Graph schemes, graph series, and modularity

Abstract

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert–Poincaré 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 A7 and A8 in terms of “sum of tails” series, and of type D4 and D5 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 mixed quantum modular forms and also mixed mock modular forms.