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Stephen Kudla : Theta integrals and generalized error functions

Recently Alexandrov, Banerjee, Manschot and Pioline [ABMP] constructed generalizations of Zwegers theta functions for lattices of signature (n-2,2). They also suggested a generalization to the case of arbitrary signature (n-q,q) and this case was subsequently proved by Nazaroglu. Their functions, which depend on certain collections $\CC$ of negative vectors, are obtained by `completing' a non-modular holomorphic generating series by means of a non-holomorphic theta type series involving generalized error functions. In joint work with Jens Funke, we show that their completed modular series arises as integrals of the q-form valued theta functions, defined in old joint work of the author and John Millson, over a certain singular $q$-cube determined by the data $\CC$. This gives an alternative construction of such series and a conceptual basis for their modularity. If time permits, I will discuss the simplicial case and a curious `convexity' problem for Grassmannians that arises in this context.

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