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1. 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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2. A heterotic Kähler gravity and the distance conjecture

   https://doi.org/10.1007/JHEP01(2025)168  

   Murgas_Ibarra, Javier José; Oehlmann, Paul-Konstantin; Ruehle, Fabian; Svanes, Eirik Eik (January 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> Deformations of the heterotic superpotential give rise to a topological holomorphic theory with similarities to both Kodaira-Spencer gravity and holomorphic Chern-Simons theory. Although the action is cubic, it is only quadratic in the complex structure deformations (the Beltrami differential). Treated separately, for large fluxes, or alternatively at large distances in the background complex structure moduli space, these fields can be integrated out to obtain a new field theory in the remaining fields, which describe the complexified hermitian and gauge degrees of freedom. We investigate properties of this new holomorphic theory, and in particular connections to the swampland distance conjecture in the context of heterotic string theory. In the process, we define a new type of symplectic cohomology theory, where the background complex structure Beltrami differential plays the role of the symplectic form. 

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3. Heegaard Floer homology and cosmetic surgeries in $S^3$

   https://doi.org/10.4171/JEMS/1218  

   Hanselman, Jonathan (March 2022, Journal of the European Mathematical Society)

   If a knot K in S^3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either ±2 or ±1/q for some value of q that is explicitly determined by the knot Floer homology of K. Moreover, in the former case the genus of K must be 2, and in the latter case there is a bound relating q to the genus and the Heegaard Floer thickness of K. As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to 2. We also show that the conjecture holds for any knot K for which each prime summand of K has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes ±1 and ±2 on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access. 

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4. On the Frøyshov invariant and monopole Lefschetz number

   https://doi.org/10.4310/jdg/1683307008  

   Lin, Jianfeng; Ruberman, Daniel; Saveliev, Nikolai (March 2023, Journal of Differential Geometry)

   Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz num- ber of the map induced by this involution in the reduced mono- pole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic ar- gument, making use of an exact triangle in monopole Floer homol- ogy, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants as- sociated to spin structures on Y . We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology. 

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5. Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities

   https://doi.org/10.1090/jams/962  

   Kebekus, Stefan; Schnell, Christian (April 2021, Journal of the American Mathematical Society)

   null (Ed.)

   We investigate under what conditions holomorphic forms defined on the regular locus of a reduced complex space extend to holomorphic (or logarithmic) forms on a resolution of singularities. We give a simple necessary and sufficient condition for this, whose proof relies on the Decomposition Theorem and Saito’s theory of mixed Hodge modules. We use it to generalize the theorem of Greb-Kebekus-Kovács-Peternell to complex spaces with rational singularities, and to prove the existence of a functorial pull-back for reflexive differentials on such spaces. We also use our methods to settle the “local vanishing conjecture” proposed by Mustaţă, Olano, and Popa. 

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