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1. On finite energy monopoles on $$C x \Sigma$$.

   Wang, Donghao (July 2019, ArXiv.org)

   Let $$X=\mathbb{C}\times\Sigma$$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $$X$$ with finite analytic energy. The spin bundle $$S^+\to X$$ splits as $$L^+\oplus L^-$$. When $$2-2g\leq c_1(S^+)[\Sigma]<0$$, the moduli space is in bijection with the moduli space of pairs $$((L^+,\bar{\partial}), f)$$ where $$(L^+,\bar{\partial})$$ is a holomorphic structure on $L^+$ and $$f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$$ is a polynomial map. Moreover, the solution has analytic energy $$-4\pi^2d\cdot c_1(S^+)[\Sigma]$$ if $$f$$ has degree $$d$$. When $$c_1(S^+)=0$$, all solutions are reducible and the moduli space is the space of flat connections on $$\bigwedge^2 S^+$$. We also estimate the decay rate at infinity for these solutions. 

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2. Sheaves of maximal intersection and multiplicities of stable log maps

   https://doi.org/10.1007/s00029-021-00671-0  

   Choi, Jinwon; van Garrel, Michel; Katz, Sheldon; Takahashi, Nobuyoshi (September 2021, Selecta Mathematica)

   Abstract A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps. 

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3. Sasakians and the geometry of a mass term

   https://doi.org/10.1142/S0129055X2140002X  

   Nair, V. P. (September 2021, Reviews in Mathematical Physics)

   A gauge-invariant mass term for nonabelian gauge fields in two dimensions can be expressed as the Wess–Zumino–Witten (WZW) action. Hard thermal loops in the gauge theory in four dimensions at finite temperatures generate a screening mass for some components of the gauge field. This can be expressed in terms of the WZW action using the bundle of complex structures (for Euclidean signature) or the bundle of lightcones over Minkowski space. We show that a dynamically generated mass term in three dimensions can be put within the same general framework using the bundle of Sasakian structures. 

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4. Curvature operators and rational cobordism

   https://doi.org/10.1016/j.aim.2024.109995  

   Bettiol, Renato G; Goodman, McFeely Jackson (December 2024, Advances in Mathematics)

   We determine linear inequalities on the eigenvalues of curvature operators that imply vanishing of the twisted A-hat genus on a closed Riemannian spin manifold, where the twisting bundle is any prescribed parallel bundle of tensors. These inequalities yield surgery-stable curvature conditions tailored to annihilate further rational cobordism invariants, such as the Witten genus, elliptic genus, signature, and even the rational cobordism class itself. 

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5. Deformation Theory of Log Pseudo-holomorphic Curves and Logarithmic Ruan–Tian Perturbations

   https://doi.org/10.1007/s42543-023-00069-1  

   Farajzadeh-Tehrani, Mohammad (May 2023, Peking Mathematical Journal)

   In a previous paper (Farajzadeh-Tehrani in Geom Topol 26:989–1075, 2022), for any logarithmic symplectic pair (X, D) of a symplectic manifold X and a simple normal crossings symplectic divisor D, we introduced the notion of log pseudo-holomorphic curve and proved a compactness theorem for the moduli spaces of stable log curves. In this paper, we introduce a natural Fredholm setup for studying the deformation theory of log (and relative) curves. As a result, we obtain a logarithmic analog of the space of Ruan–Tian perturbations for these moduli spaces. For a generic compatible pair of an almost complex structure and a log perturbation term, we prove that the subspace of simple maps in each stratum is cut transversely. Such perturbations enable a geometric construction of Gromov–Witten type invariants for certain semi-positive pairs (X, D) in arbitrary genera. In future works, we will use local perturbations and a gluing theorem to construct log Gromov–Witten invariants of arbitrary such pair (X, D). 

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