Type IIB parabolic ($p,q$)-strings from M2-branes with fluxes

We extend the work of Schwarz [1] to show that bound states of type IIB supersymmetric ($p, q$)-strings on a circle are associated with M2-branes irreducibly wrapped on $T^2$, or equivalently with nontrivial worldvolume fluxes. Beyond this extension we consider the Hamiltonian of an M2-brane with $C_{\pm}$ fluxes formulated on a symplectic torus bundle with monodromy. In particular, we analyze the relevant case when the monodromy is parabolic. We show that the Hamiltonian is defined in terms of the coinvariant module. We also find that the mass operator is invariant under transformations between inequivalent coinvariants. These coinvariants classify the inequivalent classes of twisted torus bundles with nontrivial monodromy for a given flux. We obtain their associated ($p,q$)-strings via double dimensional reduction, which are invariant under a parabolic subgroup of $SL(2,\mathbb{Q})$. This is the origin of the gauge symmetry of the associated gauged supergravity. These bound states could also be related to the parabolic Scherk-Schwarz reductions of type IIB string theory.