Indefinite theta series and generalized error functions

@article{Alexandrov2018IndefiniteTS,
  title={Indefinite theta series and generalized error functions},
  author={Sergei Yu. Alexandrov and Sibasish Banerjee and J. F. M. Manschot and Boris Pioline},
  journal={Selecta Mathematica},
  year={2018},
  volume={24},
  pages={3927-3972}
}
Theta series for lattices with indefinite signature $$(n_+,n_-)$$(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 ($$n_+=1$$n+=1), but have remained obscure when $$n_+\ge 2$$n+≥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… 
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$r$-Tuple Error Functions and Indefinite Theta Series of Higher-Depth
TLDR
This work constructs an indefinite theta series for signature $(r,n-r)$ lattices and shows they can be completed to modular forms by using these $r-tuple error functions.
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Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature ($$n-2,2$$n-2,2). Their functions, which depend on two pairs of…
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In this paper we construct indefinite theta series for lattices of arbitrary signature $(p,q)$ as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second…
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Vafa–Witten (VW) theory is a topologically twisted version of $${\mathcal{N}=4}$$N=4 supersymmetric Yang–Mills theory. S-duality suggests that the partition function of VW theory with gauge group…
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