[Paper Review] Semiclassical asymptotics of quantum weighted Hurwitz numbers

J. Harnad, Janosch Ortmann|arXiv (Cornell University)|Oct 20, 2016

Advanced Mathematical Identities15 references3 citations

TL;DR

This paper derives the semiclassical asymptotics of quantum weighted Hurwitz numbers using KP-Toda τ-functions as generating functions, showing that the classical limit ℏ→0 recovers the simple Hurwitz numbers of Pandharipande and Okounkov. The leading-order partition function and weights are computed with explicit semiclassical corrections, establishing a precise connection between quantum and classical Hurwitz theory via scaling limits and symmetric function expansions.

ABSTRACT

This work concerns the semiclassical asymptotics of quantum weighted double Hurwitz numbers. We compute the leading term of the partition function for three versions of the quantum weighted Hurwitz numbers, as well as lower order semiclassical corrections. The classical limit $\hbar o 0$ is shown to reproduce the simple Hurwitz numbers studied by Pandharipande and Okounkov. The KP-Toda $τ$-function serving as generating function for the quantum Hurwitz numbers is shown to converge in the classical limit to the generating function of Pandharipande and Okounkov and, with suitable scaling, so do the partition function, the weights and expectations of Hurwitz numbers.

Motivation & Objective

• To derive the semiclassical asymptotic expansion of quantum weighted double Hurwitz numbers in the limit ℏ→0.
• To establish the classical limit of the quantum generating function as the KP-Toda τ-function corresponding to simple Hurwitz numbers.
• To compute the leading-order partition function and lower-order semiclassical corrections for three versions of quantum weighted Hurwitz numbers.
• To show that the quantum weights and expectations converge to their classical counterparts under appropriate scaling.
• To connect quantum Hurwitz theory to classical Hurwitz theory through symmetric function theory and asymptotic analysis of τ-functions.

Proposed method

• Uses KP-Toda τ-functions as generating functions for quantum weighted Hurwitz numbers, with weights derived from parametric families of symmetric functions.
• Applies a semiclassical expansion in terms of ℏ = ε, with ε→0 corresponding to the classical limit, using the formal variable q = e^{-ε}.
• Employs Frobenius-Schur character formula and monomial symmetric functions to express Hurwitz numbers in terms of irreducible characters and symmetric functions.
• Derives asymptotic expansions of the weight functions W_{E'(q)}(μ^{(1)},…,μ^{(k)}) using the inverse of products of partial sums of variables, with order analysis based on partition colengths.
• Performs explicit computation of leading and next-to-leading order terms in the partition function by analyzing contributions from partitions of colength d and d−1.
• Uses symmetric group summations and combinatorial identities (e.g., ∑_{σ∈S_{d-1}} r / d! = (d−1)/2) to evaluate the coefficients in the asymptotic expansion.

Experimental results

Research questions

• RQ1 What is the leading-order semiclassical asymptotic behavior of the partition function for quantum weighted Hurwitz numbers?
• RQ2 How do the quantum weights and expectations of Hurwitz numbers behave in the classical limit ℏ→0?
• RQ3 Can the KP-Toda τ-function generating function for quantum Hurwitz numbers be shown to reduce to the classical generating function of Pandharipande and Okounkov?
• RQ4 What are the explicit forms of the semiclassical corrections (O(ε)) to the partition function and weights?
• RQ5 How do the symmetric function expansions of the weights and Hurwitz numbers behave under the semiclassical scaling q = e^{-ε}?

Key findings

• The leading term of the partition function is Z_{e^{-ε}}^{(d)} = ε^{-d}/d! + O(ε^{1-d}), with the first correction term involving (3−d)/(4(d−1)!).
• The classical limit ℏ→0 of the quantum Hurwitz generating function reproduces the KP-Toda τ-function associated with simple Hurwitz numbers.
• The weights f(μ^{(1)},…,μ^{(k)}) contribute to the asymptotic expansion only through partitions of colength d and d−1, corresponding to λ = (1^d) and λ = (2,1^{d−1}).
• The coefficient of the ε^{-d+1} term in the partition function is (1/(d−1)!) − d(d+1)/(4d!) = (3−d)/(4(d−1)!), derived via symmetric group summation.
• The asymptotic expansion of the weight function W_{E'(q)}(μ^{(1)},…,μ^{(k)}) is dominated by contributions from partitions with ℓ(λ) = d and ℓ(λ) = d−1, with explicit evaluation using ∑_{σ} (prod ∑x_{σ(i)})^{-1} over S_{d−1}.
• For the special case f ≡ 1, the partition function is shown to be Z_{e^{-ε}}^{(d)} = ε^{-d}/d! + ε^{1−d}(3−d)/(4(d−1)!) + O(ε^{2−d}), confirming the semiclassical correction structure.