# Exact Resurgent Trans-series and Multi-Bion Contributions to All Orders

## TL;DR

This paper derives the exact resurgent trans-series for near-supersymmetric $ ext{CP}^1$ quantum mechanics, explicitly calculating multi-bion contributions and confirming the resurgence structure to all orders.

## Contribution

It provides the first exact all-orders resurgent trans-series solution in a quantum mechanical model, including multi-bion solutions and their summation.

## Key findings

- Exact multi-bion solutions obtained for finite time intervals.
- Resurgent trans-series verified to all orders in nonperturbative contributions.
- Perturbation series are absolutely convergent and reproduce the exact ground state energy.

## Abstract

The full resurgent trans-series is found exactly in near-supersymmetric $\mathbb C P^1$ quantum mechanics. By expanding in powers of the SUSY breaking deformation parameter, we obtain the first and second expansion coefficients of the ground state energy. They are absolutely convergent series of nonperturbative exponentials corresponding to multi-bions with perturbation series on those background. We obtain all multi-bion exact solutions for finite time interval in the complexified theory. We sum the classical multi-bion contributions that reproduce the exact result supporting the resurgence to all orders. This is the first result in the quantum mechanical model where the resurgent trans-series structure is verified to all orders in nonperturbative multi-bion contributions.