SILCC -- VIII: The impact of far-ultraviolet radiation on star formation and the interstellar medium

We present magnetohydrodynamic simulations of star formation in the multiphase interstellar medium to quantify the impact of non-ionising far-ultraviolet (FUV) radiation within the \textsc{Silcc Project} simulation framework. Our study incorporates the radiative transfer of ionising radiation and self-consistent modelling of variable FUV radiation from star clusters, advancing beyond previous studies using static or simplified FUV fields. This enables a more accurate capture of the dynamic interaction between radiation and the evolving ISM alongside other stellar feedback channels. The interstellar radiation field (ISRF) near young star clusters can reach $G_0 \approx 10^4$ (in Habing units), far exceeding the solar neighbourhood value of $G_0 = 1.7$. Despite these high intensities, FUV radiation minimally impacts the integrated star formation rate compared to ionising radiation, stellar winds, and supernovae. A slight reduction in star formation burstiness is linked to increased photoelectric (PE) heating efficiency by the variable FUV field. photoelectric (PE) heating efficiency by the variable FUV field. Dust near star-forming regions can be heated up to 60 K via the PE effect, with a broad temperature distribution. PE heating rates in variable FUV models exhibit higher peaks but lower averages than static ISRF models. Simulations under solar neighbourhood conditions without stellar winds or ionising radiation but with supernovae yield unexpectedly high star formation rates of $\sim 0.1 \mathrm{M_\odot~yr^{-1}~kpc^{-2}}$. Our analysis reveals increased cold neutral medium (CNM) volume-filling factors (VFF) outside stellar clusters, reduced thermally unstable gas, and sharper warm-cold gas separation. The variable FUV field also promotes a cold diffuse gas phase with a molecular component, exhibiting a VFF of $\sim5-10$~per cent.