Indefinite theta series and generalized error functions

Abstract : Theta series for lattices with indefinite signature $(n_+,n_-)$ arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their modular properties are well understood in the Lorentzian case ($n_+=1$), but have remained obscure when $n_+\geq 2$. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of `conformal' holomorphic theta series ($n_+=2$). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice ${\operatorname A}_2$, which arose in the study of rank 3 vector bundles on $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.
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Sergey Alexandrov, Sibasish Banerjee, Jan Manschot, Boris Pioline. Indefinite theta series and generalized error functions. Selecta Mathematica (New Series), Springer Verlag (Germany), 2018, 24 (5), pp.3927-3972. ⟨10.1007/s00029-018-0444-9⟩. ⟨hal-01334181⟩

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