Carroll Limit of $\mathcal{O}(F^2)$ $\text{D}_p$-branes from Kaluza--Klein-like Null reduction in the Polyakov Formulation

Abstract: We construct the electric and magnetic Carroll limits of the $\mathcal{O}(F<sup>2)$-truncated $\text{D}<em>p$-brane via the Polyakov--KK route, which combines a single-mode Kaluza--Klein-like reduction at fixed light-cone momentum with a Dirac classification that keeps the auxiliary worldvolume frame. Unlike the $U(1)$-free ILST string, for $p\geq2$ the Carrollian D$</em>{p}$-brane cannot be treated as a consistent fixed-frame constrained system after the Dirac classification. In the electric family the obstruction is directly produced by the worldvolume $U(1)$ matter sector, while in the magnetic family the frame retention is forced by the scalar kinetic's density-weight structure rather than by the $U(1)$ currents. The frame must therefore be retained through the classification and eliminated afterwards by Dirac brackets. Freezing it beforehand is not a legitimate step in the Dirac procedure. The embedding scalar sector inherits the Carroll--Weyl $χ$ structure at the level of the ILST kinetic density, which is $χ$-invariant, whereas the full scalar action is only $χ$-covariant. The local extension of $χ$ to the full matter sector is obstructed by the $U(1)$ sector through the Gauss, circle and $A_-$-gradient brackets. The electric family keeps the global rescaling as a weak symmetry, while the magnetic family fails even for a global parameter. For $P_{-}\neq0$ the electric and magnetic families have $d-2$ and $d-3$ degrees of freedom, and the restored local circle at $P_{-}=0$ removes one further degree of freedom in each family. These features identify the resulting theories as constrained systems distinct from both the null string and the parent $\mathcal{O}(F<sup>2)$ DBI theory, and we also discuss their on-shell interpretation and relation to tensionless/null strings.