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1. Pursuing Quantum Difference Equations II: 3D mirror symmetry

   https://doi.org/10.1093/imrn/rnac196  

   Kononov, Yakov; Smirnov, Andrey (July 2022, International Mathematics Research Notices)

   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]. 

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2. Gromov-Witten Invariants in Complex and Morava-Local K-Theories

   https://doi.org/10.1007/s00039-024-00697-4  

   Abouzaid, Mohammed; McLean, Mark; Smith, Ivan (December 2024, Geometric and Functional Analysis)

   Abstract Given a closed symplectic manifoldX, we construct Gromov-Witten-type invariants valued both in (complex)K-theory and in any complex-oriented cohomology theory$$\mathbb{K}$$which isK_p(n)-local for some MoravaK-theoryK_p(n). We show that these invariants satisfy a version of the Kontsevich-Manin axioms, extending Givental and Lee’s work for the quantumK-theory of complex projective algebraic varieties. In particular, we prove a Gromov-Witten type splitting axiom, and hence define quantumK-theory and quantum$$\mathbb{K}$$-theory as commutative deformations of the corresponding (generalised) cohomology rings ofX; the definition of the quantum product involves the formal group of the underlying cohomology theory. The key geometric input of these results is a construction of global Kuranishi charts for moduli spaces of stable maps of arbitrary genus toX. On the algebraic side, in order to establish a common framework covering both ordinaryK-theory andK_p(n)-local theories, we introduce a formalism of ‘counting theories’ for enumerative invariants on a category of global Kuranishi charts. 

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3. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter; Zeitlin, Anton M. (September 2023, Communications in Mathematical Physics)

   Abstract We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$ $C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$) 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$${\mathbb {C}}^\times _\hbar $$ $C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$ $X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$ ${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$ $l⟶\infty$and 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$${\mathbb {P}}^2$$ $P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual. 

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4. Algebraic approximation and the decomposition theorem for Kähler Calabi–Yau varieties

   https://doi.org/10.1007/s00222-022-01096-y  

   Bakker, Benjamin; Guenancia, Henri; Lehn, Christian (June 2022, Inventiones mathematicae)

   Abstract We extend the decomposition theorem for numerically K -trivial varieties with log terminal singularities to the Kähler setting. Along the way we prove that all such varieties admit a strong locally trivial algebraic approximation, thus completing the numerically K -trivial case of a conjecture of Campana and Peternell. 

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5. Which Hessenberg varieties are GKM?

   https://doi.org/10.4310/PAMQ.2023.v19.n4.a8  

   Goldin, Rebecca; Tymoczko, Julianna (September 2023, Pure and Applied Mathematics Quarterly)

   Karshon, Yael; Melrose, Richard; Uhlmann, Gunther; Uribe, Alejandro (Ed.)

   Hessenberg varieties H(X,H) form a class of subvarieties of the flag variety G/B, parameterized by an operator X and certain subspaces H of the Lie algebra of G. We identify several families of Hessenberg varieties in type A_{n−1} that are T -stable subvarieties of G/B, as well as families that are invariant under a subtorus K of T. In particular, these varieties are candidates for the use of equivariant methods to study their geometry. Indeed, we are able to show that some of these varieties are unions of Schubert varieties, while others cannot be such unions. Among the T-stable Hessenberg varieties, we identify several that are GKM spaces, meaning T acts with isolated fixed points and a finite number of one-dimensional orbits, though we also show that not all Hessenberg varieties with torus actions and finitely many fixed points are GKM. We conclude with a series of open questions about Hessenberg varieties, both in type A_{n−1} and in general Lie type. 

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