$G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

TL;DR

This paper links the generating functions of G-invariant Hilbert scheme Euler characteristics on K3 surfaces to modular forms, providing explicit eta product formulas and extending to refined invariants.

Contribution

It establishes a novel connection between G-invariant Hilbert scheme invariants on K3 surfaces and modular forms, with explicit eta product formulas for all cases.

Findings

Z_{X,G}(q)^{-1} is a modular cusp form of weight 1/2 e(X/G)

Explicit eta product formulas for 82 cases

Extension to elliptic genus, Chi-y genus, and motivic classes

Abstract

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) = \sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$ be the generating function for the Euler characteristics of the Hilbert schemes of $G$-invariant length $n$ subschemes. We show that its reciprocal, $Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight $\frac{1}{2} e(X/G)$ for the congruence subgroup $\Gamma_{0}(|G|)$. We give an explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82 possible $(X,G)$. The key intermediate result we prove is of independent interest: it establishes an eta product identity for a certain shifted theta function of the root lattice of a simply laced root system. We extend our results to various refinements of the Euler characteristic, namely the Elliptic genus, the Chi-$y$ genus, and the motivic class.

Tables

Table 1. Table 1. Table of the modular forms Z X , G − 1 superscript subscript 𝑍 𝑋 𝐺 1 Z_{X,G}^{-1} for all symplectic G 𝐺 G actions.

$\#$	$\left|G\right|$	Singularities of $X/G$	The modular form $Z_{X,G}^{-1}$	Weight
0	1		$\eta {\left(\tau \right)}^{24}$	12
1	2	$8A_1$	$\eta {\left(2\tau \right)}^8\eta {\left(\tau \right)}^8$	8
2	3	$6A_2$	$\eta {\left(3\tau \right)}^6\eta {\left(\tau \right)}^6$	6
3	4	$12A_1$	$\eta {\left(2\tau \right)}^{12}$	6
4	4	$2A_1+4A_3$	$\eta {\left(4\tau \right)}^4\eta {\left(2\tau \right)}^2\eta {\left(\tau \right)}^4$	5
5	5	$4A_4$	$\eta {\left(5\tau \right)}^4\eta {\left(\tau \right)}^4$	4
6	6	$8A_1+3A_2$	$\frac{\eta {\left(3\tau \right)}^8\eta {\left(2\tau \right)}^3}{\eta \left(6\tau \right)}$	5
7	6	$2A_1+2A_2+2A_5$	$\eta {\left(6\tau \right)}^2\eta {\left(3\tau \right)}^2\eta {\left(2\tau \right)}^2\eta {\left(\tau \right)}^2$	4
8	7	$3A_6$	$\eta {\left(7\tau \right)}^3\eta {\left(\tau \right)}^3$	3
9	8	$14A_1$	$\frac{\eta {\left(4\tau \right)}^{14}}{\eta {\left(8\tau \right)}^4}$	5
10	8	$9A_1+2A_3$	$\frac{\eta {\left(4\tau \right)}^9\eta {\left(2\tau \right)}^2}{\eta {\left(8\tau \right)}^2}$	9/2
11	8	$4A_1+4A_3$	$\eta {\left(4\tau \right)}^4\eta {\left(2\tau \right)}^4$	4
12	8	$3A_3+2D_4$	$\frac{\eta {\left(\tau \right)}^2\eta {\left(4\tau \right)}^6}{\eta \left(2\tau \right)}$	7/2
13	8	$A_1+4D_4$	$\frac{\eta {\left(4\tau \right)}^{13}\eta {\left(\tau \right)}^4}{\eta {\left(8\tau \right)}^2\eta {\left(2\tau \right)}^8}$	7/2
14	8	$A_1+A_3+2A_7$	$\eta {\left(8\tau \right)}^2\eta \left(4\tau \right)\eta \left(2\tau \right)\eta {\left(\tau \right)}^2$	3
15	9	$8A_2$	$\eta {\left(3\tau \right)}^8$	4
16	10	$8A_1+2A_4$	$\frac{\eta {\left(5\tau \right)}^8\eta {\left(2\tau \right)}^2}{\eta {\left(10\tau \right)}^2}$	4
17	12	$4A_1+6A_2$	$\frac{\eta {\left(6\tau \right)}^4\eta {\left(4\tau \right)}^6}{\eta {\left(12\tau \right)}^2}$	4
18	12	$9A_1+A_2+A_5$	$\frac{\eta {\left(6\tau \right)}^9\eta \left(4\tau \right)\eta \left(2\tau \right)}{\eta {\left(12\tau \right)}^3}$	4
19	12	$3A_1+3A_5$	$\eta {\left(6\tau \right)}^3\eta {\left(2\tau \right)}^3$	3
20	12	$A_2+2A_3+2D_5$	$\frac{\eta {\left(4\tau \right)}^3\eta {\left(3\tau \right)}^2\eta {\left(\tau \right)}^2\eta {\left(6\tau \right)}^4}{\eta \left(12\tau \right)\eta {\left(2\tau \right)}^4}$	3
21	16	$15A_1$	$\frac{\eta {\left(8\tau \right)}^{15}}{\eta {\left(16\tau \right)}^6}$	9/2
22	16	$10A_1+2A_3$	$\frac{\eta {\left(8\tau \right)}^{10}\eta {\left(4\tau \right)}^2}{\eta {\left(16\tau \right)}^4}$	4
23	16	$5A_1+4A_3$	$\frac{\eta {\left(8\tau \right)}^5\eta {\left(4\tau \right)}^4}{\eta {\left(16\tau \right)}^2}$	7/2
24	16	$6A_1+A_3+2D_4$	$\frac{\eta {\left(8\tau \right)}^{12}\eta {\left(2\tau \right)}^2}{\eta {\left(16\tau \right)}^4\eta {\left(4\tau \right)}^3}$	7/2
25	16	$6A_3$	$\eta {\left(4\tau \right)}^6$	3
26	16	$4A_1+A_3+A_7+D_4$	$\frac{\eta {\left(8\tau \right)}^7\eta {\left(2\tau \right)}^2}{\eta {\left(16\tau \right)}^2\eta \left(4\tau \right)}$	3
27	16	$2A_1+4D_4$	$\frac{\eta {\left(8\tau \right)}^{14}\eta {\left(2\tau \right)}^4}{\eta {\left(4\tau \right)}^8\eta {\left(16\tau \right)}^4}$	3
28	16	$2A_1+A_3+2A_7$	$\eta {\left(8\tau \right)}^2\eta \left(4\tau \right)\eta {\left(2\tau \right)}^2$	5/2
29	16	$A_3+D_4+2D_6$	$\frac{\eta \left(4\tau \right)\eta {\left(8\tau \right)}^7\eta {\left(\tau \right)}^2}{\eta {\left(16\tau \right)}^2\eta {\left(2\tau \right)}^3}$	5/2
30	18	$8A_1+4A_2$	$\frac{\eta {\left(9\tau \right)}^8\eta {\left(6\tau \right)}^4}{\eta {\left(18\tau \right)}^4}$	4
31	18	$2A_1+3A_2+2A_5$	$\frac{\eta {\left(9\tau \right)}^2\eta {\left(6\tau \right)}^3\eta {\left(3\tau \right)}^2}{\eta \left(18\tau \right)}$	3
32	20	$2A_1+4A_3+A_4$	$\frac{\eta {\left(10\tau \right)}^2\eta {\left(5\tau \right)}^4\eta \left(4\tau \right)}{\eta \left(20\tau \right)}$	3
33	21	$6A_2+A_6$	$\frac{\eta {\left(7\tau \right)}^6\eta \left(3\tau \right)}{\eta \left(21\tau \right)}$	3
34	24	$5A_1+3A_2+2A_3$	$\frac{\eta {\left(12\tau \right)}^5\eta {\left(8\tau \right)}^3\eta {\left(6\tau \right)}^2}{\eta {\left(24\tau \right)}^3}$	7/2
35	24	$4A_1+2A_2+2A_5$	$\frac{\eta {\left(12\tau \right)}^4\eta {\left(8\tau \right)}^2\eta {\left(4\tau \right)}^2}{\eta {\left(24\tau \right)}^2}$	3
36	24	$5A_1+A_3+A_5+D_5$	$\frac{\eta {\left(12\tau \right)}^7\eta \left(6\tau \right)\eta \left(2\tau \right)\eta \left(8\tau \right)}{\eta {\left(24\tau \right)}^3\eta \left(4\tau \right)}$	3
37	24	$2A_2+A_5+D_4+E_6$	$\frac{\eta {\left(8\tau \right)}^4\eta \left(4\tau \right)\eta \left(3\tau \right)\eta {\left(12\tau \right)}^4\eta \left(\tau \right)}{\eta {\left(6\tau \right)}^2\eta {\left(24\tau \right)}^2\eta {\left(2\tau \right)}^2}$	5/2
38	24	$2A_2+A_3+2E_6$	$\frac{\eta {\left(8\tau \right)}^6\eta \left(6\tau \right)\eta {\left(\tau \right)}^2\eta {\left(12\tau \right)}^2}{\eta {\left(2\tau \right)}^4\eta {\left(24\tau \right)}^2}$	5/2
39	32	$8A_1+3A_3$	$\frac{\eta {\left(16\tau \right)}^8\eta {\left(8\tau \right)}^3}{\eta {\left(32\tau \right)}^4}$	7/2
40	32	$9A_1+2D_4$	$\frac{\eta {\left(16\tau \right)}^{15}\eta {\left(4\tau \right)}^2}{\eta {\left(32\tau \right)}^6\eta {\left(8\tau \right)}^4}$	7/2
41	32	$3A_1+5A_3$	$\frac{\eta {\left(16\tau \right)}^3\eta {\left(8\tau \right)}^5}{\eta {\left(32\tau \right)}^2}$	3
42	32	$4A_1+2A_3+2D_4$	$\frac{\eta {\left(16\tau \right)}^{10}\eta {\left(4\tau \right)}^2}{\eta {\left(32\tau \right)}^4\eta {\left(8\tau \right)}^2}$	3
43	32	$5A_1+2A_7$	$\frac{\eta {\left(16\tau \right)}^5\eta {\left(4\tau \right)}^2}{\eta {\left(32\tau \right)}^2}$	5/2
44	32	$2A_1+2A_3+A_7+D_4$	$\frac{\eta {\left(16\tau \right)}^5\eta {\left(4\tau \right)}^2}{\eta {\left(32\tau \right)}^2}$	5/2
45	32	$3A_1+D_4+2D_6$	$\frac{\eta {\left(16\tau \right)}^{10}\eta {\left(2\tau \right)}^2}{\eta {\left(32\tau \right)}^4\eta {\left(4\tau \right)}^3}$	5/2
46	36	$2A_1+2A_2+4A_3$	$\frac{\eta {\left(18\tau \right)}^2\eta {\left(12\tau \right)}^2\eta {\left(9\tau \right)}^4}{\eta {\left(36\tau \right)}^2}$	3
47	36	$A_1+6A_2+A_5$	$\frac{\eta \left(18\tau \right)\eta {\left(12\tau \right)}^6\eta \left(6\tau \right)}{\eta {\left(36\tau \right)}^2}$	3
48	36	$6A_1+A_2+2A_5$	$\frac{\eta {\left(18\tau \right)}^6\eta \left(12\tau \right)\eta {\left(6\tau \right)}^2}{\eta {\left(36\tau \right)}^3}$	3
49	48	$5A_1+6A_2$	$\frac{\eta {\left(24\tau \right)}^5\eta {\left(16\tau \right)}^6}{\eta {\left(48\tau \right)}^4}$	7/2
50	48	$6A_2+2A_3$	$\frac{\eta {\left(16\tau \right)}^6\eta {\left(12\tau \right)}^2}{\eta {\left(48\tau \right)}^2}$	3
51	48	$5A_1+A_2+2A_3+A_5$	$\frac{\eta {\left(24\tau \right)}^5\eta \left(16\tau \right)\eta {\left(12\tau \right)}^2\eta \left(8\tau \right)}{\eta {\left(48\tau \right)}^3}$	3
52	48	$4A_1+3A_5$	$\frac{\eta {\left(24\tau \right)}^4\eta {\left(8\tau \right)}^3}{\eta {\left(48\tau \right)}^2}$	5/2
53	48	$A_1+A_2+2A_3+2D_5$	$\frac{\eta {\left(24\tau \right)}^5\eta {\left(16\tau \right)}^3\eta {\left(12\tau \right)}^2\eta {\left(4\tau \right)}^2}{\eta {\left(48\tau \right)}^3\eta {\left(8\tau \right)}^4}$	5/2
54	48	$4A_1+A_2+A_7+E_6$	$\frac{\eta {\left(24\tau \right)}^5\eta {\left(16\tau \right)}^3\eta \left(6\tau \right)\eta \left(2\tau \right)}{\eta {\left(48\tau \right)}^3\eta {\left(4\tau \right)}^2}$	5/2
55	60	$4A_1+3A_2+2A_4$	$\frac{\eta {\left(30\tau \right)}^4\eta {\left(20\tau \right)}^3\eta {\left(12\tau \right)}^2}{\eta {\left(60\tau \right)}^3}$	3
56	64	$5A_1+3A_3+D_4$	$\frac{\eta {\left(32\tau \right)}^8\eta \left(16\tau \right)\eta \left(8\tau \right)}{\eta {\left(64\tau \right)}^4}$	3
57	64	$6A_1+3D_4$	$\frac{\eta {\left(32\tau \right)}^{15}\eta {\left(8\tau \right)}^3}{\eta {\left(64\tau \right)}^6\eta {\left(16\tau \right)}^6}$	3
58	64	$3A_1+3A_3+A_7$	$\frac{\eta {\left(32\tau \right)}^3\eta {\left(16\tau \right)}^3\eta \left(8\tau \right)}{\eta {\left(64\tau \right)}^2}$	5/2
59	64	$5A_3+D_4$	$\frac{\eta {\left(32\tau \right)}^3\eta {\left(16\tau \right)}^3\eta \left(8\tau \right)}{\eta {\left(64\tau \right)}^2}$	5/2
60	64	$4A_1+A_3+2D_6$	$\frac{\eta {\left(32\tau \right)}^8\eta {\left(16\tau \right)}^3\eta {\left(4\tau \right)}^2}{\eta {\left(64\tau \right)}^4\eta {\left(8\tau \right)}^4}$	5/2
61	72	$4A_1+3A_2+A_3+D_5$	$\frac{\eta {\left(36\tau \right)}^6\eta {\left(24\tau \right)}^4\eta \left(18\tau \right)\eta \left(6\tau \right)}{\eta {\left(72\tau \right)}^4\eta {\left(12\tau \right)}^2}$	3
62	72	$3A_1+2A_3+2A_5$	$\frac{\eta {\left(36\tau \right)}^3\eta {\left(18\tau \right)}^2\eta {\left(12\tau \right)}^2}{\eta {\left(72\tau \right)}^2}$	5/2
63	72	$A_2+3A_3+2D_4$	$\frac{\eta \left(24\tau \right)\eta {\left(9\tau \right)}^2\eta {\left(36\tau \right)}^6}{\eta {\left(72\tau \right)}^3\eta \left(18\tau \right)}$	5/2
64	80	$3A_1+4A_4$	$\frac{\eta {\left(40\tau \right)}^3\eta {\left(16\tau \right)}^4}{\eta {\left(80\tau \right)}^2}$	5/2
65	96	$3A_1+3A_2+3A_3$	$\frac{\eta {\left(48\tau \right)}^3\eta {\left(32\tau \right)}^3\eta {\left(24\tau \right)}^3}{\eta {\left(96\tau \right)}^3}$	3
66	96	$2A_1+2A_2+A_3+2A_5$	$\frac{\eta {\left(48\tau \right)}^2\eta {\left(32\tau \right)}^2\eta \left(24\tau \right)\eta {\left(16\tau \right)}^2}{\eta {\left(96\tau \right)}^2}$	5/2
67	96	$2A_1+3A_2+A_7+D_4$	$\frac{\eta {\left(48\tau \right)}^5\eta {\left(32\tau \right)}^3\eta {\left(12\tau \right)}^2}{\eta {\left(96\tau \right)}^3\eta {\left(24\tau \right)}^2}$	5/2
68	96	$3A_1+2A_3+A_5+D_5$	$\frac{\eta {\left(48\tau \right)}^5\eta {\left(24\tau \right)}^2\eta \left(8\tau \right)\eta \left(32\tau \right)}{\eta {\left(96\tau \right)}^3\eta \left(16\tau \right)}$	5/2
69	96	$3A_1+2A_2+2E_6$	$\frac{\eta {\left(48\tau \right)}^5\eta {\left(32\tau \right)}^6\eta {\left(4\tau \right)}^2}{\eta {\left(96\tau \right)}^4\eta {\left(8\tau \right)}^4}$	5/2
70	120	$2A_1+A_2+2A_3+A_4+A_5$	$\frac{\eta {\left(60\tau \right)}^2\eta \left(40\tau \right)\eta {\left(30\tau \right)}^2\eta \left(24\tau \right)\eta \left(20\tau \right)}{\eta {\left(120\tau \right)}^2}$	5/2
71	128	$3A_1+2A_3+D_4+D_6$	$\frac{\eta {\left(64\tau \right)}^8\eta \left(32\tau \right)\eta \left(8\tau \right)}{\eta {\left(128\tau \right)}^4\eta \left(16\tau \right)}$	5/2
72	144	$A_1+4A_2+2A_5$	$\frac{\eta \left(72\tau \right)\eta {\left(48\tau \right)}^4\eta {\left(24\tau \right)}^2}{\eta {\left(144\tau \right)}^2}$	5/2
73	160	$2A_1+3A_3+2A_4$	$\frac{\eta {\left(80\tau \right)}^2\eta {\left(40\tau \right)}^3\eta {\left(32\tau \right)}^2}{\eta {\left(160\tau \right)}^2}$	5/2
74	168	$A_1+3A_2+2A_3+A_6$	$\frac{\eta \left(84\tau \right)\eta {\left(56\tau \right)}^3\eta {\left(42\tau \right)}^2\eta \left(24\tau \right)}{\eta {\left(168\tau \right)}^2}$	5/2
75	192	$2A_1+6A_2+D_4$	$\frac{\eta {\left(96\tau \right)}^5\eta {\left(64\tau \right)}^6\eta \left(24\tau \right)}{\eta {\left(192\tau \right)}^4\eta {\left(48\tau \right)}^2}$	3
76	192	$2A_1+A_2+2A_3+A_5+D_4$	$\frac{\eta {\left(96\tau \right)}^5\eta \left(64\tau \right)\eta \left(32\tau \right)\eta \left(24\tau \right)}{\eta {\left(192\tau \right)}^3}$	5/2
77	192	$2A_1+A_2+3A_3+E_6$	$\frac{\eta {\left(96\tau \right)}^3\eta {\left(64\tau \right)}^3\eta {\left(48\tau \right)}^3\eta \left(8\tau \right)}{\eta {\left(192\tau \right)}^3\eta {\left(16\tau \right)}^2}$	5/2
78	288	$2A_1+2A_2+A_3+2D_5$	$\frac{\eta {\left(144\tau \right)}^6\eta {\left(96\tau \right)}^4\eta \left(72\tau \right)\eta {\left(24\tau \right)}^2}{\eta {\left(288\tau \right)}^4\eta {\left(48\tau \right)}^4}$	5/2
79	360	$A_1+2A_2+2A_3+2A_4$	$\frac{\eta \left(180\tau \right)\eta {\left(120\tau \right)}^2\eta {\left(90\tau \right)}^2\eta {\left(72\tau \right)}^2}{\eta {\left(360\tau \right)}^2}$	5/2
80	384	$A_1+3A_2+2A_3+D_6$	$\frac{\eta {\left(192\tau \right)}^3\eta {\left(128\tau \right)}^3\eta {\left(96\tau \right)}^3\eta \left(24\tau \right)}{\eta {\left(384\tau \right)}^3\eta {\left(48\tau \right)}^2}$	5/2
81	960	$A_1+3A_2+2A_4+D_4$	$\frac{\eta {\left(480\tau \right)}^4\eta {\left(320\tau \right)}^3\eta {\left(192\tau \right)}^2\eta \left(120\tau \right)}{\eta {\left(960\tau \right)}^3\eta {\left(240\tau \right)}^2}$	5/2

Equations

$$Z_{s}cX,G$$

$${\mathrm{Hilb}}^n(X{)}^G\subset {\mathrm{Hilb}}^n(X).$$

$$Z_{X,G}(q)=\text{sumop}{\text{displaylimits}}_{n=0}^{\infty }e\left({\mathrm{Hilb}}^n(X{)}^G\right)q^{n-1}$$

$$q=\exp \left(2\pi i\tau \right)$$

$$\eta (\tau )=q^{\frac{1}{24}}\text{prodop}{\text{displaylimits}}_{n=1}^{\infty }(1-q^n)$$

$$Z(q)=\text{sumop}{\text{displaylimits}}_{n=0}^{\infty }e\left({\mathrm{Hilb}}^n({\mathbb{C}}^2{)}^G\right)q^{n-\frac{1}{24}}.$$

$$Z_{A_n}(q)=\frac{1}{\eta (\tau )}$$

$$Z(q)=\frac{{\eta }^2(2\tau )\eta (4E\tau )}{\eta (\tau )\eta (2E\tau )\eta (2F\tau )\eta (2V\tau )}$$

$$(E,F,V)=\begin{cases}(n-2,2,n-2),\\ (6,4,4),\\ (12,8,6),\\ (30,20,12),\end{cases}\text{if}\quad\begin{array}[]{l}\Delta=D_{n}\\ \Delta=E_{6}\\ \Delta=E_{7}\\ \Delta=E_{8}\end{array}$$

$$Z(q)=\frac{\theta (\tau )}{\eta (k\tau {)}^{n+1}}$$

$$\theta (\tau )=\text{sumop}{\text{displaylimits}}_{\mathbf{m}\in M}{q}^{\frac{k}{2}\left(\mathbf{m}+\frac{1}{k}\mathbf{\zeta }|\mathbf{m}+\frac{1}{k}\mathbf{\zeta }\right)}$$

$${\theta }_{A_n}(\tau )=\frac{{\eta }^{n+1}((n+1)\tau )}{\eta (\tau )}$$

$$\theta (\tau )=\frac{{\eta }^2(2\tau ){\eta }^{n+2}(4E\tau )}{\eta (\tau )\eta (2E\tau )\eta (2F\tau )\eta (2V\tau )}$$

$$a=e(X/G)-r=\frac{24}{k}-\text{sumop}{\text{displaylimits}}_{i=1}^r\frac{1}{k_i}.$$

$$Z_{X,G}(q)={\eta }^{-a}(k\tau )\text{prodop}{\text{displaylimits}}_{i=1}^r{Z}_{{}_i}\left(\frac{k\tau }{k_i}\right).$$

$$(X\times E)/G$$

$$f_X(q)=\text{sumop}{\text{displaylimits}}_{n=1}^{\infty }{a}_n{q}^n$$

$$Z_{X,G}(q{)}^{-1}=f_X(q).$$

$$Z_{X,G}^{\mathsf{bir}}(q)=\text{sumop}{\text{displaylimits}}_{n=0}^{\infty }[{\mathrm{Hilb}}^n(X{)}^G{]}_{\mathsf{bir}}{q}^{n-1}$$

$$Z_{X,G}^{\mathsf{bir}}(q)=Z_Y^{\mathsf{bir}}(q^k)\cdot Z_{X,G}(q)\cdot \Delta (k\tau )$$

$$Z_{X,G}^{\mathsf{Ell}}(Q,q,y)=\text{sumop}{\text{displaylimits}}_{n=0}^{\infty }{\mathrm{Ell}}_{q,y}\left({\mathrm{Hilb}}^n(X{)}^G\right)Q^{n-1}$$

$$Z_{X,G}^{\mathsf{Ell}}(Q,q,y)=\frac{{\varphi }_{10,1}(\tau ,z)}{{\chi }_{10}(k\sigma ,\tau ,z)}\cdot Z_{X,G}(q)\cdot \Delta (k\tau )$$

$${\overline{\chi }}_{-y}(M)$$

$$=y^{-\frac{1}{2}\dim M}\text{sumop}{\text{displaylimits}}_{p,q}(-1{)}^{p+q}{y}^q\dim H^{p,q}(M)$$

$$Z_{X,G}^{\overline{\chi }}(q,y)=\text{sumop}{\text{displaylimits}}_{n=0}^{\infty }{\overline{\chi }}_{-y}\left({\mathrm{Hilb}}^n(X{)}^G\right)q^{n-1}.$$

$$Z_{X,G}^{\overline{\chi }}(q,y)=y^{-1}(1-y{)}^2\frac{Z_{X,G}(q)}{{\varphi }_{-2,1}(q^k,y)}$$

$$y^{-1}(1-y{)}^2{Z}_{X,G}^{\overline{\chi }}(q,y{)}^{-1}=\frac{{\varphi }_{-2,1}(q^k,y)}{Z_{X,G}(q)}$$

$$\frac{1}{2}e(X/G)-2=10-\frac{1}{2}\text{sumop}{\text{displaylimits}}_{i=1}^r\mathrm{rank}{}_i$$

$$Z_{X,G}^{K_0}(q)=\text{sumop}{\text{displaylimits}}_{n=0}^{\infty }[{\mathrm{Hilb}}^n(X{)}^G{]}_{K_0}{q}^{n-1}$$

$$Z_{X,G}^{K_0}(q)=q^{-1}\cdot \text{prodop}{\text{displaylimits}}_{m=1}^{\infty }{\left(1-{\mathbb{L}}^{m-1}{q}^{km}\right)}^{-[Y]}\cdot Z_{X,G}(q)\cdot \Delta (k\tau )$$

$$(X/G{)}^o=X/G\setminus \{p_1,\dots ,p_r\},$$

Full text

1. 1 Introduction
2. 2 The global partition function
3. 3 The local partition function
4. 4 The local partition function as a theta function via Nakajima
5. 5 Proof of Theorem 1.11
6. Appendix A.  Another Strange Formula
7. Apppendix B.  Table of eta products

1.  Introduction

Let $X$ be a complex $K3$ surface with an effective action of a group $G$ which preserves the holomorphic symplectic form. Mukai showed that such $G$ are precisely the subgroups of the Mathieu group $M_{23}\subset M_{24}$ such that the induced action on the set $\{1,\dots,24\}$ has at least five orbits [Muk88]. Xiao classified all possible actions into 82 possible topological types of the quotient $X/G$ [Xia96].

The $G$-fixed Hilbert scheme111Some authors call this the $G$-equivariant Hilbert scheme or the $G$-invariant Hilbert scheme. of $X$ parameterizes $G$-invariant length $n$ subschemes $Z\subset X$. It can be identified with the $G$-fixed point locus in the Hilbert scheme of points:

[TABLE]

We define the corresponding $G$-fixed partition function of $X$ by

[TABLE]

where $e(-)$ is the topological Euler characteristic.

Throughout this paper we set

[TABLE]

so that we may regard $Z_{X,G}$ as a function of $\tau\in\mathbb{H}$ where $\mathbb{H}$ is the upper half-plane.

1.1.  The Main Results.

Our main result is the following:

Theorem 1.1**.**

The function $Z_{X,G}(q)^{-1}$ is a modular cusp form222By cusp form, we mean that the order of vanishing at $q=0$ is at least 1. Modular forms of half integral weight transform with respect to a multiplier system. We refer to [Köh11] for definitions. of weight $\frac{1}{2}e(X/G)$ for the congruence subgroup ${}_{0}(|G|)$.

Our theorem specializes in the case where $G$ is the trivial group to a famous result of Göttsche [Göt90]. The case where $G$ is a cyclic group was proved in [BO18]. An analogous result for the case where $X$ is an Abelian surface acted on symplectically by a finite group $G$ has been recently proven by Pietromonaco [Pie20].

We give an explicit formula for $Z_{X,G}(q)$ in terms of the Dedekind eta function

[TABLE]

as follows. Let $p_{1},\dots,p_{r}$ be the singular points of $X/G$ and let $G_{1},\dots,G_{r}$ be the corresponding stabilizer subgroups of $G$. The singular points are necessarily of ADE type: they are locally given by ${\mathbb C}^{2}/G_{i}$ where $G_{i}\subset\operatorname{SU}(2)$. Finite subgroups of $\operatorname{SU}(2)$ have an ADE classification and we let ${}_{1},\dots,{}_{r}$ denote the corresponding ADE root systems.

For any finite subgroup $G\subset\operatorname{SU}(2)$ with associated root system we define the local $G$-fixed partition function by

[TABLE]

The main geometric result we prove is the following.

Theorem 1.2**.**

The local partition function for of type $A_{n}$ is given by

[TABLE]

and for type $D_{n}$ and $E_{n}$ by

[TABLE]

where $(E,F,V)$ are given by:

[TABLE]

Remark 1.3*.*

For of type $D_{n}$ or $E_{n}$, the group $H=G/\{\pm 1\}\subset\operatorname{SO}(3)$ is the symmetry group of a polyhedral decomposition of $S^{2}\cong\mathbb{P}^{1}$ into isomorphic regular spherical polygons. Then $E$, $F$, and $V$ are the number of edges, faces, and vertices of the polyhedron. The key idea in proving the above theorem is to show that $\operatorname{Hilb}({\mathbb C}^{2})^{G}$ is deformation equivalent to $\operatorname{Hilb}(Y)^{H}$ where $Y=\operatorname{Tot}(K_{\mathbb{P}^{1}})$ is the minimal resolution of ${\mathbb C}^{2}/\{\pm 1\}$ (see Section 3).

Using the work of Nakajima, we will also prove in Lemma 4.2 that

[TABLE]

where

[TABLE]

is a shifted theta function for $M$, the root lattice of . Here $n$ is the rank of the root system, $k=|G|$, and $\bm{\zeta}$ is dual to the longest root (see Section 4 and Equation (4.1) for details).

Theorem 1.2 then yields an eta product identity for the theta function $\theta(\tau)$ reminiscent of the MacDonald identities:

Theorem 1.4**.**

The shifted theta function $\theta(\tau)$ defined above $($cf. § 4 and Equation (4.1)$)$ is given by an eta product as follows:

[TABLE]

for of type $A_{n}$ and

[TABLE]

for of type $D_{n}$ or $E_{n}$, where $E,F,V$ are as in Theorem 1.2.

Remark 1.5*.*

Kac found that the Macdonald identities could be interpreted in terms of the character formula for highest weight representations of Kac-Moody algebras (cf. [Kac94, § 10]). It would be very interesting to find such an interpretation of the new identities in Theorem 1.4.

The 82 possible collections of ADE root systems ${}_{1},\dots,{}_{r}$ associated to $(X,G)$ a $K3$ surface with a symplectic $G$ action, are given in Appendix Apppendix B.  Table of eta products, Table LABEL:table:_list_of_eta_products. We let $k=|G|$, $k_{i}=|G_{i}|$, and

[TABLE]

The global series $Z_{X,G}(q)$ can be expressed as a product of local contributions (and thus via Theorem 1.2 as an explicit eta product) by our next result:

Theorem 1.6**.**

With the above notation we have

[TABLE]

Theorem 1.1 then immediately follows from Theorem 4.1 and Theorem 1.6 using the formulas for the weight and level of an eta product given in [Köh11, § 2.1].

In Appendix Apppendix B.  Table of eta products, Table LABEL:table:_list_of_eta_products we have listed explicitly the eta product of the modular form $Z_{X,G}(q)^{-1}$ for all 82 possible cases of $(X,G)$.

1.2.  Consequences of the Main Results.

Having obtained explicit eta product expressions for $Z_{X,G}(q)$ allows us to make several observational corollaries:

Corollary 1.7**.**

If $G$ is a finite subgroup of an elliptic curve $E$, i.e. $G$ is isomorphic to a product of one or two cyclic groups, then $Z_{X,G}(q)^{-1}$ is a Hecke eigenform. In Table LABEL:table:_list_of_eta_products these are the 13 cases having Xiao number in the set $\{0,1,2,3,4,5,7,8,11,14,15,19,25\}$. Moreover, in each of these cases, the dimension of the Hecke eigenspace is one.

We remark that in these cases, we may form a Calabi-Yau threefold called a CHL model by taking the free group quotient

[TABLE]

Then the partition function $Z_{X,G}(q)$ gives the Donaldson-Thomas invariants of $(X\times E)/G$ in curve classes which have degree zero over $X/G$ (cf. [BO18]).

Remark 1.8*.*

Hecke eigenforms of weight 3 arise in the arithmetic of $K3$ surfaces: if $X$ is a $K3$ surface defined over ${\mathbb Q}$ and has $\rho(X)=\operatorname{rk}\operatorname{NS(X)}=20$, then there is a weight 3 Hecke eigenform

[TABLE]

such that for almost all primes $p$, $a_{p}$ is the trace of the $p$-th Frobenius morphism acting on $H^{2}(X)/NS(X)$. There are four cases where $Z_{X,G}(q)^{-1}$ is a weight three Hecke eigenform and they correspond to the cases where $G$ is ${\mathbb Z}_{7}$, ${\mathbb Z}_{8}$, ${\mathbb Z}_{2}\times{\mathbb Z}_{6}$, or ${\mathbb Z}_{4}\times{\mathbb Z}_{4}$ (numbered 8, 14, 19, 25 on Table LABEL:table:_list_of_eta_products). If $X$ admits a symplectic $G$ action for one of these four groups, then we may take $X$ to be defined over ${\mathbb Q}$, have $\rho(X)=20$, and then remarkably

[TABLE]

Indeed, in each of these cases, we may take $X$ to be elliptically fibered over $\mathbb{P}^{1}$ and have $G$ as its group of sections (thus giving rise to the symplectic $G$ action). Moreover, $X$ is then the universal curve over the modular curve parameterizing $(E,G)$, an elliptic curve $E$ with a subgroup $G\subset E$. We thank Shuai Wang and Noam Elkies for noticing and elucidating this phenomenon.

For any eta product expression of a modular form, one may easily compute the order of vanishing (or pole) at any of the cusps [Köh11, Corollary 2.2]. Performing this computation on the 82 cases yields the following:

Corollary 1.9**.**

The modular form $Z_{X,G}(q)^{-1}$ always vanishes with order 1 at the cusps $i\infty$ and [math]. Moreover, $Z_{X,G}(q)^{-1}$ is holomorphic at all cusps except for the two cases with Xiao number 38 or 69, which have poles at the cusps $1/2$ and $1/8$ respectively. These are precisely the cases where $X/G$ has two singularities of type $E_{6}$.

Remark 1.10*.*

The integers $e\left(\operatorname{Hilb}^{n}(X)^{G}\right)$ should have enumerative significance: they can be interpreted as virtual counts of $G$-invariant curves, whose quotient is rational, in a complete linear series of dimension $n$ on $X$. This generalizes the famous Yau-Zaslow formula [YZ96] in the case where $G$ is the trivial group. The precise nature between the virtual count and the actual count is expected to be subtle for the case of general $G$. This has been recently explored in [Zha19] and also in the case of $G$ acting on an Abelian surface in [Pie20].

1.3.  Refinements of the Euler Characteristic.

We can extend our results to various refinements of the Euler characteristic, namely the elliptic genus, the $\chi_{y}$ genus, and the motivic class. These refinements all stem from the next result which we prove in Section 5. Let

[TABLE]

be a formal series whose coefficients we regard as birational equivalence classes of projective hyperkahler manifolds. Such equivalence classes form a semi-ring under disjoint union and Cartesian product.

Theorem 1.11**.**

Let $Y$ be the minimal resolution of $X/G$, then

[TABLE]

where $k=|G|$, $\Delta(\tau)=\eta(\tau)^{24}$, and we have suppressed the trivial group from the notation in the series $Z^{\mathsf{bir}}_{Y}(q^{k})$.

A famous theorem of Huybrechts [Huy99, Theorem 4.6] asserts that birational projective hyperkahler manifolds are deformation equivalent. Moreover, combining Huybrechts’ theorem with [NS17, Proposition 3.21] it follows that birational projective hyperkahler manifolds are equal in $K_{0}(\operatorname{Var}_{{\mathbb C}})$, the Grothendieck group of varieties.

Thus we may specialize the series $Z^{\mathsf{bir}}_{X,G}(q)$ to Elliptic genus, motivic class, and $\chi_{y}$ genus since these are all well defined on birational equivalence classes of projective hyperkahler manifolds. These specializations are all well known for the series $Z^{\mathsf{bir}}_{Y}$ and hence we easily get the following corollaries.

Corollary 1.12**.**

Let $Q=\exp\left(2\pi i\sigma\right)$, $q=\exp\left(2\pi i\tau\right)$, $y=\exp\left(2\pi iz\right)$, and let

[TABLE]

where $\operatorname{Ell}_{q,y}(-)$ is elliptic genus. Then

[TABLE]

where $\phi_{10,1}(q,y)$ is the unique Jacobi cusp form of weight 10 and index 1 and $\chi_{10}(\sigma,\tau,z)$ is Igusa’s genus 2 Siegel cusp form of weight 10.

We refer the reader to [Pie18, § 5, § 6, and Equation 6.9.8] for definitions of $\operatorname{Ell}_{q,y}$, $\phi_{10,1}$, $\chi_{10}$, and the formula for the elliptic genera of $\operatorname{Hilb}^{n}(Y)$.

A further specialization of particular interest is the (normalized) $\chi_{y}$ genus. Let

[TABLE]

and we note that $\overline{\chi}_{-y}(M)=\operatorname{Ell}_{q,y}(M)|_{q=0}$.

Corollary 1.13**.**

Let

[TABLE]

Then

[TABLE]

where $\phi_{-2,1}$ is the unique weak Jacobi form of weight $-2$ and index $1$. In particular,

[TABLE]

is a Jacobi form of index 1 and weight

[TABLE]

for the congruence subgroup ${}_{1}(k)$.

We note that for $G$ cyclic, the series $Z_{X,G}(q)/\phi_{-2,1}(q^{k},y)$ is the leading coefficient in the expansion of the Donaldson-Thomas partition function of $(X\times E)/G$ in the variable tracking the curve class in $X$ (see [BO18, Theorem 0.1]).

We also get a formula for the motivic classes of the $G$-fixed Hilbert schemes:

Corollary 1.14**.**

Let

[TABLE]

where $[\operatorname{Hilb}^{n}(X)^{G}]_{K_{0}}\in K_{0}(\operatorname{Var}_{{\mathbb C}})$ denotes the motivic class of the $G$-fixed Hilbert scheme. Then

[TABLE]

where $\mathbb{L}=[\mathbb{A}^{1}_{{\mathbb C}}]\in K_{0}(\operatorname{Var}_{{\mathbb C}})$.

We refer the reader to [GZLMH04] for the meaning of $[Y]$ in the exponent and the formula for the motivic class of $\operatorname{Hilb}^{n}(Y)$. The above series has further specializations giving formulas for the Hodge polynomials and Poincare polynomials of the $G$-fixed Hilbert schemes.

1.4.  Structure of paper.

In Section 2 we express the global partition function in terms of the local partition functions and deduce Theorem 1.6. In Section 3 we prove our main geometric result Theorem 1.2 which gives the eta product expression of the local partition functions. In Section 4 we express the local partition functions in terms of certain theta functions and thus prove our Theorem 1.4 which gives us the new theta function identities. In Section 5 we obtain the enhanced result of Theorem 1.11 on the partition function birational equivalence class of the $G$-fixed Hilbert schemes. Appendix Appendix A.  Another Strange Formula contains a proof of a root theoretic identity we need and Appendix Apppendix B.  Table of eta products contains a table listing the modular form $Z^{-1}_{X,G}$ in all 82 topological types of symplectic actions on a $K3$ surface.

Acknowledgements.

The authors warmly thank Jenny Bryan, Noam Elkies, Federico Amadio Guidi, Georg Oberdieck, Ken Ono, Stephen Pietromonaco, Balázs Szendrői, Shuai Wang, and Alex Weekes for helpful comments and/or technical help. We would also like to thank the anonymous referee for helping us to fix and greatly simplify the proof of Proposition 3.1.

2.  The global partition function

As in the introduction, let $X$ be a $K3$ surface with a symplectic action of a finite group $G$. Recall that $p_{1},\dotsc,p_{r}\in X/G$ are the singular points of $X/G$ with corresponding stabilizer subgroups $G_{i}\subset G$ of order $k_{i}$ and ADE type i. Let $\{x_{i}^{1},\dotsc,x_{i}^{k/k_{i}}\}$ be the orbit of $G$ in $X$ corresponding to the point $p_{i}$ (recall that $k=|G|$). We may stratify $\operatorname{Hilb}(X)^{G}$ according to the orbit types of subscheme as follows.

Let $Z\subset X$ be a $G$-invariant subscheme of length $nk$ whose support lies on free orbits. Then $Z$ determines and is determined by a length $n$ subscheme of

[TABLE]

i.e. a point in $\operatorname{Hilb}^{n}((X/G)^{o})$.

On the other hand, suppose $Z\subset X$ is a $G$-invariant subscheme of length $\frac{nk}{k_{i}}$ supported on the orbit $\{x_{i}^{1},\dotsc,x_{i}^{k/k_{i}}\}$. Then $Z$ determines and is determined by the length $n$ component of $Z$ supported on a formal neighborhood of one of the points, say $x_{i}^{1}$. Choosing a $G_{i}$-equivariant isomorphism of the formal neighborhood of $x_{i}^{1}$ in $X$ with the formal neighborhood of the origin in ${\mathbb C}^{2}$, we see that $Z$ determines and is determined by a point in $\operatorname{Hilb}_{0}^{n}({\mathbb C}^{2})^{G_{i}}$, where $\operatorname{Hilb}_{0}^{n}({\mathbb C}^{2})\subset\operatorname{Hilb}^{n}({\mathbb C}^{2})$ is the punctual Hilbert scheme parameterizing subschemes supported on a formal neighborhood of the origin in ${\mathbb C}^{2}$.

By decomposing an arbitrary $G$-invariant subscheme into components of the above types, we obtain a stratification of $\operatorname{Hilb}(X)^{G}$ into strata which are given by products of $\operatorname{Hilb}((X/G)^{o})$ and $\operatorname{Hilb}_{0}({\mathbb C}^{2})^{G_{1}},\dotsc,\operatorname{Hilb}_{0}({\mathbb C}^{2})^{G_{r}}$. Then using the fact that Euler characteristic is additive under stratifications and multiplicative under products, we arrive at the following equation of generating functions:

[TABLE]

As in the introduction, let $a=e(X/G)-r=e\left((X/G)^{o}\right)$. Then by Göttsche’s formula [Göt90],

[TABLE]

We also note that $e\left(\operatorname{Hilb}_{0}^{n}({\mathbb C}^{2})^{G_{i}}\right)=e\left(\operatorname{Hilb}^{n}({\mathbb C}^{2})^{G_{i}}\right)$ since the natural ${\mathbb C}^{*}$ action on both $\operatorname{Hilb}_{0}^{n}({\mathbb C}^{2})^{G_{i}}$ and $\operatorname{Hilb}^{n}({\mathbb C}^{2})^{G_{i}}$ have the same fixed points. Thus we may write

[TABLE]

Multiplying Equation (2.1) by $q^{-1}$ and substituting the above formulas, we find that

[TABLE]

From the following Euler characteristic calculation, we see that the exponent of $q$ in the above equation is zero:

[TABLE]

This completes the proof of Theorem 1.6. ∎

3.  The local partition function

Recall that the local partition function is defined by

[TABLE]

where $G\subset\operatorname{SU}(2)$ is the finite subgroup corresponding to the ADE root system . In this section, we prove Theorem 1.2 which provides an explicit formula for $Z(q)$ in terms of the Dedekind eta function. We regard this as the main geometric result of this paper.

3.1.  Proof of Theorem 1.2 in the $A_{n}$ case.

We wish to prove

[TABLE]

which is equivalent to the statement

[TABLE]

The action of ${\mathbb Z}/(n+1)$ on ${\mathbb C}^{2}$ commutes with the action of ${\mathbb C}^{*}\times{\mathbb C}^{*}$ on ${\mathbb C}^{2}$ and consequently, the Euler characteristics on the left hand side may be computed by counting the ${\mathbb C}^{*}\times{\mathbb C}^{*}$-fixed subschemes, namely those given by monomial ideals. Such subschemes of length $m$ have a well known bijection with integer partitions of $m$, whose generating function is given by the right hand side.∎

3.2.  Proof of Theorem 1.2 in the $D_{n}$ and $E_{n}$ cases.

Our proof of Theorem 1.2 in the $D_{n}$ and $E_{n}$ cases uses a trick exploiting the fact that the Hilbert schemes of the stack $\mathfrak{X}=[{\mathbb C}^{2}/\{\pm 1\}]$ and the Hilbert schemes of the space $Y=\operatorname{Tot}(K_{\mathbb{P}^{1}})$ can both be realized as moduli spaces of quiver representations of the $A_{1}$ Nakajima quiver variety.

Let $G\subset\operatorname{SU}(2)$ be a subgroup where the corresponding root system is of $D$ or $E$ type. Then $\{\pm 1\}\subset G$ and let $H\subset\operatorname{SO}(3)$ be the quotient

[TABLE]

The induced action of $H$ on $\mathbb{P}^{1}\cong S^{2}$ is by rotations. Indeed, $H$ is the symmetry group of a regular polyhedral decomposition of $S^{2}$ which is given by the platonic solids in the $E_{n}$ case and the decomposition into two hemispherical $(n-2)$-gons in the $D_{n}$ case. $H$ is generated by rotations of order $\mathsf{p}$, $\mathsf{q}$, $\mathsf{r}$, obtained by rotating about the center of an edge, a face, or a vertex respectively. The group $H$ has the following presentation:

[TABLE]

Let $M=|H|$ be the order of $H$ and let $E,F,V$ be the number of edges, faces, and vertices respectively. Then

[TABLE]

and since the stabilizer of an edge is always of order 2 we have $\mathsf{p}=2$ and so $M=2E$. Then since $F+V-E=2$ we find

[TABLE]

We summarize this information below:

[TABLE]

Now let $\mathfrak{X}$ be the quotient stack

[TABLE]

and let

[TABLE]

be the minimal resolution of the singular space $X={\mathbb C}^{2}/\{\pm 1\}$.

The quotient stack $[\mathbb{P}^{1}/H]$ has three stacky points with stabilizers of order $\mathsf{p},\mathsf{q},\mathsf{r}$, and consequently the stack quotient $[Y/H]$ has three orbifold points locally of the form $[{\mathbb C}^{2}/{\mathbb Z}_{\mathsf{a}}]$ for $\mathsf{a}\in\{\mathsf{p},\mathsf{q},\mathsf{r}\}$.

We observe that

[TABLE]

and consequently

[TABLE]

The scheme $\operatorname{Hilb}^{n}(\mathfrak{X})$ decomposes into components $\operatorname{Hilb}^{m_{0},m_{1}}(\mathfrak{X})$ with $n=m_{0}+m_{1}$ where the corresponding $\{\pm 1\}$ invariant subschemes $Z\subset{\mathbb C}^{2}$ have the property that as a $\{\pm 1\}$-representation, $H^{0}(\mathcal{O}_{Z})$ has $m_{0}$ copies of the trivial representation and $m_{1}$ copies of the non-trivial representation.

Proposition 3.1**.**

$\operatorname{Hilb}^{m_{0},m_{1}}(\mathfrak{X})^{H}$* is deformation equivalent to and hence diffeomorphic to $\operatorname{Hilb}^{m_{0}-(m_{0}-m_{1})^{2}}(Y)^{H}$. In particular*

[TABLE]

Proof.

It is known that both $\operatorname{Hilb}^{m_{0},m_{1}}(\mathfrak{X})$ and $\operatorname{Hilb}^{n}(Y)$ can be realised as moduli spaces of quiver representations of the $A_{1}$ Nakajima quiver variety with dimension vectors $(m_{0},m_{1})$ and $(n,n)$ respectively and framing vector $(1,0)$ for both (see [CGGS21, Proposition 5.2] and [Kuz07]). By the dimension formula for quiver varieties [Nak94, Equation (2.6)] (see also Equation (5.2)), $\dim\operatorname{Hilb}^{m_{0},m_{1}}(\mathfrak{X})=2n$ where $n=m_{0}-(m_{1}-m_{0})^{2}$. Consequently, both $\operatorname{Hilb}^{m_{0},m_{1}}(\mathfrak{X})$ and $\operatorname{Hilb}^{n}(Y)$ are crepant projective resolutions of $\operatorname{Sym}^{n}({\mathbb C}^{2}/\{\pm 1\})$. By [BC20, Corollary 1.3], any two projective crepant resolutions of $\operatorname{Sym}^{n}({\mathbb C}^{2}/\{\pm 1\})$ can be realized as moduli spaces of quiver representations with the same dimension and framing vectors (namely $(n,n)$ and $(1,0)$) but with different stability conditions. Then by Nakajima [Nak94, Corollary 4.2], $\operatorname{Hilb}^{m_{0},m_{1}}(\mathfrak{X})$ is deformation equivalent to, and hence diffeomorphic to $\operatorname{Hilb}^{m_{0}-(m_{1}-m_{0})^{2}}(Y)$. Moreover, as $H$ acts symplectically on both resolutions, the hyperkähler reduction providing the deformation equivalence in the proof of [Nak94, Corollary 4.2] can be performed $H$-equivariantly to obtain a deformation equivalence between the $H$-fixed points of the two resolutions. ∎

Let

[TABLE]

We then can compute:

[TABLE]

The following identity follows easily from the Jacobi triple product formula:

[TABLE]

Substituting this into the previous equation multiplied by $q^{\frac{1}{8}}$ we find

[TABLE]

We can now compute the summation factor in the above equation by the same method we used to compute the global series in Section 2. Here we use the fact that the singularities of $Y/H$ are all of type $A$ and we have already proven our formula for the local series in the $A_{n}$ case. Indeed, the quotient $[Y/H]$ has three stacky points with stabilizers ${\mathbb Z}_{\mathsf{p}}$, ${\mathbb Z}_{\mathsf{q}}$, and ${\mathbb Z}_{\mathsf{r}}$ and the complement of those points $(Y/H)^{o}$ has Euler characteristic $-1$. Proceeding then by the same argument we used in Section 2 to get Equation (2.1), we obtain

[TABLE]

Substituting into the previous equation and cancelling the factors of $q^{\frac{1}{6}}$, we have thus proved

[TABLE]

which completes the proof of Theorem 1.2 in the general case. ∎

4.  The local partition function as a theta function via Nakajima

The local partition functions $Z(q)$ considered in this paper are obtained from a specialization of the partition functions of the stack $[{\mathbb C}^{2}/G]$. Using the work of Nakajima [Nak02], the partition function of the Euler characteristics of the Hilbert scheme of points on the stack quotient $[{\mathbb C}^{2}/G]$ was computed explicitly in [GNS18] in terms of the root data of . We use this to express $Z(q)$ in terms of $\theta(\tau)$, a shifted theta function for the root lattice of . As a byproduct we obtain an eta product formula for the associated shifted theta function (Theorem 1.4).

A zero-dimensional substack $Z\subset[{\mathbb C}^{2}/G]$ may be regarded as a $G$ invariant, zero-dimensional subscheme of ${\mathbb C}^{2}$. Consequently, we may identify the Hilbert scheme of points on the stack $[{\mathbb C}^{2}/G]$ with the $G$ fixed locus of the Hilbert scheme of points on ${\mathbb C}^{2}$:

[TABLE]

This Hilbert scheme has components indexed by representations $\rho$ of $G$ as follows

[TABLE]

Let $\{\rho_{0},\dotsc,\rho_{n}\}$ be the irreducible representations of $G$ where $\rho_{0}$ is the trivial representation. We note that $n$ is also the rank of . We define

[TABLE]

Recall that our local partition function $Z(q)$ is defined by

[TABLE]

We then readily see that

[TABLE]

where

[TABLE]

The following formula is given explicitly in [GNS18, Theorem 1.3], but its content is already present in the work of Nakajima [Nak02]:

Theorem 4.1**.**

Let $C$ be the Cartan matrix of the root system , then

[TABLE]

where $Q=q_{0}^{d_{0}}q_{1}^{d_{1}}\dotsb q_{n}^{d_{n}}$.

We note that under the specialization $q_{i}=q^{d_{i}}$,

[TABLE]

where $k=|G|$ is the order of the group $G$.

We then obtain

[TABLE]

where $\bm{d}=(d_{1},\dotsc,d_{n})$.

Let $M$ be the root lattice of which we identify with ${\mathbb Z}^{n}$ via the basis given by $\alpha_{1},\dotsc,\alpha_{n}$, the simple positive roots of . Under this identification, the standard Weyl invariant bilinear form is given by

[TABLE]

and $\bm{d}$ is identified with the longest root. We define

[TABLE]

so that

[TABLE]

We may then write

[TABLE]

where

[TABLE]

and $\theta(\tau)$ is the shifted theta function:

[TABLE]

where as throughout this paper we have identified $q=\exp\left(2\pi i\tau\right)$.

In Appendix Appendix A.  Another Strange Formula, we will prove the following formula which for $\Delta=A_{n}$ coincides with the “strange formula” of Freudenthal and de Vries [FdV69]:

[TABLE]

It follows that $A=0$ and we obtain the following:

Lemma 4.2**.**

The local series $Z(q)$ is given by

[TABLE]

5.  Proof of Theorem 1.11

Let $\mathcal{Z}=[X/G]$ be the quotient stack of $X$ by $G$ and let $Y\to X/G$ be the minimal resolution. The Hilbert scheme of zero dimensional substacks of $\mathcal{Z}$ is naturally identified with the $G$-fixed Hilbert scheme of $X$:

[TABLE]

We emphasize that $\operatorname{Hilb}(\mathcal{Z})$ is itself a scheme, not just a stack, as the objects it parameterizes (substacks $V\subset\mathcal{Z}$) do not have automorphisms (see [OS03] or [BCY12, § 2.3]). Components of $\operatorname{Hilb}(\mathcal{Z})$ are indexed by the numerical $K$-theory class of $\mathcal{O}_{Z}$ for $Z\subset\mathcal{Z}$. The $K$-theory class of $\mathcal{O}_{Z}$ can be written in a basis for $K$-theory as follows:

[TABLE]

where $p\in\mathcal{Z}$ is a generic point and $p_{1},\dotsc,p_{r}\in\mathcal{Z}$ are the orbifold points. The local group of $\mathcal{Z}$ at $p_{i}$ is $G_{\Delta(i)}\subset\operatorname{SU}(2)$ and has corresponding root system $\Delta(i)$ of rank $n(i)$, and has irreducible representations $\rho_{0}(i),\rho_{1}(i),\dotsc,\rho_{n(i)}(i)$ where $\rho_{0}(i)$ is the trivial representation. We note that we do not need to include $[\mathcal{O}_{p_{i}}\otimes\rho_{0}(i)]$ in our basis for $K$-theory because of the following relation in $K$-theory which holds for all $i$:

[TABLE]

where $\rho_{\mathsf{reg}}(i)$ is the regular representation of $G_{\Delta(i)}$.

We abbreviate the data $\left\{m_{j}(i)\right\}$ appearing in the $K$-theory class above by the symbol $\mathfrak{m}$ and we denote by

[TABLE]

the corresponding component. Let

[TABLE]

where $E_{1}(i),\dotsc,E_{n(i)}(i)$ are the exceptional curves over $p_{i}$. We can organize the data $\mathfrak{m}=\left\{m_{j}(i)\right\}$ into $\bm{m}(i)\in M_{\Delta(i)}$, i.e. the vectors in the root lattice of $\Delta(i)$ having components $m_{1}(i),\dotsc,m_{n(i)}(i)$. Under this identification

[TABLE]

since the intersection form of the exceptional curves over $p_{i}$ is the negative of the corresponding Cartan matrix $C_{\Delta(i)}$.

Proposition 5.1**.**

$\operatorname{Hilb}^{n,\mathfrak{m}}(\mathcal{Z})$* is birational to $\operatorname{Hilb}^{n+\frac{1}{2}D_{\mathfrak{m}}^{2}}(Y)$.*

To prove this we will first need the following lemma.

Lemma 5.2**.**

1. (1)

*Let ** be a rank $n$ ADE root system, let $M$ be the corresponding root lattice, and let $G\subset\operatorname{SU}(2)$ the corresponding finite subgroup. To any element $\bm{m}\in M$ there is a unique rigid 333By definition, a substack $Z$ is rigid if it corresponds to an isolated point in the Hilbert scheme. * substack $Z_{\bm{m}}\subset[\mathbb{C}^{2}/G]$ with $K$-theory class

[TABLE]

where $p\in[\mathbb{C}^{2}/G]$ is a generic point. 2. (2)

For every datum $\mathfrak{m}$ there is a unique rigid substack $Z_{\mathfrak{m}}\subset\mathcal{Z}$ with $K$-theory class

[TABLE]

*where $p\in\mathcal{Z}$ is a generic point. *

Proof.

Part (2) is implied by Part (1) since we can take the union of the rigid subschemes supported at the orbifold points $p_{1},\dots,p_{r}\in\mathcal{Z}$. So we need only prove the local case.

To prove Part (1) we need to show that component of $\operatorname{Hilb}([{\mathbb C}^{2}/G])$ corresponding to substacks with $K$-theory class

[TABLE]

is a single isolated point. This component corresponds to the coefficient of

[TABLE]

in Theorem 4.1. It follows immediately from the formula in Theorem 4.1 that this coefficient is 1, and thus to prove this component is a single point, we need only prove that it has dimension 0.

By Equation (5.1), we have

[TABLE]

and so the component in question is

[TABLE]

where

[TABLE]

We define

[TABLE]

so that our $\bm{v}$ of interest may be written

[TABLE]

Nakajima has shown [Nak02, § 2] that

[TABLE]

where $\bm{w}=(1,0,\dotsc,0)$ and $M(\bm{v},\bm{w})$ is the Nakajima quiver variety associated to the affine Dynkin diagram of with framing vector $\bm{w}$ and dimension vector $\bm{v}$. By [Nak94, Equation (2.6)] we have

[TABLE]

where $\left\langle\cdot,\cdot\right\rangle$ is the inner product given by the Cartan matrix associated to the affine Dynkin diagram.

We also have

[TABLE]

The first follows directly from our definitions, and the later two are well known properties of the vector $\bm{\delta}$. Using the above we compute the dimension of the Hilbert scheme of interest:

[TABLE]

We thus can conclude that $\operatorname{Hilb}^{v_{0}\rho_{0}+\dotsb+v_{n}\rho_{n}}([{\mathbb C}^{2}/G])=M(\bm{v},\bm{w})$ is a single point which finishes the proof of the lemma. ∎

Proof of Proposition 5.1.

Let $U=\mathcal{Z}\setminus\{p_{1},\dotsc,p_{r}\}$ be the Zariski open part with trivial stabilizers. Let $V\subset Y$ be the complement of the exceptional divisors. Let furthermore $Z_{\mathfrak{m}}\subset\mathcal{Z}$ be the rigid substack corresponding to the $K$-theory datum $\mathfrak{m}$ provided by Lemma 5.2. The Zariski open substack of $\mathrm{Hilb}^{n,\mathfrak{m}}(\mathcal{Z})$ parameterizing substacks of $\mathcal{Z}$ of the form $P\cup Z_{\mathfrak{m}}$ where $P$ is a colength $n+\frac{1}{2}D_{\mathfrak{m}}^{2}$ subscheme of $U$ is isomorphic to $\mathrm{Hilb}^{n+\frac{1}{2}D_{\mathfrak{m}}^{2}}(U)$. This is because $Z_{\mathfrak{m}}$ was rigid and it had the $K$-theory class

[TABLE]

On the other hand, the Zariski open subset of $\mathrm{Hilb}^{n+\frac{1}{2}D_{\mathfrak{m}}^{2}}(Y)$ parameterizing subschemes supported on $V\subset Y$ is isomorphic to $\mathrm{Hilb}^{n+\frac{1}{2}D_{\mathfrak{m}}^{2}}(V)$. Finally, $\mathrm{Hilb}^{n+\frac{1}{2}D_{\mathfrak{m}}^{2}}(U)\cong\mathrm{Hilb}^{n+\frac{1}{2}D_{\mathfrak{m}}^{2}}(V)$ since $U$ and $V$ are canonically isomorphic. ∎

With Proposition 5.1, we can now prove Theorem 1.11. Using the identification

[TABLE]

and identifying discrete parameters we get

[TABLE]

where (recalling that $d_{j}(i)=\dim\rho_{j}(i)$),

[TABLE]

Let

[TABLE]

Then

[TABLE]

with

[TABLE]

where $\bm{\zeta}(i)\in M_{\Delta(i)}\otimes{\mathbb Q}$ is as in Section 4.

Completing the square and using the formula

[TABLE]

which follows from Lemma A.1, we get

[TABLE]

It then follows that

[TABLE]

where

[TABLE]

Since

[TABLE]

we see that $A=0$.

Thus we have

[TABLE]

where we used Theorem 1.4, Theorem 1.6, and we set

[TABLE]

The previous equation which showed that $A=0$ also shows that $B=24$. Then since $\Delta(\tau)=\eta(\tau)^{24}$, we see that Theorem 1.11 follows. ∎

Appendix A.  Another Strange Formula

We recall the notation from Section 4. Let be an ADE root system of rank $n$. Let $\alpha_{1},\dotsc\alpha_{n}$ be a system of positive simple roots and let

[TABLE]

be the largest root. Let $(\cdot|\cdot)$ be the Weyl invariant bilinear form with $(\alpha_{i}|\alpha_{i})=2$ and let $\bm{\zeta}$ be the dual vector to $\bm{d}$ in the sense that

[TABLE]

Let

[TABLE]

The identity of the following lemma coincides with Freudenthal and de Vries’s “strange formula” when is $A_{n}$.

Lemma A.1**.**

Let $k$, $n$, and $\bm{\zeta}$ be as above. Then,

[TABLE]

Proof.

The case of $\Delta=A_{n}$.— For any ADE root system we have $(\bm{\rho}|\alpha)=1$ for all positive roots where $\bm{\rho}=\frac{1}{2}\sumop\displaylimits_{\alpha\in R^{+}}\alpha$ is half the sum of the positive roots. Since for $A_{n}$, $d_{i}=1$, it follows from Equation (A.1) that $\bm{\zeta}=\bm{\rho}$, and it follows from Equation (A.2) that $k=n+1=h$ is the Coxeter number. The lemma is then

[TABLE]

Since the Lie algebra associated to $A_{n}$, namely $\mathfrak{sl}_{n+1}$, has dimension $(n+1)^{2}-1$ and the Killing form satisfies $\kappa(\cdot,\cdot)=\frac{1}{2h}(\cdot|\cdot)$, the lemma may be rewritten as

[TABLE]

which is Freudenthal and de Vries’s “Strange Formula” [FdV69, § 47.11].

The case of $\Delta=D_{n}$.— Let $e_{1},\dotsc,e_{n}$ be the standard orthonormal basis of ${\mathbb R}^{n}$. Then the collection $\{\pm e_{i}\pm e_{j},i<j\}$ is a $D_{n}$ root system and we may take

[TABLE]

as a system of simple positive roots. Then the fundamental weights $\omega_{i}$, which are defined by the condition $(\omega_{i}|\alpha_{j})=\delta_{ij}$, are given by [Kna96, Appendix C]

[TABLE]

Then since

[TABLE]

we have

[TABLE]

and so

[TABLE]

Finally since $k=1+\sumop\displaylimits_{i=1}^{n}d_{i}=4(n-2)$ the lemma becomes

[TABLE]

which is readily verified.

The case of $\Delta=E_{6},E_{7},E_{8}$.— These three individual cases are easily checked one by one. ∎

Apppendix B.  Table of eta products

The following table provides the list of the modular forms $Z_{X,G}^{-1}$, expressed as eta products, for each of the 82 possible symplectic actions of a group $G$ on a $K3$ surface $X$. Our numbering matches Xiao’s [Xia96] whose table we refer to for a description of each group.