Non-Planar String Networks on Tori

Type II strings in D=5 contain particle-like 1/8 supersymmetric BPS states. In this note we give a string-network representation of such states by considering (periodic) non-planar $(p,q,r)$-string networks of eight dimensional type II string theory on $T^3$. We obtain the BPS mass formula of such states, in terms of charges and generating-vectors of the torus, and show its invariance under an $SL(3, Z)\times SL(3, Z)$ group of transformations. Results are then generalized to string-networks associated with the $SL(5, Z)$ $U$-duality in seven dimensions. We also discuss reinterpretation of the above $(D=5)$ mass formula in terms of BPS states in world-volume theories of $U2$-branes in D=8.