%0 Journal Article
%T Indefinite theta series and generalized error functions
%+ Laboratoire Charles Coulomb (L2C)
%A Alexandrov, Sergey
%A Banerjee, Sibasish
%A Manschot, Jan
%A Pioline, Boris
%Z 30 pages, 2 figures
%< avec comité de lecture
%Z L2C:16-078
%@ 1022-1824
%J Selecta Mathematica (New Series)
%I Springer Verlag
%V 24
%N 5
%P 3927-3972
%8 2018-11
%D 2018
%Z 1606.05495
%R 10.1007/s00029-018-0444-9
%Z Mathematics [math]/Number Theory [math.NT]
%Z Physics [physics]/High Energy Physics - Theory [hep-th]
%Z Mathematics [math]/Algebraic Geometry [math.AG]Journal articles
%X 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.
%G English
%L hal-01334181
%U https://hal.science/hal-01334181
%~ CNRS
%~ LPTHE
%~ L2C
%~ MIPS
%~ UNIV-MONTPELLIER
%~ SORBONNE-UNIVERSITE
%~ SU-SCIENCES
%~ SU-TI
%~ ALLIANCE-SU
%~ UM-2015-2021