skip to main content


Title: Quantum K-theory of quiver varieties and many-body systems
Abstract We define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.  more » « less
Award ID(s):
2054527
NSF-PAR ID:
10326519
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
Selecta Mathematica
Volume:
27
Issue:
5
ISSN:
1022-1824
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$C2($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$Cħ×-action. First, we find a two-parameter family$$X_{k,l}$$Xk,lof self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$Hilbk[C2]is obtained via direct limit$$l\longrightarrow \infty $$land by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$ħ-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$P2with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

     
    more » « less
  2. Abstract

    Let $\textsf {X}$ and $\textsf {X}^{!}$ be a pair of symplectic varieties dual with respect to 3D mirror symmetry. The $K$-theoretic limit of the elliptic duality interface is an equivariant $K$-theory class $\mathfrak {m} \in K(\textsf {X}\times \textsf {X}^{!})$. We show that this class provides correspondences $$ \begin{align*} & \Phi_{\mathfrak{m}}: K(\textsf{X}) \leftrightarrows K(\textsf{X}^{!}) \end{align*}$$mapping the $K$-theoretic stable envelopes to the $K$-theoretic stable envelopes. This construction allows us to relate various representation theoretic objects of $K(\textsf {X})$, such as action of quantum groups, quantum dynamical Weyl groups, $R$-matrices, etc., to those for $K(\textsf {X}^{!})$. In particular, we relate the wall $R$-matrices of $\textsf {X}$ to the $R$-matrices of the dual variety $\textsf {X}^{!}$. As an example, we apply our results to $\textsf {X}=\textrm {Hilb}^{n}({{\mathbb {C}}}^2)$—the Hilbert scheme of $n$ points in the complex plane. In this case, we arrive at the conjectures of Gorsky and Negut from [10].

     
    more » « less
  3. null (Ed.)
    The jeu-de-taquin-based Littlewood-Richardson rule of H. Thomas and A. Yong (2009) for minuscule varieties has been extended in two orthogonal directions, either enriching the cohomology theory or else expanding the family of varieties considered. In one direction, A. Buch and M. Samuel (2016) developed a combinatorial theory of 'unique rectification targets' in minuscule posets to extend the Thomas-Yong rule from ordinary cohomology to $K$-theory. Separately, P.-E. Chaput and N. Perrin (2012) used the combinatorics of R. Proctor's '$d$-complete posets' to extend the Thomas-Yong rule from minuscule varieties to a broader class of Kac-Moody structure constants. We begin to address the unification of these theories. Our main result is the existence of unique rectification targets in a large class of $d$-complete posets. From this result, we obtain conjectural positive combinatorial formulas for certain $K$-theoretic Schubert structure constants in the Kac-Moody setting. 
    more » « less
  4. Abstract A plausible origin of the seismically observed mid-lithospheric discontinuity (MLD) in the subcontinental lithosphere is mantle metasomatism. The metasomatized mantle is likely to stabilize hydrous phases such as amphiboles. The existing electrical conductivity data on amphiboles vary significantly. The electrical conductivity of hornblendite is much higher than that of tremolite. Thus, if hornblendite truly represents the amphibole varieties in MLD regions, then it is likely that amphibole will cause high electrical conductivity anomalies at MLD depths. However, this is inconsistent with the magnetotelluric observations across MLD depths. Hence, to better understand this discrepancy in electrical conductivity data of amphiboles and to evaluate whether MLD could be caused by metasomatism, we determined the electrical conductivity of a natural metasomatized rock sample. The metasomatized rock sample consists of ~87% diopside pyroxene, ~9% sodium-bearing tremolite amphibole, and ~3% albite feldspar. We collected the electrical conductivity data at ~3.0 GPa, i.e., the depth relevant to MLD. We also spanned a temperature range between 400 to 1000 K. We found that the electrical conductivity of this metasomatized rock sample increases with temperature. The temperature dependence of the electrical conductivity exhibits two distinct regimes. At low temperatures <700 K, the electrical conductivity is dominated by the conduction in the solid state. At temperatures >775 K, the conductivity increases, and it is likely to be dominated by the conduction of aqueous fluids due to partial dehydration. The main distinction between the current study and the prior studies on the electrical conductivity of amphiboles or amphibole-bearing rocks is the sodium (Na) content in amphiboles of the assemblage. Moreover, it is likely that the higher Na content in amphiboles leads to higher electrical conductivity. Pargasite and edenite amphiboles are the most common amphibole varieties in the metasomatized mantle, and our study on Na-bearing tremolite is the closest analog of these amphiboles. Comparison of the electrical conductivity results with the magnetotelluric observations constrains the amphibole abundance at MLD depths to <1.5%. Such a low-modal proportion of amphiboles could only reduce the seismic shear wave velocity by 0.4–0.5%, which is significantly lower than the observed velocity reduction of 2–6%. Thus, it might be challenging to explain both seismic and magnetotelluric observations at MLD simultaneously. 
    more » « less
  5. Abstract We consider a pair of quiver varieties $(X;X^{\prime})$ related by 3D mirror symmetry, where $X =T^*{Gr}(k,n)$ is the cotangent bundle of the Grassmannian of $k$-planes of $n$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $X \times X^{\prime} $ (the mother function) whose restrictions to $X$ and $X^{\prime} $ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $X$ and $X^{\prime}$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side. 
    more » « less