[Paper Review] Rationality of capped descendent vertex in $K$-theory

Andrey Smirnov|arXiv (Cornell University)|Dec 4, 2016

Algebraic structures and combinatorial models10 citations

TL;DR

This paper establishes the rationality of the $K$-theoretic 1-leg capped descendent vertex for the moduli space of instantons on $\mathbb{P}^2$ by analyzing the asymptotic behavior of the capping operator in the limit of vanishing equivariant framing parameters $a_i \to 0$. Using a $K$-theoretic quantum difference equation and factorization theorems for the capping operator and bare vertex, the authors derive rationality as a direct consequence, providing a novel, simplified proof distinct from prior cohomological approaches.

ABSTRACT

In this paper we analyze the fundamental solution of the extit{quantum difference equation} (qde) for the moduli space of instantons on two-dimensional projective space. The qde is a $K$-theoretic generalization of the quantum differential equation in quantum cohomology. As in the quantum cohomology case, the fundamental solution of qde provides the capping operator in $K$-theory (the rubber part of the capped vertex). We study the dependence of the capping operator on the equivariant parameters $a_i$ of the torus acting on the instanton moduli space by changing the framing. We prove that the capping operator factorizes at $a_i o 0$. The rationality of the $K$-theoretic 1-leg capped descendent vertex follows from factorization of the capping operator as a simple corollary.

Motivation & Objective

• To establish the rationality of the $K$-theoretic 1-leg capped descendent vertex in $K$-theory for the moduli space of instantons on $\mathbb{P}^2$.
• To generalize the analysis beyond the Hilbert scheme case ($r=1$) to arbitrary rank $r$ instanton moduli spaces $\mathscr{M}(n,r)$.
• To prove that the capping operator and bare vertex factorize in the limit $a_i \to 0$, where $a_i$ are framing torus equivariant parameters.
• To show that rationality of the capped vertex follows as a corollary from these factorization theorems, offering a new, simpler proof compared to cohomological methods.
• To unify the $K$-theoretic vertex function with quantum difference equations and quantum toroidal algebra structures, providing a broader geometric and algebraic framework.

Proposed method

• Analyzes the fundamental solution of the $K$-theoretic quantum difference equation (qde) for $\mathscr{M}(n,r)$, which defines the capping operator.
• Introduces a universal form of the qde solution using Heisenberg subalgebras of the quantum toroidal algebra $U_\hbar(\widehat{\mathfrak{gl}}_1)$.
• Studies the asymptotic behavior of the capping operator and vertex functions as the framing parameters $a_i \to 0$, proving factorization theorems for both the capping operator and the bare vertex.
• Uses localization techniques on the fixed points of the torus action to compute explicit expressions for the bare vertex and its contributions.
• Applies the coproduct structure of the quantum toroidal algebra to decompose vertex contributions into independent parts in the $a_i \to 0$ limit.
• Demonstrates that the vertex function in the limit splits into a product of lower-rank vertex functions, with correction factors involving $\hbar$ and $q$, leading to rationality.

Experimental results

Research questions

• RQ1 Does the $K$-theoretic 1-leg capped descendent vertex exhibit rationality in the presence of descendents, as conjectured in enumerative geometry of 3-folds?
• RQ2 How does the capping operator in $K$-theory behave under the limit $a_i \to 0$ of the framing torus parameters?
• RQ3 Can the factorization of the capping operator and bare vertex in the $a_i \to 0$ limit be used to deduce rationality of the capped vertex?
• RQ4 Is there a unified algebraic framework—via quantum toroidal algebras and qde—that explains the rationality of $K$-theoretic vertex functions?
• RQ5 How does the $K$-theoretic vertex function relate to the cohomological vertex function in the $q \to 1$ limit, and does this relationship preserve rationality?

Key findings

• The capping operator factorizes in the limit $a_i \to 0$, decomposing into a product of independent operators associated with sub-tori.
• The bare $K$-theoretic 1-leg vertex function also factorizes in the $a_i \to 0$ limit, splitting into contributions from two disjoint sub-vertices.
• The vertex function in the limit $a \to 0$ is given by $V_{{\overrightarrow{\lambda}}_1}^{(r_1), (\tau)}(z \hbar^{r_2/2}) V^{(r_2), (1)}_{{\overrightarrow{\lambda}}_2}(z \hbar^{-r_1/2} q^{-r_1})$, showing a multiplicative structure.
• The rationality of the $K$-theoretic 1-leg capped descendent vertex follows directly from the factorization theorems, as stated in Theorem 2.
• The $K$-theoretic vertex function in the $r=1$ case reduces to the cohomological vertex function in the $q \to 1$ limit, and rationality in this case is recovered via the same mechanism.
• The contribution of cross-terms between different framing components vanishes or becomes constant in the $a \to 0$ limit, leading to a decoupled product structure with explicit correction factors of order $(-\hbar q)^{1/2}$.