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 so-called mixed quantum modular forms.

Generalized multiple q-zeta values and characters of vertex algebras

“…In this work, we are concerned with q-MZVs in connection to characters of vertex algebras. Our approach is motivated on one hand by our previous work on arc spaces and graph series [29,30,11,35] (see also [34]), and on the other hand by character formulas for certain vertex algebras introduced by Arakawa [4] coming from S-class theories in physics [9,13].…”

“…The concepts of associated variety and associated schemes has been quite useful in studying intrinsic geometric properties of vertex algebras, including their characters. As already pointed out in [30,11,35,34] the arc algebra C[J ∞ (X)], X = Spec(R), where R is the edge algebra of a simple graph Γ coincides with the character of a certain principal subspace W Γ inside an appropriate lattice vertex algebra [41,31,16] (after Feigin and Stoyanovsky [24]). Their Hilbert series is what we call graph series:…”

“…Essentially the same type of (q, x)-series appeared in socalled "knots-quivers" correspondences; see [23,32]. In this paper we only consider the special case b = (1, 1, ..., 1) (standard grading), thus recovering the original graph series introduced in [11,29,30]. With this choice, the graph series is denoted by H Γ (q).…”

Koszul algebras and Donaldson-Thomas invariants

“…Remark 3.2. In a particular case where for all i , j ∈ Q 0 there is at most one arrow from i to j (to such quiver we may associate in an obvious way a graph Γ without multiple edges), the algebra A Q was previously studied in the literature as the coordinate algebra defining the arc scheme of the graph scheme of Γ, see, for example, [7,8,20,26,27].…”

“…It is interesting to remark that the corresponding commutative vertex algebras and vertex Lie algebras are free in an appropriate sense (for certain locality functions on the sets of generators [4,35]), and so the Koszul duality between the algebras A Q and g Q appears to mimic the classical boson-fermion correspondence in a way that appears quite different from that recently described in the recent paper [9]. It is also worth noting that if Q is obtained from a simple graph G by "doubling" (putting an arrow i → j whenever i and j are connected by an edge), the algebra A Q may be interpreted as the coordinate algebra of the arc space [19,31] of the variety defined by quadratic monomials corresponding to edges of G. Those algebras are studied in the recent papers [7,8,20,26,27] with lattice vertex algebras as one of the main tools. It would be desirable to examine the algebras A Q for other quivers in the context of the study of relationships between arc spaces and vertex algebras [1].…”