From closed to open strings: the tensionless route in Kalb-Ramond background and noncommutativity

We study tensionless bosonic strings propagating in the presence of a constant Kalb--Ramond background and show how closed strings undergo a transition into open strings. Working in the intrinsically tensionless theory, we show that the Carrollian limit of the closed-string worldsheet induces a universal gluing of `left'- and `right'-moving oscillators, which is deformed in the presence of the background $B$-field. From the action we derive the mixed boundary conditions, construct the corresponding gluing matrix, and obtain the induced vacuum as a squeezed boundary state. This gives a first-principles realization of the closed-to-open string transition in the tensionless regime. We further extend the analysis to toroidal compactification and show that the worldsheet Bose--Einstein-like condensation mechanism continues to hold in the presence of the $B$-field. Second, we analyze boundary noncommutativity in a unified symplectic framework, both in the tensile theory and in the tensionless regime. In the tensile strings, we show that the boundary symplectic form reproduces the Seiberg--Witten noncommutative parameter. We then study the tensionless limit of this construction and show that the boundary $B$-field term remains as the unique surviving source of the symplectic structure. We derive the same structure directly in the intrinsically tensionless strings, where the inverse boundary symplectic form defines the noncommutative parameter of the null string.