Generating weighted Hurwitz numbers

M. Bertola; J. Harnad; B. Runov

arXiv:1901.10038·math-ph·March 4, 2021

Generating weighted Hurwitz numbers

M. Bertola, J. Harnad, B. Runov

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TL;DR

This paper introduces a method to generate weighted Hurwitz numbers using multicurrent correlators linked to KP tau-functions, enabling the computation of various types of weighted Hurwitz numbers through formal series expansions.

Contribution

It presents a novel approach connecting multicurrent correlators and KP tau-functions to generate and compute weighted Hurwitz numbers in a unified framework.

Findings

01

Derived formal Taylor series for weighted Hurwitz numbers

02

Computed simple, rational, and quantum weighted Hurwitz numbers

03

Established a new link between correlators and Hurwitz number generation

Abstract

Multicurrent correlators associated to KP $τ$ -functions of hypergeometric type are used as generating functions for weighted Hurwitz numbers. These are expressed as formal Taylor series and used to compute generic, simple, rational and quantum weighted single Hurwitz numbers.

Tables13

Table 1. Table 1: ρ a b 1 ( 𝐠 ) superscript subscript 𝜌 𝑎 𝑏 1 𝐠 \rho_{ab}^{1}({\bf g}) for 0 ≤ a , b , ≤ 4 0\leq a,b,\leq 4

$a \ b$	$\|$	$0$	$1$	$2$	$3$	$4$
$0$	$\|$	$0$	$\frac{1}{2} g_{1}$	$- \frac{1}{2} g_{1}$	$\frac{1}{4} g_{1}$	$- \frac{1}{12} g_{1}$
1	—	$\frac{1}{2} g_{1}$	$0$	$- \frac{1}{4} g_{1}$	$\frac{1}{6} g_{1}$	$- \frac{1}{16} g_{1}$
2	—	$\frac{1}{2} g_{1}$	$- \frac{1}{4} g_{1}$	$0$	$\frac{1}{24} g_{1}$	$- \frac{1}{48} g_{1}$
3	—	$\frac{1}{4} g_{1}$	$- \frac{1}{6} g_{1}$	$\frac{1}{24} g_{1}$	$0$	$- \frac{1}{288} g_{1}$
4	—	$\frac{1}{12} g_{1}$	$- \frac{1}{16} g_{1}$	$\frac{1}{48} g_{1}$	$- \frac{1}{288} g_{1}$	$0$

Table 2. Table 2: ρ a b 2 ( 𝐠 ) superscript subscript 𝜌 𝑎 𝑏 2 𝐠 \rho_{ab}^{2}({\bf g}) for 0 ≤ a , b ≤ 4 formulae-sequence 0 𝑎 𝑏 4 0\leq a,b\leq 4 . The last column ρ a 4 2 ( 𝐠 ) subscript superscript 𝜌 2 𝑎 4 𝐠 \rho^{2}_{a4}({\bf g}) is determined by applying the symmetry property ( 2.67 ) and setting ρ 4 , 4 2 = − 5 864 ( g 1 2 − 2 g 2 ) subscript superscript 𝜌 2 4 4 5 864 superscript subscript 𝑔 1 2 2 subscript 𝑔 2 \rho^{2}_{4,4}=-{5\over 864}(g_{1}^{2}-2g_{2}) .

$a \ b$	$\|$	$0$	$1$	$2$	$3$
$0$	$\|$	$0$	$- \frac{1}{2} g_{2}$	$\frac{1}{6} (2 g_{1}^{2} + 5 g_{2})$	$- \frac{1}{24} (11 g_{1}^{2} + 14 g_{2})$
$1$	—	$\frac{1}{2} g_{2}$	$\frac{1}{3} (g_{1}^{2} - 2 g_{2})$	$\frac{1}{8} (- g_{1}^{2} + 6 g_{2})$	$- \frac{1}{6} (g_{1}^{2} + 3 g_{2})$
$2$	—	$\frac{1}{6} (2 g_{1}^{2} + 5 g_{2})$	$\frac{1}{8} (g_{1}^{2} - 6 g_{2})$	$\frac{1}{4} (- g_{1}^{2} + 2 g_{2})$	$\frac{1}{72} (5 g_{1}^{2} - 19 g_{2})$
$3$	—	$\frac{1}{24} (11 g_{1}^{2} + 14 g_{2})$	$- \frac{1}{6} (g_{1}^{2} + 3 g_{2})$	$\frac{1}{72} (- 5 g_{1}^{2} + 19 g_{2})$	$\frac{1}{18} (g_{1}^{2} - 2 g_{2})$
$4$	—	$\frac{1}{24} (7 g_{1}^{2} + 6 g_{2})$	$- \frac{1}{144} (25 g_{1}^{2} + 31 g_{2})$	$\frac{1}{48} (g_{1}^{2} + 5 g_{2})$	$\frac{1}{576} (7 g_{1}^{2} - 22 g_{2})$

Table 3. Table 3: ρ a b 3 ( 𝐠 ) superscript subscript 𝜌 𝑎 𝑏 3 𝐠 \rho_{ab}^{3}({\bf g}) for 0 ≤ a , b ≤ 4 formulae-sequence 0 𝑎 𝑏 4 0\leq a,b\leq 4 . The last column ρ a 4 2 ( 𝐠 ) subscript superscript 𝜌 2 𝑎 4 𝐠 \rho^{2}_{a4}({\bf g}) is determined by applying the symmetry property ( 2.67 ) and setting ρ 4 , 4 3 = 0 subscript superscript 𝜌 3 4 4 0 \rho^{3}_{4,4}=0 .

$a \ b$	$\|$	$0$	$1$	$2$	$3$
$0$	$\|$	$0$	$\frac{1}{2} g_{3}$	$- \frac{1}{6} (6 g_{1} g_{2} - 9 g_{3})$	$\frac{1}{4} (\frac{g_{1}^{3} + 8 g_{1} g_{2}}{+ 6 g_{3}})$
$1$	—	$\frac{1}{2} g_{3}$	$0$	$\frac{1}{4} (\frac{g_{1}^{3} - 2 g_{2} g_{2}}{- 4 g_{3}})$	$\frac{1}{6} (\frac{- g_{1}^{3} + 8 g_{1} g_{2}}{+ 7 g_{3}})$
$2$	—	$\frac{1}{6} (6 g_{1} g_{2} + 9 g_{3})$	$\frac{1}{4} (\frac{g_{1}^{3} - 2 g_{1} g_{2}}{- 4 g_{3}})$	$0$	$\frac{1}{24} (\frac{- 5 g_{1}^{3} + 10 g_{1} g_{2}}{+ 9 g_{3}})$
$3$	—	$\frac{1}{4} (\frac{g_{1}^{3} + 8 g_{1} g_{2}}{+ 6 g_{3}})$	$\frac{1}{6} (\frac{g_{1}^{3} - 8 g_{1} g_{2}}{- 7 g_{3}})$	$\frac{1}{24} (\frac{- 5 g_{1}^{3} + 10 g_{1} g_{2}}{+ 9 g_{3}})$	$0$
$4$	—	$\frac{5}{12} (\frac{g_{1}^{3} + 4 g_{1} g_{2}}{+ 2 g_{3}})$	$- \frac{1}{48} (\frac{5 g_{1}^{3} + 60 g_{1} g_{2}}{+ 33 g_{3}})$	$\frac{1}{48} (\frac{- 5 g_{1}^{3} + 22 g_{1} g_{2}}{+ 13 g_{3}})$	$\frac{1}{144} (\frac{7 g_{1}^{3} - 14 g_{1} g_{2}}{- 8 g_{3}})$

Table 4. Table 4: Nonconnected weighted Hurwitz numbers H G d ( μ ) subscript superscript 𝐻 𝑑 𝐺 𝜇 H^{d}_{G}(\mu) with ℓ ( μ ) = 1 ℓ 𝜇 1 \ell(\mu)=1 or 2 2 2 , N = 2 , 3 𝑁 2 3 N=2,3 or 4 4 4

$N$	$μ$	$H_{G}^{N + ℓ (μ) - 2}$	$H_{G}^{N + ℓ (μ)}$
2	(2)	$\frac{1}{2} g_{1}$	$\frac{1}{2} g_{3}$
3	(3)	$\frac{1}{3} (g_{1}^{2} + g_{2})$	$\frac{1}{3} (g_{2}^{2} + 4 g_{1} g_{3} + 5 g_{4})$
3	(21)	$g_{1} g_{2} + \frac{3}{2} g_{3}$	$3 g_{1} g_{4} + 2 g_{2} g_{3} + \frac{11}{2} g_{5}$
4	(4)	$\frac{1}{4} (g_{1}^{3} + 3 g_{1} g_{2} + g_{3})$	$\frac{5}{2} g_{1}^{2} g_{3} + \frac{5}{4} g_{1} g_{2}^{2} + \frac{25}{4} g_{4} g_{1} + \frac{15}{4} g_{2} g_{3} + \frac{15}{4} g_{5}$
4	(31)	$g_{1}^{2} g_{2} + \frac{4}{3} g_{2}^{2} + \frac{10}{3} g_{1} g_{3} + \frac{8}{3} g_{4}$	$g_{1} (6 g_{1} g_{4} + 8 g_{2} g_{3} + 24 g_{5}) + g_{2} (g_{2}^{2} + 16 g_{4}) + 7 g_{3}^{2} + 22 g_{6}$
4	(22)	$\frac{1}{2} g_{1}^{2} g_{2} + \frac{1}{4} g_{2}^{2} + g_{1} g_{3} + \frac{1}{2} g_{4}$	$g_{1} (\frac{11}{4} g_{1} g_{4} + \frac{13}{4} g_{2} g_{3} + 9 g_{5}) + g_{2} (\frac{1}{4} g_{2}^{2} + \frac{19}{4} g_{4}) + \frac{11}{4} g_{3}^{2} + \frac{25}{4} g_{6}$

Table 5. Table 5: Connected weighted Hurwitz numbers H ~ G d ( μ ) subscript superscript ~ 𝐻 𝑑 𝐺 𝜇 \tilde{H}^{d}_{G}(\mu) with ℓ ( μ ) = 1 ℓ 𝜇 1 \ell(\mu)=1 or 2 2 2 , N = 2 , 3 𝑁 2 3 N=2,3 or 4 4 4

$N$	$μ$	${\tilde{H}}_{G}^{N + ℓ (μ) - 2}$	${\tilde{H}}_{G}^{N + ℓ (μ)}$
2	(2)	$\frac{1}{2} g_{1}$	$\frac{1}{2} g_{3}$
3	(3)	$\frac{1}{3} (g_{1}^{2} + g_{2})$	$\frac{1}{3} (g_{2}^{2} + 4 g_{1} g_{3} + 5 g_{4})$
3	(21)	$g_{1} g_{2} + g_{3}$	$3 g_{1} g_{4} + 2 g_{2} g_{3} + 5 g_{5}$
4	(4)	$\frac{1}{4} (g_{1}^{3} + 3 g_{1} g_{2} + g_{3})$	$\frac{5}{2} g_{1}^{2} g_{3} + \frac{5}{4} g_{1} g_{2}^{2} + \frac{25}{4} g_{4} + \frac{15}{4} g_{2} g_{3} + \frac{15}{4} g_{5}$
4	(31)	$g_{1}^{2} g_{2} + g_{2}^{2} + 2 g_{1} g_{3} + g_{4}$	$g_{1} (6 g_{1} g_{4} + 8 g_{2} g_{3} + 20 g_{5}) + g_{2} (g_{2}^{2} + 14 g_{4}) + 6 g_{3}^{2} + 15 g_{6}$
4	(22)	$\frac{1}{2} g_{1}^{2} g_{2} + \frac{1}{4} g_{2}^{2} + g_{1} g_{3} + \frac{1}{2} g_{4}$	$g_{1} (\frac{11}{4} g_{1} g_{4} + \frac{13}{4} g_{2} g_{3} + \frac{35}{4} g_{5}) + g_{2} (\frac{1}{4} g_{2}^{2} + \frac{19}{4} g_{4}) + \frac{5}{2} g_{3}^{2} + \frac{25}{4} g_{6}$

Table 6. Table 6: Nonconnected weighted Hurwitz numbers H G d ( μ ) subscript superscript 𝐻 𝑑 𝐺 𝜇 H^{d}_{G}(\mu) with ℓ ( μ ) = 3 ℓ 𝜇 3 \ell(\mu)=3 , N = 3 , 4 , 5 𝑁 3 4 5 N=3,4,5 or 6 6 6

$N$	$μ$	$d$	$H_{G}^{d} (μ)$
3	(1,1,1)	$2$	$\frac{1}{2} g_{2}$
4	(2,1,1)	$3$	$\frac{5}{4} (g_{1} g_{2} + g_{3})$
5	(2,2,1)	$2$	$\frac{1}{8} g_{1}^{2}$
5	(2,2,1)	$4$	$g_{1}^{2} g_{2} + \frac{1}{4} g_{2}^{2} + \frac{7}{4} g_{4}$
6	(3,2,1)	$3$	$1 / 6 (g_{1}^{3} + g_{1} g_{2})$
6	(3,2,1)	$5$	$1 / 6 (11 g_{1}^{3} g_{2} + 31 g_{1}^{2} g_{3} + 15 g_{2} g_{3} + 2 g_{1} (9 g_{2}^{2} + 13 g_{4})) + g_{5}$

Table 7. Table 7: Connected weighted Hurwitz numbers H ~ G d ( μ ) subscript superscript ~ 𝐻 𝑑 𝐺 𝜇 \tilde{H}^{d}_{G}(\mu) with ℓ ( μ ) = 3 ℓ 𝜇 3 \ell(\mu)=3 , N = 3 , 4 , 5 𝑁 3 4 5 N=3,4,5 or 6 6 6 . By the Riemann-Hurwitz formula, H ~ G d ( 1 , 1 , 1 ) = 0 subscript superscript ~ 𝐻 𝑑 𝐺 1 1 1 0 \tilde{H}^{d}_{G}(1,1,1)=0 for d < 4 𝑑 4 d<4 , H ~ G d ( 2 , 1 , 1 ) = 0 subscript superscript ~ 𝐻 𝑑 𝐺 2 1 1 0 \tilde{H}^{d}_{G}(2,1,1)=0 for d < 5 𝑑 5 d<5 , H ~ G d ( 2 , 2 , 1 ) = 0 subscript superscript ~ 𝐻 𝑑 𝐺 2 2 1 0 \tilde{H}^{d}_{G}(2,2,1)=0 for d < 6 𝑑 6 d<6 and H ~ G d ( 2 , 2 , 2 ) = H ~ G d ( 3 , 2 , 1 ) = 0 subscript superscript ~ 𝐻 𝑑 𝐺 2 2 2 subscript superscript ~ 𝐻 𝑑 𝐺 3 2 1 0 \tilde{H}^{d}_{G}(2,2,2)=\tilde{H}^{d}_{G}(3,2,1)=0 for d < 7 𝑑 7 d<7 .

$N$	$μ$	$d$	${\tilde{H}}_{G}^{d} (μ)$
3	(1,1,1)	$4$	$\frac{1}{3} (g_{2}^{2} + g_{1} g_{3} + 2 g_{4})$
4	(2,1,1)	$5$	$\frac{1}{2} (2 g_{1}^{2} g_{3} + 7 g_{2} g_{3} + g_{1} (3 g_{2}^{2} + 7 g_{4}) + 5 g_{5})$
4	(2,1,1)	$7$	$5 g_{2}^{2} g_{3} + 22 g_{3} g_{4} + 10 g_{1}^{2} g_{5} + 15 g_{2} (g_{1} g_{4} + 2 g_{5}) + 5 g_{1} (g_{3}^{2} + 8 g_{6}) + 35 g_{7}$
5	(2,2,1)	$6$	$g_{2}^{3} + g_{1}^{3} g_{3} + 5 g_{2} g_{4} + g_{1}^{2} (2 g_{2}^{2} + 5 g_{4}) + g_{1} (9 g_{2} g_{3} + 7 g_{5}) + 3 (g_{3}^{2} + g_{6})$
6	(2,2,2)	$7$	$1 / 6 (2 g_{1}^{4} g_{3} + 11 g_{2}^{2} g_{3} + 17 g_{3} g_{4} + g_{1}^{3} (5 g_{2}^{2} + 13 g_{4}) + 13 g_{2} g_{5}$
			$+ 3 g_{1}^{2} (11 g_{2} g_{3} + 9 g_{5}) + g_{1} (7 g_{2}^{3} + 22 g_{3}^{2} + 36 g_{2} g_{4} + 23 g_{6}) + 7 g_{7})$
6	(3,2,1)	$7$	$2 g_{1}^{4} g_{3} + 18 g_{2}^{2} g_{3} + 17 g_{3} g_{4} + g_{1}^{3} (5 g_{2}^{2} + 13 g_{4}) + 18 g_{2} g_{5}$
			$+ g_{1}^{2} (35 g_{2} g_{3} + 27 g_{5}) + g_{1} (10 g_{2}^{3} + 22 g_{3}^{2} + 43 g_{2} g_{4} + 23 g_{6}) + 7 g_{7}$

Table 8. Table 8: Connected and nonconnected simple Hurwitz numbers H exp d ( μ ) subscript superscript 𝐻 𝑑 𝜇 H^{d}_{\exp}(\mu) with ℓ ( μ ) = 1 ℓ 𝜇 1 \ell(\mu)=1 or 2 2 2 , N = 2 , 3 𝑁 2 3 N=2,3 or 4 4 4

$N$	$μ$	${\tilde{H}}_{\exp}^{N + ℓ (μ) - 2}$	${\tilde{H}}_{\exp}^{N + ℓ (μ)}$	$H_{\exp}^{N + ℓ (μ) - 2}$	$H_{\exp}^{N + ℓ (μ)}$
2	(2)	$\frac{1}{2}$	$\frac{1}{12}$	$\frac{1}{2}$	$\frac{1}{12}$
3	(3)	$\frac{1}{2}$	$\frac{3}{8}$	$\frac{1}{2}$	$\frac{3}{8}$
3	(21)	$\frac{3}{4}$	$\frac{1}{3}$	$\frac{3}{4}$	$\frac{27}{80}$
4	(4)	$\frac{2}{3}$	$\frac{4}{3}$	$\frac{2}{3}$	$\frac{4}{3}$
4	(31)	$\frac{9}{8}$	$\frac{27}{16}$	$\frac{3}{4}$	$\frac{9}{5}$
4	(22)	$\frac{1}{2}$	$\frac{2}{3}$	$\frac{13}{24}$	$\frac{121}{180}$

Table 9. Table 9: Connected and nonconnected simple Hurwitz numbers H exp d ( μ ) subscript superscript 𝐻 𝑑 𝜇 H^{d}_{\exp}(\mu) with ℓ ( μ ) = 3 ℓ 𝜇 3 \ell(\mu)=3 , N = 3 , 4 , 5 𝑁 3 4 5 N=3,4,5 or 6 6 6

$N$	$μ$	$d$	${\tilde{H}}_{\exp}^{d} (μ)$	$H_{\exp}^{d} (μ)$
$3$	$(1, 1, 1)$	$4$	$\frac{1}{6}$	$\frac{3}{16}$
$4$	$(2, 1, 1)$	$5$	$1$	$\frac{41}{30}$
$4$	$(2, 1, 1)$	$7$	$\frac{13}{12}$	$\frac{73}{63}$
$5$	$(2, 2, 1)$	$6$	$2$	$\frac{521}{180}$
$6$	$(2, 2, 2)$	$7$	$\frac{4}{3}$	$\frac{9853}{5760}$
$6$	$(3, 2, 1)$	$7$	$9$	$\frac{2511}{160}$

Table 10. Table 10: Nonconnected quantum weighted Hurwitz numbers H G H q d ( μ ) subscript superscript 𝐻 𝑑 subscript 𝐺 subscript 𝐻 𝑞 𝜇 H^{d}_{G_{H_{q}}}(\mu) with ℓ ( μ ) = 1 ℓ 𝜇 1 \ell(\mu)=1 or 2 2 2 , N = 2 , 3 𝑁 2 3 N=2,3 or 4 4 4

$N$	$μ$	$H_{G_{H_{q}}}^{N + ℓ (μ) - 2}$	$H_{G_{H_{q}}}^{N + ℓ (μ)}$
2	(2)	$\frac{1}{2 {(q; q)}_{1}}$	$\frac{1}{2 {(q; q)}_{3}}$
3	(3)	$\frac{2 + q}{3 {(q; q)}_{3}}$	$\frac{10 + 5 q + 6 q^{2} + 5 q^{3} + q^{4}}{3 {(q; q)}_{4}}$
3	(21)	$\frac{5 + 2 q + 2 q^{2}}{2 {(q; q)}_{3}}$	$\frac{21 + 10 q + 14 q^{2} + 14 q^{3} + 14 q^{4} + 4 q^{5} + 4 q^{6}}{2 {(q; q)}_{5}}$
4	(4)	$\frac{5 + 5 q + 5 q^{2} + q^{3}}{4 {(q; q)}_{3}}$	$\frac{5 (14 + 14 q + 21 q^{2} + 24 q^{3} + 25 q^{4} + 14 q^{5} + 11 q^{6} + 4 q^{7} + q^{8})}{4 {(q; q)}_{5}}$
4	(31)	$\frac{25 + 20 q + 27 q^{2} + 23 q^{3} + 10 q^{4} + 3 q^{5}}{3 {(q; q)}_{4}}$	$\frac{\frac{84 + 77 q + 125 q^{2} + 156 q^{3} + 198 q^{4} + 191 q^{5} + 163 q^{6}}{+ 124 q^{7} + 94 q^{8} + 52 q^{9} + 21 q^{10} + 10 q^{11} + q^{12}}}{{(q; q)}_{6}}$
4	(22)	$\frac{10 + 10 q + 13 q^{2} + 12 q^{3} + 5 q^{4} + 2 q^{5}}{4 {(q; q)}_{4}}$	$\frac{\frac{231 + 231 q + 370 q^{2} + 469 q^{3} + 589 q^{4} + 579 q^{5} + 489 q^{6}}{+ 378 q^{7} + 281 q^{8} + 161 q^{9} + 62 q^{10} + 30 q^{11} + 2 q^{12}}}{8 {(q; q)}_{6}}$

Table 11. Table 11: Connected quantum weighted Hurwitz numbers, H ~ G H q d ( μ ) subscript superscript ~ 𝐻 𝑑 subscript 𝐺 subscript 𝐻 𝑞 𝜇 \tilde{H}^{d}_{G_{H_{q}}}(\mu) with ℓ ( μ ) = 1 ℓ 𝜇 1 \ell(\mu)=1 or 2 2 2 , N = 2 , 3 𝑁 2 3 N=2,3 or 4 4 4

$N$	$μ$	${\tilde{H}}_{G_{H_{q}}}^{N + ℓ (μ) - 2}$	${\tilde{H}}_{G_{H_{q}}}^{N + ℓ (μ)}$
2	(2)	$\frac{1}{2 {(q; q)}_{1}}$	$\frac{1}{2 {(q; q)}_{2}}$
3	(3)	$\frac{2 + q}{3 {(q; q)}_{3}}$	$\frac{10 + 5 q + 6 q^{2} + 5 q^{3} + q^{4}}{3 {(q; q)}_{4}}$
3	(21)	$\frac{2 + q + q^{2}}{{(q; q)}_{3}}$	$\frac{10 + 5 q + 7 q^{2} + 7 q^{3} + 7 q^{4} + 2 q^{5} + 2 q^{6}}{{(q; q)}_{5}}$
4	(4)	$\frac{5 + 5 q + 5 q^{2} + q^{3}}{4 {(q; q)}_{3}}$	$\frac{5 (14 + 14 q + 21 q^{2} + 24 q^{3} + 25 q^{4} + 14 q^{5} + 11 q^{6} + 4 q^{7} + q^{8})}{4 {(q; q)}_{5}}$
4	(31)	$\frac{5 + 5 q + 7 q^{2} + 6 q^{3} + 3 q^{4} + q^{5}}{{(q; q)}_{4}}$	$\frac{\frac{70 + 70 q + 115 q^{2} + 145 q^{3} + 185 q^{4} + 180 q^{5} + 156 q^{6}}{+ 120 q^{7} + 91 q^{8} + 51 q^{9} + 21 q^{10} + 10 q^{11} + q^{12}}}{{(q; q)}_{6}}$
4	(22)	$\frac{9 + 9 q + 12 q^{2} + 11 q^{3} + 5 q^{4} + 2 q^{5}}{4 {(q; q)}_{4}}$	$\frac{\frac{114 + 114 q + 183 q^{2} + 232 q^{3} + 292 q^{4} + 287 q^{5} + 243 q^{6}}{+ 188 q^{7} + 140 q^{8} + 80 q^{9} + 31 q^{10} + 15 q^{11} + q^{12}}}{4 {(q; q)}_{6}}$

Table 12. Table 12: Nonconnected quantum weighted Hurwitz numbers H G H q d ( μ ) subscript superscript 𝐻 𝑑 subscript 𝐺 subscript 𝐻 𝑞 𝜇 H^{d}_{G_{H_{q}}}(\mu) with ℓ ( μ ) = 3 ℓ 𝜇 3 \ell(\mu)=3 , N = 4 , 5 𝑁 4 5 N=4,5 or 6 6 6

$N$	$μ$	$d$	$H_{G_{H_{q}}}^{d}$
3	(1,1,1)	$2$	$\frac{1}{2 {(q; q)}_{2}}$
4	(2,1,1)	$3$	$\frac{5}{4} \frac{2 + q + q^{2}}{{(q; q)}_{3}}$
5	(2,2,1)	$2$	$\frac{1 + q}{8 {(q; q)}_{2}}$
5	(2,2,1)	$4$	$\frac{14 + 16 q + 21 q^{2} + 20 q^{3} + 9 q^{4} + 4 q^{5}}{4 {(q; q)}_{4}}$
6	(3,2,1)	$3$	$\frac{(2 + q) (1 + q + q^{2})}{6 {(q; q)}_{3}}$
6	(3,2,1)	$5$	$\frac{107 + 172 q + 287 q^{2} + 369 q^{3} + 409 q^{4} + 319 q^{5} + 248 q^{6} + 133 q^{7} + 51 q^{8} + 11 q^{9}}{6 {(q; q)}_{5}}$

Table 13. Table 13: Connected quantum weighted Hurwitz numbers H ~ G H q d ( μ ) subscript superscript ~ 𝐻 𝑑 subscript 𝐺 subscript 𝐻 𝑞 𝜇 \tilde{H}^{d}_{G_{H_{q}}}(\mu) with ℓ ( μ ) = 3 ℓ 𝜇 3 \ell(\mu)=3 , N = 4 , 5 𝑁 4 5 N=4,5 or 6 6 6 .

$N$	$μ$	$d$	${\tilde{H}}_{G_{H_{q}}}^{d}$
3	(1,1,1)	$4$	$\frac{4 + 2 q + 3 q^{2} + 2 q^{3} + q^{4}}{3 {(q; q)}_{4}}$
4	(2,1,1)	$5$	$\frac{24 + 24 q + 39 q^{2} + 44 q^{3} + 47 q^{4} + 28 q^{5} + 23 q^{6} + 8 q^{7} + 3 q^{8}}{2 {(q; q)}_{5}}$
4	(2,1,1)	$7$	$\frac{\frac{162 + 162 q + 279 q^{2} + 371 q^{3} + 518 q^{4} + 593 q^{5} + 685 q^{6} + 598 q^{7} + 598 q^{8}}{+ 491 q^{9} + 404 q^{10} + 257 q^{11} + 182 q^{12} + 90 q^{13} + 50 q^{14} + 15 q^{15} + 5 q^{16}}}{{(q; q)}_{7}}$
5	(2,2,1)	$6$	$\frac{\frac{36 + 54 q + 99 q^{2} + 141 q^{3} + 189 q^{4} + 207 q^{5} + 204 q^{6} + 179 q^{7}}{+ 143 q^{8} + 97 q^{9} + 53 q^{10} + 28 q^{11} + 8 q^{12} + 2 q^{13}}}{{(q; q)}_{6}}$
6	(2,2,2)	$7$	$\frac{\frac{216 + 432 q + 891 q^{2} + 1485 q^{3} + 2295 q^{4} + 3099 q^{5} + 3922 q^{6} + 4377 q^{7} + 4689 q^{8} + 4567 q^{9} + 4157 q^{10}}{+ 3435 q^{11} + 2655 q^{12} + 1837 q^{13} + 1173 q^{14} + 638 q^{15} + 301 q^{16} + 117 q^{17} + 29 q^{18} + 5 q^{19}}}{6 {(q; q)}_{7}}$
6	(3,2,1)	$7$	$\frac{\frac{240 + 480 q + 1002 q^{2} + 1664 q^{3} + 2584 q^{4} + 3484 q^{5} + 4416 q^{6} + 4927 q^{7} + 5288 q^{8} + 5139 q^{9} + 4687 q^{10}}{+ 3863 q^{11} + 2990 q^{12} + 2063 q^{13} + 1319 q^{14} + 711 q^{15} + 338 q^{16} + 128 q^{17} + 32 q^{18} + 5 q^{19}}}{{(q; q)}_{7}}$

Equations182

I_{N} = h_{1} \dots h_{k}

I_{N} = h_{1} \dots h_{k}

h_{i} \in cyc (μ^{(i)}), i = 1, \dots, k .

h_{i} \in cyc (μ^{(i)}), i = 1, \dots, k .

M : π_{1} (P^{1} / {Q^{(1)}, \dots, Q^{(k)})} \to S_{N}

M : π_{1} (P^{1} / {Q^{(1)}, \dots, Q^{(k)})} \to S_{N}

H (μ^{(1)}, \dots, μ^{(k)}) = λ, ∣ λ ∣ = N \sum (h (λ))^{k - 2} i = 1 \prod k \frac{χ _{λ} ( μ ^{(i)} )}{z _{μ^{(i)}}},

H (μ^{(1)}, \dots, μ^{(k)}) = λ, ∣ λ ∣ = N \sum (h (λ))^{k - 2} i = 1 \prod k \frac{χ _{λ} ( μ ^{(i)} )}{z _{μ^{(i)}}},

z_{μ} = i = 1 \prod μ_{1} m_{i} (μ))! i^{m_{i} (μ)}

z_{μ} = i = 1 \prod μ_{1} m_{i} (μ))! i^{m_{i} (μ)}

G (z) = 1 + i = 1 \sum \infty g_{i} z^{i}

G (z) = 1 + i = 1 \sum \infty g_{i} z^{i}

G (z) = i = 1 \prod \infty (1 + c_{i} z)

G (z) = i = 1 \prod \infty (1 + c_{i} z)

\tilde{G} (z) = i = 1 \prod \infty (1 - d_{i} z)^{- 1},

\tilde{G} (z) = i = 1 \prod \infty (1 - d_{i} z)^{- 1},

\tilde{G} (z) = 1 + i = 1 \sum \infty \tilde{g}_{i} z^{i},

\tilde{G} (z) = 1 + i = 1 \sum \infty \tilde{g}_{i} z^{i},

g_{i} = e_{i} (c), \tilde{g}_{i} = h_{i} (d), i \in N^{+},

g_{i} = e_{i} (c), \tilde{g}_{i} = h_{i} (d), i \in N^{+},

G (z) = exp (z) = e^{z}

G (z) = exp (z) = e^{z}

μ^{(i)} = (2, (1)^{N - 2}), i = 1, \dots, k

μ^{(i)} = (2, (1)^{N - 2}), i = 1, \dots, k

G_{c, d} (z) := \frac{\prod _{i = 1}^{L} ( 1 + c _{i} z )}{\prod _{j = 1}^{M} ( 1 - d _{j} z )},

G_{c, d} (z) := \frac{\prod _{i = 1}^{L} ( 1 + c _{i} z )}{\prod _{j = 1}^{M} ( 1 - d _{j} z )},

g_{i} (c, d) := j = 0 \sum i e_{j} (c) h_{i - j} (d),

g_{i} (c, d) := j = 0 \sum i e_{j} (c) h_{i - j} (d),

G (z) = H_{q} (z) := i = 0 \prod \infty (1 - q^{i} z)^{- 1} =: e_{q} (z (1 - q)^{- 1}),

G (z) = H_{q} (z) := i = 0 \prod \infty (1 - q^{i} z)^{- 1} =: e_{q} (z (1 - q)^{- 1}),

g_{i} (q) := \frac{1}{( q ; q ) _{i}}, i \in N^{+},

g_{i} (q) := \frac{1}{( q ; q ) _{i}}, i \in N^{+},

(q; q)_{i} := (1 - q) \dots (1 - q^{i}) .

(q; q)_{i} := (1 - q) \dots (1 - q^{i}) .

H_{G}^{d} (μ) := k = 1 \sum d \sideset^{'} \sum_{∣ μ ^{(i)} ∣ = N , \sum _{i = 1}^{k} ℓ ^{*} ( μ ^{(i)} ) = d μ ^{(1)} , \dots μ ^{(k)} ,} W_{G} (μ^{(1)}, \dots, μ^{(k)}) H (μ^{(1)}, \dots, μ^{k)}, μ),

H_{G}^{d} (μ) := k = 1 \sum d \sideset^{'} \sum_{∣ μ ^{(i)} ∣ = N , \sum _{i = 1}^{k} ℓ ^{*} ( μ ^{(i)} ) = d μ ^{(1)} , \dots μ ^{(k)} ,} W_{G} (μ^{(1)}, \dots, μ^{(k)}) H (μ^{(1)}, \dots, μ^{k)}, μ),

ℓ^{*} (μ^{(i)}) := ∣ μ^{(i)} ∣ - ℓ (μ^{(i)}) = N - ℓ (μ^{(i)})

ℓ^{*} (μ^{(i)}) := ∣ μ^{(i)} ∣ - ℓ (μ^{(i)}) = N - ℓ (μ^{(i)})

W_{G} (μ^{(1)}, \dots, μ^{(k)})

W_{G} (μ^{(1)}, \dots, μ^{(k)})

m_{λ} (c) = \frac{1}{∣ aut ( λ ) ∣} σ \in S_{k} \sum 1 \leq b_{1} < \dots < b_{k} \sum c_{b_{σ (1)}}^{λ_{1}} \dots c_{b_{σ (k)}}^{λ_{k}},

m_{λ} (c) = \frac{1}{∣ aut ( λ ) ∣} σ \in S_{k} \sum 1 \leq b_{1} < \dots < b_{k} \sum c_{b_{σ (1)}}^{λ_{1}} \dots c_{b_{σ (k)}}^{λ_{k}},

{λ_{i}}_{i = 1, \dots k} \sim {ℓ^{*} (μ^{(i)})}_{i = 1, \dots k}, λ_{1} \geq \dots \geq λ_{k} > 0

{λ_{i}}_{i = 1, \dots k} \sim {ℓ^{*} (μ^{(i)})}_{i = 1, \dots k}, λ_{1} \geq \dots \geq λ_{k} > 0

∣ aut (λ) ∣ := i \geq 1 \prod m_{i} (λ)!

∣ aut (λ) ∣ := i \geq 1 \prod m_{i} (λ)!

\tilde{H}_{G}^{d} (μ) := k = 1 \sum d \sideset^{'} \sum_{∣ μ ^{(i)} ∣ = N , \sum _{i = 1}^{k} ℓ ^{*} ( μ ^{(i)} ) = d μ ^{(1)} , \dots μ ^{(k)} ,} W_{G} (μ^{(1)}, \dots, μ^{(k)}) \tilde{H} (μ^{(1)}, \dots, μ^{k)}, μ)

\tilde{H}_{G}^{d} (μ) := k = 1 \sum d \sideset^{'} \sum_{∣ μ ^{(i)} ∣ = N , \sum _{i = 1}^{k} ℓ ^{*} ( μ ^{(i)} ) = d μ ^{(1)} , \dots μ ^{(k)} ,} W_{G} (μ^{(1)}, \dots, μ^{(k)}) \tilde{H} (μ^{(1)}, \dots, μ^{k)}, μ)

d := i = 1 \sum k ℓ^{*} (μ^{(i)})

d := i = 1 \sum k ℓ^{*} (μ^{(i)})

χ = N + ℓ (μ) - d,

χ = N + ℓ (μ) - d,

χ = 2 - 2 g .

χ = 2 - 2 g .

H_{G}^{d} (μ) = λ : ∣ λ ∣ = d \sum m_{λ} (c) \frac{∣ aut ( λ ) ∣}{ℓ ( λ )!} {ℓ_{1}, \dots, ℓ_{k}} \in A na (λ) \sum ∣ μ ^{(j)} ∣ = ∣ μ ∣ ℓ ^{*} ( μ ^{(j))} = ℓ _{j} : \sum H (μ^{(1)}, μ^{(2)}, \dots, μ^{(k)}, μ)

H_{G}^{d} (μ) = λ : ∣ λ ∣ = d \sum m_{λ} (c) \frac{∣ aut ( λ ) ∣}{ℓ ( λ )!} {ℓ_{1}, \dots, ℓ_{k}} \in A na (λ) \sum ∣ μ ^{(j)} ∣ = ∣ μ ∣ ℓ ^{*} ( μ ^{(j))} = ℓ _{j} : \sum H (μ^{(1)}, μ^{(2)}, \dots, μ^{(k)}, μ)

F_{λ} (μ) := ∣ μ ^{(j)} ∣ = ∣ μ ∣ ℓ ^{*} ( μ ^{(j))} = ℓ _{j} \sum H (μ^{(1)}, μ^{(2)}, \dots, μ^{(k)}, μ),

F_{λ} (μ) := ∣ μ ^{(j)} ∣ = ∣ μ ∣ ℓ ^{*} ( μ ^{(j))} = ℓ _{j} \sum H (μ^{(1)}, μ^{(2)}, \dots, μ^{(k)}, μ),

H_{G}^{d} (μ) = λ, ∣ λ ∣ = d \sum m_{λ} (c) F_{λ} (μ) .

H_{G}^{d} (μ) = λ, ∣ λ ∣ = d \sum m_{λ} (c) F_{λ} (μ) .

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Full text

**Generating weighted Hurwitz numbers **

M. Bertola1,2,3**e-mail: [email protected], [email protected], J. Harnad1,2*†††e-mail: [email protected] and B. Runov1,2‡‡‡e-mail: [email protected]

*1**Department of Mathematics and Statistics, Concordia University

1455 de Maisonneuve Blvd. W. Montreal, QC H3G 1M8 Canada

2Centre de recherches mathématiques, Université de Montréal,

C. P. 6128, succ. centre ville, Montréal, QC H3C 3J7 Canada

3SISSA/ISAS, via Bonomea 265, Trieste, Italy *

Abstract

Multicurrent correlators associated to KP $\tau$ -functions of hypergeometric type are used as generating functions for weighted Hurwitz numbers. These are expressed as formal Taylor series and used to compute generic, simple, rational and quantum weighted single Hurwitz numbers.

1 Introduction

It is well-known that KP $\tau$ -functions of hypergeometric type may serve as combinatorial generating functions for weighted Hurwitz numbers [28, 30, 6, 15, 21, 16, 18]. An efficient way of computing the latter is to make use of the associated multicurrent correlators [4, 5]. This method is implemented in the following, without recourse to matrix integral representations [1, 2, 9] or topological recursion [10, 8, 25, 3, 4, 5].

Section 2 reviews the use of KP and $2D$ -Toda $\tau$ -functions as generating functions for weighted Hurwitz numbers. Their definition is recalled, with the special cases of simple Hurwitz numbers, rationally weighted Hurwitz numbers (2.13) and quantum Hurwitz numbers as illustrative examples. The starting point, developed in refs. [15, 21, 16], is Theorem 2.1, which, generalizing the earlier results of Okounkov [28] and Pandharipande [30] for simple Hurwitz numbers, states that KP $\tau$ -functions [31, 33] of special hypergeometric type [29] are generating functions for weighted Hurwitz numbers corresponding to arbitrary weight generating functions $G(z)$ ,

The new element in the present work is the observation that a more effective tool is provided by the multicurrent correlators $W_{g}(x_{1},\dots,x_{n})$ introduced in [4, 5] in the context of the topological recursion (TR) approach. As explained in Proposition 2.2, these provide generating functions for single weighted Hurwitz numbers $H^{d}_{G}(\mu)$ , grouped according to the length $n=\ell(\mu)$ of the partition $\mu$ giving the ramification profile over a selected branch point. Making use of the polynomial expressions for $W_{g}(x_{1},\dots,x_{n})$ in terms of pair correlators $K(x_{i},x_{j})$ given in Proposition 2.3, together with the explicit expressions (2.65), (2.63) of the latter as weighted polynomials in the Taylor coefficients $\{g_{i}\}_{i\in\mathbf{N}}$ of $G(z)$ , the multicurrent correlators are directly expressible as series with coefficients that are weighted polynomials in the $g_{i}$ ’s, without the need to solve any recursion relations.

The main result is Theorem 3.1, which provides explicit formulae for weighted Hurwitz numbers as graded polynomials in these Taylor coefficients. Using the multicurrent correlators as generating functions, the weighted single Hurwitz numbers $H^{d}_{G}(\mu)$ are determined for ramification profile $\mu$ of length $n:=\ell(\mu)$ equal to $1,2$ or $3$ . This is first done in the generic case, for arbitrary weight generating function $G(z)$ , with explicit formulae for $H^{d}_{G}(\mu)$ for small values of $N$ , $d$ and $\ell(\mu)$ given in Appendix B, Tables 4 - 7. These are specialized to various particular cases in Tables 8-13: the exponential case $G(z)=e^{z}$ , which determines simple Hurwitz numbers; polynomial and rational weight generating functions, which include, in particular, strictly and weakly monotone Hurwitz numbers as well as weighted enumeration of Belyi curves [1, 21, 34, 14, 16], and, finally, quantum Hurwitz numbers [19, 16].

2 Generating functions for weighted Hurwitz numbers

2.1 Pure and weighted Hurwitz numbers

We recall the definition of pure Hurwitz numbers [22, 11, 23, 12, 32, 26] and weighted Hurwitz numbers [15, 21, 18, 16].

Definition 2.1 (Combinatorial).

For a set of $k$ partitions $\{\mu^{(i)}\}_{i=1,\dots,k}$ of $N\in\mathbf{N}^{+}$ , the pure Hurwitz number $H(\mu^{(1)},\dots,\mu^{(k)})$ is ${1\over N!}$ times the number of distinct ways that the identity element $\mathbf{I}_{N}\in S_{N}$ in the symmetric group on $N$ elements can be expressed as a product

[TABLE]

of $k$ elements $\{h_{i}\in S_{N}\}_{i=1,\dots,k}$ , such that for each $i$ , $h_{i}$ belongs to the conjugacy class $\operatorname{cyc}(\mu^{(i)})$ whose cycle lengths are equal to the parts of $\mu^{(i)}$ :

[TABLE]

An equivalent definition involves the enumeration of branched coverings of the Riemann sphere.

Definition 2.2 (Geometric).

For a set of partitions $\{\mu^{(i)}\}_{i=1,\dots,k}$ of weight $|\mu^{(i)}|=N$ , the pure Hurwitz number $H(\mu^{(1)},\dots,\mu^{(k)})$ is defined geometrically [22, 23] as the number of inequivalent $N$ -fold branched coverings $\mathcal{C}\rightarrow\mathbf{P}^{1}$ of the Riemann sphere with $k$ branch points whose ramification profiles are given by the partitions $\{\mu^{(1)},\dots,\mu^{(k)}\}$ , normalized by the inverse $1/|\operatorname{aut}(\mathcal{C})|$ of the order of the automorphism group of the covering. The connected pure Hurwitz number $\widetilde{H}(\mu^{(1)},\dots,\mu^{(k)})$ is defined in the same way, with only connected coverings included in the enumeration.

The equivalence of the two follows from the monodromy homomorphism

[TABLE]

from the fundamental group of the Riemann sphere punctured at the branch points into $S_{N}$ obtained by lifting closed loops from the base to the covering.

The Frobenius-Schur formula determines Hurwitz numbers $H(\mu^{(1)},\dots,\mu^{(k)})$ in terms of irreducible characters of the symmetric group $S_{N}$

[TABLE]

where $\chi_{\lambda}(\mu)$ is the character of the irreducible representation corresponding to the partition $\lambda$ of $N$ evaluated on the conjugacy class $\operatorname{cyc}(\mu)\subset S_{N}$ with cycle structure given by the partition $\mu$ , $h(\lambda)$ is the product of hook lengths of $\lambda$ and

[TABLE]

is the order of the stability subgroup of any element of the conjugacy class $\operatorname{cyc}(\mu)$ , where $m_{i}(\mu)$ is the number of parts of $\mu$ equal to $i$ . Computing $H(\mu^{(1)},\dots,\mu^{(k)})$ using this formula requires the character table for $S_{N}$ , which becomes increasingly computationally complex [7] for rising $N$ and $k$ .

To define weighted Hurwitz numbers, we introduce a weight generating function $G(z)$ , either as an infinite sum

[TABLE]

or an infinite product

[TABLE]

or, in dual form

[TABLE]

which may also be developed as an infinite sum,

[TABLE]

either formally, or under suitable convergence conditions imposed upon the parameters $\{c_{i}\}$ , $\{d_{i}\}_{i\in\mathbf{N}^{+}}$ or the Taylor coefficients $\{g_{i}\}_{i\in\mathbf{N}^{+}}$ , $\{\tilde{g_{i}}\}_{i\in\mathbf{N}^{+}}$ . The independent parameters determining the weighting may be viewed as any of these. Cases (2.7) and (2.8) may be viewed as generating functions for elementary $\{e_{i}\}_{i\in\mathbf{N}}$ and complete symmetric functions $\{h_{i}\}_{i\in\mathbf{N}}$ ,, respectively, giving

[TABLE]

where ${\bf c}=(c_{1},c_{2},\dots)$ , ${\bf d}=(d_{1},d_{2},\dots)$ .

The particular case of exponential weight generating function

[TABLE]

is of special interest, since it corresponds to a Dirac measure on the space of $k$ -tuples of partitions $\{\mu^{(1)},\dots,\mu^{(k)}\}$ , supported uniformly on $k$ -tuples of $2$ -cycles

[TABLE]

which, from the viewpoint of enumeration of branched covers of the Riemann sphere, corresponds to simple branching [28, 30]. A number of other special cases of particular interest will be considered in the following.

The first of these is the family of rational weight generating functions

[TABLE]

for an arbitrary set of $L+M$ complex numbers $\{c_{1},\dots,c_{L},d_{1},\dots,d_{M}\}$ . The Taylor series coefficients for this case are

[TABLE]

where ${\bf c}=(c_{1},\dots,c_{L})$ , ${\bf d}=(d_{1},\dots,d_{M})$ . Particular cases include: linear or quadratic polynomials $(L,M)=(1,0)$ or $(2,0)$ , which correspond to two or three branch points (Belyi curves) [34, 25] and the case $(L,M)=(0,1)$ which, in the double weighted Hurwitz number case (which actually corresponds to a 2D Toda $\tau$ -function), is equivalent to the Harish-Chandra-Itzykson-Zuber matrix integral [17, 24] as generating function [14].

Also of particular interest is the case of the $q$ -exponential function

[TABLE]

which is the weight generating function for (one version of) quantum Hurwitz numbers [16, 19], whose Taylor coefficients are given by

[TABLE]

where $(q\,;q)_{i}$ is the $q$ -Pochhammer symbol evaluated at $(q,q)$ :

[TABLE]

Definition 2.3 (Weighted Hurwitz numbers).

For the case (2.7), choosing a positive integer $d$ and a fixed partition $\mu$ of weight $|\mu|=N$ , the weighted (single) Hurwitz number $H^{d}_{G}(\mu)$ is defined as the weighted sum over all $k$ -tuples $(\mu^{(1)},\dots,\mu^{(k)})$

[TABLE]

where $\sideset{}{{}^{\prime}}{\sum}$ denotes a sum over all $k$ -tuples of partitions $\{\mu^{(1)},\dots,\mu^{(k)}\}$ of $N$ other than the cycle type of the identity element $(1^{N})$ ,

[TABLE]

is the colength of the partition $\mu^{(i)}$ , and the weight factor is defined to be

[TABLE]

Here $m_{\lambda}({\bf c})$ is the monomial symmetric function of the parameters ${\bf c}:=(c_{1},c_{2},\dots)$

[TABLE]

indexed by the partition $\lambda$ of weight $|\lambda|=d$ and length $\ell(\lambda)=k$ , whose parts $\{\lambda_{i}\}$ are equal to the colengths $\{\ell^{*}(\mu^{(i)})\}$ (expressed in weakly decreasing order)

[TABLE]

and

[TABLE]

where $m_{i}(\lambda)$ is the number of parts of $\lambda$ equal to $i$ . We similarly denote by

[TABLE]

the connected weighted Hurwitz numbers corresponding to the weight generating function $G(z)$ .

Recall that the sum

[TABLE]

of the colengths of the ramification profiles $\{\mu^{(i)}\}$ of the weighted branch points determines the Euler characteristic $\chi$ of the covering curve through the Riemann-Hurwitz formula

[TABLE]

which for connected coverings $\mathcal{C}\rightarrow\mathbf{P}$ is related, as usual, to the genus $g$ by

[TABLE]

Remark 2.1.

Note that we can rearrange the sum (2.18) as follows;

[TABLE]

where $k=\ell(\lambda)$ . Here $Ana(\lambda)=\{(\ell_{1},\dots,\ell_{k})\}$ denotes the distinct rearrangements (or anagrams) of the numbers $\lambda_{1},\dots,\lambda_{k}$ . But there are precisely $\frac{|\ell(\lambda)|!}{|\operatorname{aut}(\lambda)|}$ such anagrams, so if we denote

[TABLE]

it is clear that $F_{\lambda}(\mu)$ is invarant under permutations of the $\ell_{j}$ ’s,. The counting over anagrams therfore exactly cancels the factor ${|\operatorname{aut}|\over k!}$ , and (2.18) is equivalent to

[TABLE]

The same is true, of course, for the connected version $\widetilde{H}^{d}_{G}(\mu)$ , with $F_{\lambda}(\mu)$ replaced by

[TABLE]

For the dual generating function $\tilde{G}(z)$ , the weighted Hurwitz numbers are defined as

[TABLE]

where

[TABLE]

Here $f_{\lambda}({\bf d})$ is the “forgotten” symmetric function [27] of the parameters ${\bf d}:=(d_{1},d_{2},\dots)$

[TABLE]

indexed again by the partition $\lambda$ of weight $|\lambda|=d$ and length $\ell(\lambda)=k$ , whose parts $\{\lambda_{i}\}$ are equal to the colengths $\{\ell^{*}(\mu^{(i)})\}$ , as in (2.23).

For rational weight generating functions $G_{{\bf c},{\bf d}}(z)$ as defined in (2.13), the weighted Hurwitz numbers are

[TABLE]

where the weight factor is defined to be

[TABLE]

Finally, for the case of the generating function $H_{q}(z)$ defined in (2.15), the quantum weighted Hurwitz number is

[TABLE]

where the weight factor is

[TABLE]

In order to compute the weighted Hurwitz numbers for any given weight generating function $G(z)$ , it is necessary not only to know the pure Hurwitz numbers $H(\mu^{(1)},\dots,\mu^{(k)},\mu)$ or $\tilde{H}(\mu^{(1)},\dots,\mu^{(k)},\mu)$ entering in the sums (2.18), (2.25), but also to express the weights (2.21) or (2.35) as weighted homogeneneous polynomials in the Taylor coefficients $\{g_{i}\}_{i\in\mathbf{N}^{+}}$ or $\{\tilde{g}_{i}\}_{i\in\mathbf{N}^{+}}$ of $G$ or $\tilde{G}$ , respectively. This requires the transition matrices [27] relating the various bases $\{m_{\lambda}\}$ , $\{f_{\lambda}\}$ $\{e_{\lambda}\}$ , $\{h_{\lambda}\}$ of the ring of symmetric functions which, in turn, require the Kostka matrices $K_{\lambda\mu}$ which are the transition matrices between the Schur function bases $\{s_{\lambda}\}$ and the monomial symmetric function bases $\{m_{\lambda}\}$ . The latter is readily computable for any given weight $d$ , but no known formula exists. Therefore, the computation of the weighted Hurwitz numbers $H^{d}_{G}(\mu)$ and their connected version $\tilde{H}^{d}_{G}(\mu)$ directly from their definitions is a lengthy and complex task. As will be shown in the following, however, this may all be circumvented through the use of generating functions.

Remark 2.2.

Unlike the direct computation, based on the Frobenius-Schur formula (2.4), the computation of weighted Hurwitz numbers $H^{d}_{G}(\mu)$ does not require any knowledge of the $S_{N}$ group characters or the Kostka matrices, and this is part of the reason for the more rapid computational algorithm the generating function approach provides.

However, it should be noted that knowing the weighted Hurwitz numbers $H^{d}_{G}(\mu)$ , even for the most general choice of weight generating functions $G(z)$ , does not provide sufficient information to uniquely retrieve the individual pure Hurwitz numbers $\{H(\mu^{(1)},\dots,\mu^{(k)})\}$ . This is because the only information retained in the definition of the weighted average (2.18) is the partition $\lambda$ of length $k=\ell(\mu)$ and weight $|\lambda|=d$ whose parts are the colengths $\{\ell(\mu^{i)}\}_{i=1,\dots,k}$ .

Thus, knowing the weighted Hurwitz numbers allows us only to reconstruct the values of the pure Hurwitz numbers summed, as in (2.30), over all $k$ -tuples of partitions $\{\mu^{(1)},\dots,\mu^{(k)}\}$ having the same sequence of colengths $\{\ell^{*}(\mu^{(1)}),\dots,\ell^{*}(\mu^{(k)})\}$ . These can be determined from the weighted ones only in the case of simple Hurwitz numbers, since these are the only ones in the class where the colengths are $(1,1,\dots,1)$ . In general, there is no other combination of partitions for which the same holds true.

2.2 Hypergeometric $\tau$ -functions as generating functions of weighted Hurwitz numbers

We recall the definition of KP $\tau$ -functions of hypergeometric type [29], which serve as generating functions for weighted Hurwitz numbers [16, 21, 18]. For any weight generating function $G(z)$ of the above type, and nonzero parameter $\beta$ , we define two doubly infinite sequences of numbers $\{r_{i}^{(G,\beta)},\rho_{i}\}_{i\in\mathbf{Z}}$ , labeled by integers

[TABLE]

related by

[TABLE]

where $\beta$ is viewed as a small parameter for which $G(i\beta)$ does not vanish for any integer $i\in\mathbf{Z}$ . (It is possible to extend this by requiring that, if $G(i\beta)=0$ for some smallest positive integer $i\in\mathbf{N}^{+}$ , then $\rho_{j}:=0$ for all $j\geq i$ .

For the exponential generating function $G(z)=e^{z}$ , we have

[TABLE]

For rational weight generating functions $G_{{\bf c},{\bf d}}(z)$ as defined in (2.13), the parameters $\{\rho_{i}\}_{i\in\mathbf{Z}}$ , and coefficients $\{g_{i}\}_{i\in\mathbf{N}}$ become

[TABLE]

For the quantum exponential generating function $H_{q}(z)$ defined in (2.15), the parameters $\{\rho_{i}\}_{i\in\mathbf{Z}}$ and coefficients $\{g_{i}\}_{i\in\mathbf{N}}$ become

[TABLE]

For each partition $\lambda$ of $N$ , we define the associated content product coefficient

[TABLE]

The KP $\tau$ -function of hypergeometric type associated to these parameters is defined as the Schur function series [21, 15, 16]:

[TABLE]

where $h(\lambda)$ is the product of the hook lengths of the partition $\lambda$ and ${\bf t}=(t_{1},t_{2}\dots)$ is the infinite sequence of KP flow parameters, which may be equated to the sequence of normalized power sums $(p_{1},{1\over 2}p_{2},\dots)$

[TABLE]

in a finite or infinite set of auxiliary variables $(x_{1},x_{2},\dots)$ .

Using the Schur character formula [13, 27]

[TABLE]

where $\chi_{\lambda}(\mu)$ is the irreducible character determined by $\lambda$ evaluated on the conjugacy class $\operatorname{cyc}(\mu)$ , $z_{\mu}$ is the order of the stabilizer of the elements of this conjugacy class, as given in (2.5), and

[TABLE]

is the power sum symmetric function corresponding to partition $\mu$ , we may re-express the Schur function series (2.54) as an expansion in the basis $\{p_{\mu}\}$ of power sum symmetric functions. As shown in [15, 18, 21, 16], this gives the following:

Theorem 2.1.

The $\tau$ -function $\tau^{(G,\beta)}({\bf t})$ may equivalently be expressed as

[TABLE]

It is thus a generating function for the weighted Hurwitz numbers $H^{d}_{G}(\mu)$ . This holds similarly for the dual weight generating function $\tilde{G}(z)$ and the weighted Hurwitz numbers $H^{d}_{\tilde{G}}(\mu)$

[TABLE]

Particular cases of weighted Hurwitz numbers for which hypergeometric $\tau$ -functions correspond to the weight generating functions $G(z)$ defined above include: simple Hurwitz numbers [28, 30] (both single and double), with weight generating function $G(z)=e^{z}$ ; weakly monotone Hurwitz numbers [14] (or, equivalently, signed Hurwitz numbers [21, 16]), with weight generating function $G(z)={1\over 1-z}$ ; strongly monotone Hurwitz numbers [15, 16], with weight generating function $G(z)=1+z$ (or, equivalently, weighted Hurwitz numbers for Belyi curves and dessins d’enfant [1, 25, 34]); polynomially weighted Hurwitz numbers [6, 21, 3, 4, 5]; quantum Hurwitz numbers [18, 16, 19] and multispecies Hurwitz numbers [20].

2.3 The pair correlator $K^{G}(x,y)$

An essential rôle in the computation of weighted Hurwitz numbers is played by the pair correlator associated to the $\tau$ -function, defined as

[TABLE]

where $[x]$ denotes the infinite sequence

[TABLE]

This has the following series expansion [3, 5]

[TABLE]

Defining

[TABLE]

we have

[TABLE]

and the nonsingular part of $K^{G}(x,y)$ is

[TABLE]

Taking the (formal) Taylor series expansion

[TABLE]

the coefficents $\rho_{ab}^{d}({\bf g})$ are weighted degree $d$ homogeneous polynomials in the Taylor coefficents $\{g_{i}\}_{i\in\mathbf{N}^{+}}:={\bf g}$ of $G(z)$ . It follows from (2.64) that they satisfy the symmetry conditions

[TABLE]

Their values for small values of $a,b,d$ are displayed in Tables 1- 3 of Appendix A.

In particular, for the case of the rational weight generating function $G_{{\bf c},{\bf d}}(z)$ , we have

[TABLE]

and for the quantum exponential weight generating function $G_{H_{q}}(z)$ ,

[TABLE]

2.4 The multicurrent correlator $W_{n}(x_{1},\dots,x_{n})$ as generating function for weighted Hurwitz numbers

Following [3, 4, 5], define the derivations:

Definition 2.4.

For any parameter $x$

[TABLE]

In terms of these, we introduce the following correlators

[TABLE]

which are related by

[TABLE]

As shown in [4], $W_{n}(x_{1},\dots,x_{n})$ has a fermionic representation as a multicurrent correlator. Moreover, we have the following result from refs. [3, 5], which shows that $F_{n}(x_{1},\dots,x_{n})$ and $\tilde{F}_{n}(x_{1},\dots,x_{n})$ may also serve as generating functions for weighted single Hurwitz numbers $H^{d}_{G}(\mu)$ and $\tilde{H}^{d}_{G}(\mu)$ , respectively, with ramification profile $\mu$ at the single marked branch point of length $\ell(\mu)=n$ .

Proposition 2.2 ([3, 5]).

[TABLE]

where

[TABLE]

and $m_{\mu}(x_{1},\dots,x_{n})$ is the monomial symmetric polynomial in the indeterminates $(x_{1},\dots,x_{n})$ .

Finally, we quote the following further result from [3, 5], which expresses the connected multicurrent correlators in terms of pair correlators.

Proposition 2.3.

[TABLE]

and for $n\geq 3$

[TABLE]

where the last sum is over all permutations in $S_{n}$ consisting of a single $n$ -cycle.

The corresponding nonconnected quantities are obtained from the cumulant relations [5]:

[TABLE]

or, more generally,or $n\geq 3$ ,

[TABLE]

with identical relations holding between the $\tilde{F}_{n}$ ’s and $F_{n}$ ’s .

3 Generating function computation of $\tilde{H}^{d}_{G}(\mu)$

Using the series expansion (2.69) for the pair correlator $K^{G}(x,y)$ , and expressing the connected multicurrent correlators $\tilde{W}_{n}(x_{1},\dots,x_{n})$ in terms of these via eqs. (2.81), (2.82), (2.83), we can compute all weighted connected or nonconnected single Hurwitz numbers $\tilde{H}^{d}_{G}(\mu)$ , $H^{d}_{G}(\mu)$ for coverings of any degree $N$ , Euler characteristic $\chi$ and unweighted partition $\mu$ of $N$ , as homogeneous polynomials of weighted degree $d$ in the coefficients $\{g_{i}\}_{i\in\mathbf{N}^{+}}$ of the series expansion of the weight generating function $G(z)$ .

In particular, for $\ell(\mu)\leq 3$ , we have:

Theorem 3.1.

[Weighted Hurwitz numbers for $\ell(\mu)=1,2$ or $3$ ]

The following formulae determine the connected weighted Hurwitz numbers $\tilde{H}^{d}_{G}(\mu)$ for $\mu$ of length $\ell(\mu)\leq 3$ .

$\ell((\mu_{1}))=1$ :

[TABLE]

$\ell((\mu_{1}.\mu_{2}))=2$ :

[TABLE]

where the linear part $\tilde{H}^{(d,2)}_{(G,1)}((\mu_{1},\mu_{2}))$ is given by

[TABLE]

and the quadratic part $\tilde{H}^{(d,2)}_{G,2}((\mu_{1},\mu_{2}))$ by

[TABLE]

$\ell((\mu_{1},\mu_{2},\mu_{3}))=3$ :

[TABLE]

where the linear part $\tilde{H}^{(d,3)}_{(G,1))}((\mu_{1},\mu_{2},\mu_{3}))$ is given by

[TABLE]

the quadratic part by

[TABLE]

where

[TABLE]

with $\operatorname{aut}(\mu_{1},\mu_{2},\mu_{3})$ for a composition $(\mu_{1},\mu_{2},\mu_{3})$ (i.e., an unordered set of of positive integers) defined in the same way (2.24) as for a partition, and the cubic part by

[TABLE]

Proof.

Compare the expansion (2.79) with the expressions in Proposition 2.3 for $\tilde{W}_{1}$ , $\tilde{W}_{2}$ , $\tilde{W}_{3}$ as polynomials in the pair correlators. Substitute the series expansion (2.65) for the $K(x_{i},x_{j})$ ’s and the series expansion for $\rho_{ab}^{d}$ in $\beta$ following from eq. (2.63), and use relation (2.76). Equating like powers of $\beta$ gives formulae (3.1) - (LABEL:Hd3_G3) for the weighted connected Hurwitz numbers $\tilde{H}^{d}_{G}(\mu)$ as polynomials in the Taylor coefficients $\{g_{i}\}$ of $G(z)$ . ∎

Remark 3.1.

The nonconnected versions of the weighted Hurwitz numbers are obtained by applying the cumulant relations (2.88), which give

[TABLE]

Remark 3.2.

Similar formulae, consisting of degree $n$ polynomials in the coefficients $\rho^{d}_{ab}({\bf g})$ , may be obtained for all values of $n=\ell(\mu)$ , determining thereby the corresponding weighted Hurwitz numbers $\tilde{H}^{d}_{G}(\mu)$ , $H^{d}_{G}(\mu)$ from the series expansions of $\tilde{W}_{n}(x_{1},\dots,x_{n})$ for all $n$ .

Remark 3.3.

Inserting the expressions for the coefficients $\rho_{ab}^{d}({\bf g})$ , following from (2.63), (2.66), as graded homogeneous polynomials in the Taylor coefficients $\{g_{i}\}_{i\in\mathbf{N}}$ of the generating function $G(z)$ , gives $\tilde{H}^{d}_{G}(\mu)$ as a weight $d$ graded homogeneous polynomial in the $g_{i}$ ’s.

Tables 4 - 7 of Appendix B display $H^{d}_{G}(\mu)$ and $\tilde{H}^{d}_{G}(\mu)$ for generic values of the Taylor coefficients $\{g_{i}\}_{i\in\mathbf{N}}$ , $n=1,2,3$ , and small values of $N$ and $d$ . Their evaluation for the special cases of: exponential, rational and quantum exponential weight generating functions $e^{z}$ , $G_{({\bf c,}{\bf d})}(z)$ and $H_{q}(z)$ , respectively, are displayed in Tables 8 - 13.

Appendices

Appendix A Table of coefficents $\rho_{ab}^{d}({\bf g})$

Appendix B Evaluation of weighted Hurwitz numbers

B.1 Generic case

Further examples, with $\ell(\mu)=3$ and $N=3,4,5$ or $6$ , are given in Tables 6 and 7.

B.2 Exponential case (simple Hurwitz numbers)

Choosing the exponential function $G(z)=e^{z}$ , with Taylor coefficients

[TABLE]

as weight generating function corresponds [16, 18] to a Dirac measure for the weighted Hurwitz numbers supported uniformly on simple branch points, with ramification profiles that are all $2$ -cycles:

[TABLE]

the case considered in [28, 30]. Note that for this case $d=k$ and

[TABLE]

Tables 8-9 give the evaluations of $\tilde{H}^{d}_{\exp}(\mu)$ and $H^{d}_{\exp}(\mu)$ for this case.

B.3 Rational case

The Taylor coefficients $\{g_{i}({\bf c},{\bf d})\}_{i\in\mathbf{N}^{+}}$ for rational weight generating functions $G=G_{{\bf c},{\bf d}}$ , are given in eq. (2.14) in terms of the elementary and complete symmetric polynomials $\{e_{i}({\bf c})\}$ and $\{h_{i}({\bf d})\}$ in the parameters ${\bf c}=(c_{1},\dots,c_{L})$ and ${\bf d}=(d_{1},\dots,d_{M})$ . Substituting these in Tables 4 - 7, we obtain the specialization to rationally weighted Hurwitz numbers.

In particular, the case of weakly monotone (or signed) single Hurwitz numbers ( $L=0$ , $M=1$ , $d_{1}=1$ ) is obtained by substituting the values

[TABLE]

in Tables 4 - 7. The case $L=2$ , $M=0$ , gives the weighted enumeration of Belyi curves, with three branch points, two weighted ones, with ramification profiles $(\mu^{(1)},\mu^{(2)})$ and one unweighted one, with profile $\mu$ . It is obtained by substituting

[TABLE]

in Tables 4 - 7.

B.4 Quantum case

For the case of the quantum weight generating function $G=H_{q}$ , the Taylor coefficients are given by eq. (2.16). Substituting these in Tables 4 - 7, we obtain the specialization to quantum weighted Hurwitz numbers given in Tables 10-13.

Acknowledgements. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec, Nature et technologies (FRQNT).

Bibliography34

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] J. Ambjørn and L. Chekhov, “The matrix model for dessins d’enfants”, Ann. Inst. Henri Poincaré, Comb. Phys. Interact. 1 , 337-361 (2014).
2[2] J. Ambjorn and L. Chekhov, “A matrix model for hypergeometric Hurwitz numbers”, Theor. Math. Phys.’ 181 , 1486-1498 (2014).
3[3] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad, “Weighted Hurwitz numbers and topological recursion: an overview”, J. Math. Phys. 59 , 081102: 1-20 (2018).
4[4] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad, “Fermionic approach to weighted Hurwitz numbers and topological recursion,” Commun. Math. Phys. 360 no.2, 777 (2018).
5[5] A. Alexandrov, G. Chapuy, B. Eynard and J. Harnad, “Weighted Hurwitz numbers and topological recursion” ar Xiv:1 806.09738 (to appear in Commun. Math. Phys. (2020)).
6[6] A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, “ On KP-integrable Hurwitz functions”, JHEP 1411 080 (2014).
7[7] Dan Bernstein, “The computational complexity of rules for the character table of S n subscript 𝑆 𝑛 S_{n} ”, J. of Symb. Comp. 37 , 727-748 (2014).
8[8] G. Borot, B. Eynard, M. Mulase and B. Safnuk, “A matrix model for simple Hurwitz numbers, and topological recursion”, J. Geom. Phys. 61 , 522–540 (2011).

TL;DR

Contribution

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Abstract

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Abstract

1 Introduction

2 Generating functions for weighted Hurwitz numbers

2.1 Pure and weighted Hurwitz numbers

Definition 2.1** (Combinatorial).**

Definition 2.2** (Geometric).**

Definition 2.3** (Weighted Hurwitz numbers).**

Remark 2.1**.**

Remark 2.2**.**

2.2 Hypergeometric τ\tauτ-functions as generating functions of weighted Hurwitz numbers

Theorem 2.1**.**

2.3 The pair correlator KG(x,y)K^{G}(x,y)KG(x,y)

2.4 The multicurrent correlator Wn(x1,…,xn)W_{n}(x_{1},\dots,x_{n})Wn​(x1​,…,xn​) as generating function for weighted Hurwitz numbers

Definition 2.4**.**

Proposition 2.2** ([3, 5]).**

Proposition 2.3**.**

3 Generating function computation of H~Gd(μ)\tilde{H}^{d}_{G}(\mu)H~Gd​(μ)

Theorem 3.1**.**

Proof.

Remark 3.1**.**

Remark 3.2**.**

Remark 3.3**.**

Appendices

Appendix A Table of coefficents ρabd(g)\rho_{ab}^{d}({\bf g})ρabd​(g)

Appendix B Evaluation of weighted Hurwitz numbers

B.1 Generic case

B.2 Exponential case (simple Hurwitz numbers)

B.3 Rational case

B.4 Quantum case

Definition 2.1 (Combinatorial).

Definition 2.2 (Geometric).

Definition 2.3 (Weighted Hurwitz numbers).

Remark 2.1.

Remark 2.2.

2.2 Hypergeometric $\tau$ -functions as generating functions of weighted Hurwitz numbers

Theorem 2.1.

2.3 The pair correlator $K^{G}(x,y)$

2.4 The multicurrent correlator $W_{n}(x_{1},\dots,x_{n})$ as generating function for weighted Hurwitz numbers

Definition 2.4.

Proposition 2.2 ([3, 5]).

Proposition 2.3.

3 Generating function computation of $\tilde{H}^{d}_{G}(\mu)$

Theorem 3.1.

Remark 3.1.

Remark 3.2.

Remark 3.3.

Appendix A Table of coefficents $\rho_{ab}^{d}({\bf g})$