Bions and infrared renormalons in Kähler sigma models with $U(1)$ isotropy

We study semiclassical bion contributions to the vacuum energy of Kähler sigma models with two-dimensional target spaces on R × S¹ with twisted boundary conditions. This is first illustrated in the deformed CP¹ model, where explicit real and complex bion solutions are constructed in a manner analogous to that used in its quantum-mechanical reduction [1]. Evaluating the bosonic and fermionic fluctuations at one loop, we obtain the effective action with both the coupling constant and the elongation parameter properly renormalized. Integration over the bion quasi-moduli space using Lefschetz thimbles then gives the leading nonperturbative contribution, which the real and complex bion contributions cancel at the supersymmetric point, while the first imaginary ambiguity appears when deviating from the supersymmetric point. Requiring its cancellation against the perturbative ambiguity allows us to infer the corresponding infrared-renormalon pole. This is an explicit example generalizing the resurgent structure of the CP¹ model in [2] from one coupling to two renormalized parameters. The calculation further extends to Kähler targets of one complex dimension with a $U(1)$ symmetry. At leading dilute one-loop order, the target dependence enters through the moment-map difference and the metric normalizations at the two fixed points. The ambiguity structure and infrared scaling of the bion weight are common to these targets, while the inferred pole position depends on the target metric.