Multi-centered $N=2$ black holes and Mock-Siegel-Narain theta-series

Given an indefinite lattice with signature $(1,n-1)$, two kinds of non-holomorphic theta functions with nice modular transformation properties can be defined. The first one is the Siegel-Narain theta function which has modular weight $(1,n-1)/2$. The second one is the indefinite theta function defined by Zwegers, which has weight $(0,n)/2$. This talk will discuss a theta function for a lattice with signature $(2,2n-2)$, which combines the properties of the two previously mentioned theta functions. This mock Siegel-Narain theta function appears in the context of wall-crossing for $N=2$ black holes.