[Paper Review] Permutation-equivariant quantum K-theory VI. Mirrors

Alexander Givental|arXiv (Cornell University)|Sep 25, 2015

Advanced Combinatorial Mathematics3 references3 citations

TL;DR

This paper develops a K-theoretic analogue of the Hori-Vafa mirror construction for toric manifolds, using complex oscillating integrals to represent $q$-hypergeometric functions in $K^0$-theory. It identifies Lagrangian varieties parameterized by critical points of phase functions, revealing self-similar behavior under Adams maps and linking quantum K-theory to mirror symmetry via $q$-difference operators and roots of unity limits.

ABSTRACT

We present here the K-theoretic version of mirror models of toric manifold. First, we recall the construction of cohomological mirrors for toric manifolds, i.e. representations of the toric hypergeometric functions from quantum cohomology theory by complex oscillating integrals. Then we repeat the construction in the K-theoretic situation, and obtain complex oscillating integrals representing q-hypergeometric functions from quantum K-theory of toric manifolds, bundles, and super-bundles. Finally, we examine the Lagrangian varieties parameterized by critical points of the phase functions of these oscillating integrals.

Motivation & Objective

• To extend the Hori-Vafa mirror construction from cohomological to K-theoretic quantum invariants of toric manifolds.
• To formulate $q$-hypergeometric functions in $K^0(X)$-valued oscillating integrals using phase functions and Lefschetz thimbles.
• To identify the Lagrangian varieties underlying the stationary phase asymptotics of these integrals in the quasiclassical limit near roots of unity.
• To establish a geometric link between quantum K-theory and mirror symmetry through $D_q$-modules and Adams maps.
• To generalize the mirror philosophy to include $K$-theory and $q$-difference operators via self-similar structures at roots of unity.

Proposed method

• Construct $K$-theoretic $I$-functions $I_X^K$ as $q$-hypergeometric series in $K^0(X)$, using line bundles $P_i$ and $U_j(P)$ with multiplicative relations.
• Represent $I_X^K$ via complex oscillating integrals $\mathcal{I}_X^K$ over non-compact Lefschetz thimbles in the torus $\widehat{X}_Q$ defined by constraints $\prod_j X_j^{m_{ij}} = Q_i$.
• Use Lagrange multipliers to derive critical point equations for the phase function $\sum_{k>0} \sum_j X_j^k / k(1 - q^k)$, leading to the Lagrangian variety $L_m$.
• Identify the symplectic structure on the parameter space $(P,Q)$ as $\sum_i dP_i/P_i \wedge dQ_i/Q_i$, and show $L_m$ is the pullback of $L_1$ under the Adams map $\Psi^m$.
• Analyze stationary phase asymptotics near $q = \zeta^{-1}$, a primitive $m$-th root of unity, to extract self-similar behavior of the Lagrangian varieties.
• Establish that the resulting $D_q$-module structure on $\mathcal{I}_X^K$ satisfies the same system as the $K$-theoretic $I$-function, confirming mirror symmetry.

Experimental results

Research questions

• RQ1 How can the Hori-Vafa mirror construction be generalized from cohomology to $K$-theory for toric manifolds?
• RQ2 What is the $K$-theoretic analogue of the complex oscillating integral representation of hypergeometric functions?
• RQ3 How do the critical points of the phase function in the $K$-theoretic mirror integral parameterize Lagrangian varieties in the symplectic torus?
• RQ4 What is the geometric and algebraic significance of the self-similar behavior of these Lagrangian varieties under the Adams map $\Psi^m$?
• RQ5 How does the quasiclassical limit of $q$-oscillating integrals relate to $D_q$-modules and quantum K-theory?

Key findings

• The $K$-theoretic $I$-function $I_X^K$ is represented by a complex oscillating integral $\mathcal{I}_X^K$ over Lefschetz thimbles in the torus $\widehat{X}_Q$ defined by $\prod_j X_j^{m_{ij}} = Q_i$.
• The critical points of the phase function $\sum_{k>0} \sum_j X_j^k / k(1 - q^k)$ satisfy $\sum_{l>0} X_j^{lm}/(lm) = u_j(p)$, leading to the relation $\ln(1 - X_j^m)^{1/m} = u_j(p)$.
• The Lagrangian variety $L_m$ is given by $Q_i^m = \prod_j (1 - U_j^m(P))^{m_{ij}}$, which is the inverse image of $L_1$ under the Adams map $\Psi^m$.
• For $m=1$, the Lagrangian variety $L_1$ realizes quantum deformations of Kirwan’s relations in $K^0(X)$, linking to quantum cohomology.
• The stationary phase asymptotics near $q = \zeta^{-1}$ (a root of unity) exhibit self-similarity, with $L_m$ being the pullback of $L_1$ under $\Psi^m$.
• The resulting $D_q$-module structure on $\mathcal{I}_X^K$ matches the $K$-theoretic $I$-function system, confirming mirror symmetry in $K$-theory.