effective duality invariant coupling constant $u$, which is defined on a Riemann surface $\mathbb{C}$. The coefficient function of the invariant that is independent of $S$, $T$ and $U$ is determined to all orders by resummation. The other functions can be solved as well, either algebraically or by solving differential equations whose solutions have ambiguities associated with integration constants. This determination of the topological string partition function, while interesting in its own right, reveals new qualitative features in the result for the Wilsonian action, which would be difficult to appreciate otherwise. It is demonstrated how eventually the various ambiguities are eliminated by comparing the results for the effective action and the topological string. While we only demonstrate this for the leading terms, we conjecture that this will hold in general for this model.