General Junction Condition and Casimir Effect for (1+1)-Dimensional Scalar Network CFT

Recently, BCFT and ICFT have been generalized to the CFT on networks (NCFT). A key aspect of NCFT is how we connect the CFTs across different edges at the network nodes. Previous research has primarily concentrated on a specific junction condition (JC) that requires the field to be continuous at the nodes. In this paper, we investigate the most general junction conditions for $(1+1)$-dimensional free scalars that are consistent with the variational principle and energy conservation. These general junction conditions are characterized by an $O(p)$ group, where $p$ represents the number of edges connected at a node. We provide exact realizations of two typical JCs in real physical systems. Additionally, we derive both the lower and upper bounds on the network Casimir energy for $(1+1)$-dimensional free scalar fields and extend the lower bound to encompass general NCFTs. Finally, we analyze the Casimir effect in networks composed of regular polyhedra and examine the binding energy required to construct such networks from individual components.