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1. Quivers and curves in higher dimension

   https://doi.org/10.1090/tran/9232  

   Argüz, Hülya; Bousseau, Pierrick (September 2024, Transactions of the American Mathematical Society)

   We prove a correspondence between Donaldson–Thomas invariants of quivers with potential having trivial attractor invariants and genus zero punctured Gromov–Witten invariants of holomorphic symplectic cluster varieties. The proof relies on the comparison of the stability scattering diagram, describing the wall-crossing behavior of Donaldson–Thomas invariants, with a scattering diagram capturing punctured Gromov–Witten invariants via tropical geometry. 

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2. Punctured logarithmic maps

   Abramovich, Dan; Chen, Qile; Gross, Mark; Siebert, Bernd (February 2025, EMS press)

   We introduce a variant of stable logarithmic maps, which we call punctured logarith- mic maps. They allow an extension of logarithmic Gromov–Witten theory in which marked points have a negative order of tangency with boundary divisors. As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with trop- ical geometry providing the underlying combinatorics. Punctured Gromov–Witten invariants also play a pivotal role in the intrinsic con- struction of mirror partners by the last two authors, conjecturally relating to symplec- tic cohomology, and in the logarithmic gauged linear sigma model in work of Qile Chen, Felix Janda and Yongbin Ruan. 

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3. 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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4. K-theoretic Gromov–Witten invariants of line degrees on flag varieties

   https://doi.org/10.1142/S0217751X24460138  

   Buch, Anders S; Chen, Linda; Xu, Weihong (November 2024, International Journal of Modern Physics A)

   A homology class [Formula: see text] of a complex flag variety [Formula: see text] is called a line degree if the moduli space [Formula: see text] of 0-pointed stable maps to X of degree d is also a flag variety [Formula: see text]. We prove a quantum equals classical formula stating that any n-pointed (equivariant, [Formula: see text]-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety [Formula: see text]. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags [Formula: see text]. Our formulas make it straightforward to compute the big quantum [Formula: see text]-theory ring [Formula: see text] modulo the ideal [Formula: see text] generated by degrees d larger than line degrees. 

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5. Solomon-Tukachinsky’s Versus Welschinger’s Open Gromov-Witten Invariants of Symplectic Six-Folds

   https://doi.org/10.1093/imrn/rnaa318  

   Chen, Xujia (December 2020, International Mathematics Research Notices)

   Abstract Our previous paper describes a geometric translation of the construction of open Gromov–Witten invariants by Solomon and Tukachinsky from a perspective of $$A_{\infty }$$-algebras of differential forms. We now use this geometric perspective to show that these invariants reduce to Welschinger’s open Gromov–Witten invariants in dimension 6, inline with their and Tian’s expectations. As an immediate corollary, we obtain a translation of Solomon–Tukachinsky’s open WDVV equations into relations for Welschinger’s invariants. 

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