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1. 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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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. Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, Proceedings of symposia in pure mathematics)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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4. Title: String-Math 2022 Article: Holomorphic Floer theory and Donaldson-Thomas invariants

   Bousseau, Pierrick (April 2024, AMS)

   Donag, Ron; Langer, Adrian; Sułkowski, Piotr; Wendland, Katrin (Ed.)

   We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relating holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds. More generally, we conjecture that the BPS spectrum of a 4-dimensional N = 2 quantum field theory can be recovered from the holomorphic Floer theory of the corresponding Seiberg-Witten integrable system. 

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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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