Graph Schemes, Graph Series, and Modularity

TL;DR

This paper explores the properties of graph series associated with simple graphs, revealing new formulas and modular behaviors, including connections to theta functions and quantum modular forms, with implications for algebraic and number theoretic structures.

Contribution

It introduces new $q$-representations of graph series, derives explicit formulas for types $A_7$, $A_8$, $D_4$, and $D_5$, and demonstrates their modular properties.

Findings

New formulas for graph series of type A7 and A8

Representation of D4 and D5 series as indefinite theta functions

Proof that certain graph series are mixed quantum modular forms

Abstract

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert--Poincar\'e series of a certain infinite jet scheme. We study new $q$-representations and examine modular properties of several examples including Dynkin diagrams of finite and affine type. Notably, we obtain new formulas for graph series of type $A_7$ and $A_8$ in terms of "sum of tails" series, and of type $D_4$ and $D_5$ in the form of indefinite theta functions of signature $(1,1)$. We also study examples related to sums of powers of divisors corresponding to $5$-cycles. For several examples of graphs, we prove that graph series are so-called mixed quantum modular forms.