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 $n_{+}\ge 2$. 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 $A_2$, which arose in the study of rank 3 vector bundles on ${\mathbb{P}}^2$. The extension of our method to $n_{+}>2$ is outlined.

arXiv comment32 pages, 2 figures; v2: discussed ${\Delta }_{12}=0$ case at end of section 3, added subsection 4.4 on $C_1=C_2$ case (relevant for signature (2,1)), and added several references; v3: published version in Selecta Mathematica (with apologies for not uploading it earlier)

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Cited by in corpus (21)

• Multiple D3-instantons and mock modular forms I
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• S-duality and refined BPS indices
• Multiple D3-instantons and mock modular forms II
• Higher depth quantum modular forms and plumbed $3$-manifolds
• Vafa-Witten theory and iterated integrals of modular forms
• Black holes and higher depth mock modular forms
• Vafa-Witten invariants from modular anomaly
• Rank $N$ Vafa-Witten invariants, modularity and blow-up
• Scaling Black Holes and Modularity
• Theta integrals and generalized error functions, II
• Higher depth quantum modular forms, multiple Eichler integrals, and ${\mathfrak{sl}}_3$ false theta functions
• Mock modularity at work, or black holes in a forest
• Refinement and modularity of immortal dyons
• Modular anomaly of BPS black holes
• Siegel theta series for indefinite quadratic forms
• An exact formula for $\mathrm{U}(3)$ Vafa-Witten invariants on ${\mathbb{P}}^2$
• Modularity of counting functions of convex planar polygons with rationality conditions
• Higher depth mock theta functions and $q$-hypergeometric series