Dualities, DT invariants and higher depth mock modular forms

Using dualities of string theory, I derive modular constraints onthe generating function of MSW invariants, which coincide withgeneralized Donaldson-Thomas invariants of a Calabi-Yau threefoldat the large volume attractor point in the moduli space. Thefunction is labeled by a divisor and when the divisor isirreducible, it is known to be modular. I show that in the case ofa reducible divisor, the modularity is lost, but can be restored

in a non-holomorphic completion. I find a general form of thecompletion for arbitrary degree of reducibility and provide its explicit expression up to degree 5.