Indefinite theta series: the case of an $N$-gon

In this note, we provide a construction of the indefinite theta series attached to $N$-gons in the symmetric space of an indefinite inner product space of signature $(m-2,2)$ following the suggestions of Section C in the recent paper of Alexandrov, Banerjee, Manschot, and Pioline, [2]. We prove the termwise absolute convergence of the holomorphic mock modular part of these series and also obtain an interpretation of the coefficients of this part as linking numbers. Thus, we prove the convergence conjecture of [2] provided none of the vectors in the collection $C = {\lbrace C_1, \dotsc, C_N \rbrace}$ is a null vector. It should be noted that the use of linking numbers and a homotopy argument eliminates the need for an explicit parametrization of a surface $S$ spanning the $N$-gon that was used in an essential way in our previous work [4]. We indicate how our method could be carried over to a more general situation for signature $(m-q,q)$ where higher homotopy groups are now involved. In the last section, we apply the method to the case of a dodecahedral cell in the symmetric space of a quadratic form of signature $(m-3,3)$.

Keywords

indefinite theta series, mock modular forms

2010 Mathematics Subject Classification

11F27, 11F37

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Stephen Kudla is supported by an NSERC Grant.

Received 22 September 2021

Received revised 9 July 2022

Accepted 18 July 2022

Published 3 April 2023