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1. Compact quantum metric spaces from free graph algebras

   https://doi.org/10.1142/S0129167X22500732  

   Aguilar, Konrad; Hartglass, Michael; Penneys, David (September 2022, International Journal of Mathematics)

   Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance. 

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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. Splitting of Gromov–Witten invariants with toric gluing strata

   https://doi.org/10.1090/jag/826  

   Wu, Yixian (November 2023, Journal of Algebraic Geometry)

   We prove a splitting formula that reconstructs the logarithmic Gromov–Witten invariants of simple normal crossing varieties from the punctured Gromov–Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov–Witten theory developed by Abramovich, Chen, Gross, and Siebert. 

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4. Computing a categorical Gromov–Witten invariant

   https://doi.org/10.1112/S0010437X20007174  

   Căldăraru, Andrei; Tu, Junwu (July 2020, Compositio Mathematica)

   null (Ed.)

   We compute the $g=1$ , $n=1$ B-model Gromov–Witten invariant of an elliptic curve $$E$$ directly from the derived category $$\mathsf{D}_{\mathsf{coh}}^{b}(E)$$ . More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $$\mathscr{A}_{\infty }$$ model of $$\mathsf{D}_{\mathsf{coh}}^{b}(E)$$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf. 

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5. Quantum K theory of Grassmannians, Wilson line operators and Schur bundles

   https://doi.org/10.1017/fms.2025.10088  

   Gu, Wei; Mihalcea, Leonardo; Sharpe, Eric; Zou, Hao (January 2025, Forum of Mathematics, Sigma)

   Abstract We prove a ‘Whitney’ presentation, and a ‘Coulomb branch’ presentation, for the torus equivariant quantum K theory of the Grassmann manifold$$\mathrm {Gr}(k;n)$$, inspired from physics, and stated in an earlier paper. The first presentation is obtained by quantum deforming the product of the Hirzebruch$$\lambda _y$$classes of the tautological bundles. In physics, the$$\lambda _y$$classes arise as certain Wilson line operators. The second presentation is obtained from the Coulomb branch equations involving the partial derivatives of a twisted superpotential from supersymmetric gauge theory. This is closest to a presentation obtained by Gorbounov and Korff, utilizing integrable systems techniques. Algebraically, we relate the Coulomb and Whitney presentations utilizing transition matrices from the (equivariant) Grothendieck polynomials to the (equivariant) complete homogeneous symmetric polynomials. Along the way, we calculate K-theoretic Gromov-Witten invariants of wedge powers of the tautological bundles on$$\mathrm {Gr}(k;n)$$, using the ‘quantum=classical’ statement. 

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