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Motivated by the S-duality conjecture, we study the Donaldson-Thomasinvariants of the 2 dimensional Gieseker stable sheaves on a threefold. Thesesheaves are supported on the fibers of a nonsingular threefold X fibered over anonsingular curve. In the case where X is a K3 fibration, we express theseinvariants in terms of the Euler characteristic of the Hilbert scheme of pointson the K3 fiber and the Noether-Lefschetz numbers of the fibration. We provethat a certain generating function of these invariants is a vector modular formof weight -3/2 as predicted in S-duality.

Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms