A digital twin (DT), with the components of a physics-based model, a data-driven model, and a machine learning (ML) enabled efficient surrogate, behaves as a virtual twin of the real-world physical process. In terms of Laser Powder Bed Fusion (L-PBF) based additive manufacturing (AM), a DT can predict the current and future states of the melt pool and the resulting defects corresponding to the input laser parameters, evolve itself by assimilating in-situ sensor data, and optimize the laser parameters to mitigate defect formation. In this paper, we present a deep neural operator enabled computational framework of the DT for closed-loop feedback control of the L-PBF process. This is accomplished by building a high-fidelity computational model to accurately represent the melt pool states, an efficient surrogate model to approximate the melt pool solution field, followed by an physics-based procedure to extract information from the computed melt pool simulation that can further be correlated to the defect quantities of interest (e.g., surface roughness). In particular, we leverage the data generated from the high-fidelity physics-based model and train a series of Fourier neural operator (FNO) based ML models to effectively learn the relation between the input laser parameters and the corresponding full temperature field of the melt pool. Subsequently, a set of physics-informed variables such as the melt pool dimensions and the peak temperature can be extracted to compute the resulting defects. An optimization algorithm is then exercised to control laser input and minimize defects. On the other hand, the constructed DT can also evolve with the physical twin via offline finetuning and online material calibration. Finally, a probabilistic framework is adopted for uncertainty quantification. The developed DT is envisioned to guide the AM process and facilitate high-quality manufacturing.