One-loop vacuum amplitude for D-branes in constant electromagnetic field

Following Polchinskiʹs approach we calculate the one-loop vacuum amplitude for two parallel D-branes connected by open bosonic (neutral or charged) string in a constant uniform electromagnetic (EM) field. For neutral string, an external EM field contribution appears as a multiplier (Born-Infeld type action) of the vacuum amplitude without external EM field. Hence, this gives an alternative way to see the inducing of Born-Infeld type action for the description of D-branes. For charged string the situation is more complicated, and may indicate the necessity to modify the induced D-branes action in this case.