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1. Mass deformations of brane brick models

   https://doi.org/10.1007/JHEP09(2023)176  

   Franco, Sebastián; Ghim, Dongwook; Goulas, Georgios P; Seong, Rak-Kyeong (September 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We investigate a class of mass deformations that connect pairs of 2d(0,2) gauge theories associated to different toric Calabi-Yau 4-folds. These deformations are generalizations to 2dof the well-known Klebanov-Witten deformation relating the 4dgauge theories for the ℂ²/ℤ₂× ℂ orbifold and the conifold. We investigate various aspects of these deformations, including their connection to brane brick models and the relation between the change in the geometry and the pattern of symmetry breaking triggered by the deformation. We also explore how the volume of the Sasaki-Einstein 7-manifold at the base of the Calabi-Yau 4-fold varies under deformation, which leads us to conjecture that it quantifies the number of degrees of freedom of the gauge theory and its dependence on the RG scale. 

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2. Deformations of Calabi–Yau varieties with k -liminal singularities

   https://doi.org/10.1017/fms.2024.44  

   Friedman, Robert; Laza, Radu (January 2024, Forum of Mathematics, Sigma)

   Abstract The goal of this paper is to describe certain nonlinear topological obstructions for the existence of first-order smoothings of mildly singular Calabi–Yau varieties of dimension at least$$4$$. For nodal Calabi–Yau threefolds, a necessary and sufficient linear topological condition for the existence of a first-order smoothing was first given in [Fri86]. Subsequently, Rollenske–Thomas [RT09] generalized this picture to nodal Calabi–Yau varieties of odd dimension by finding a necessary nonlinear topological condition for the existence of a first-order smoothing. In a complementary direction, in [FL22a], the linear necessary and sufficient conditions of [Fri86] were extended to Calabi–Yau varieties in every dimension with$$1$$-liminal singularities (which are exactly the ordinary double points in dimension$$3$$but not in higher dimensions). In this paper, we give a common formulation of all of these previous results by establishing analogues of the nonlinear topological conditions of [RT09] for Calabi–Yau varieties with weighted homogeneousk-liminal hypersurface singularities, a broad class of singularities that includes ordinary double points in odd dimensions. 

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3. BFT2: a general class of 2d $$ \mathcal{N} $$ = (0, 2) theories, 3-manifolds and toric geometry

   https://doi.org/10.1007/JHEP08(2022)277  

   Franco, Sebastián; Yu, Xingyang (August 2022, Journal of High Energy Physics)

   A bstract We introduce and initiate the study of a general class of 2 d $$ \mathcal{N} $$ N = (0, 2) quiver gauge theories, defined in terms of certain 2-dimensional CW complexes on oriented 3-manifolds. We refer to this class of theories as BFT 2 ’s. They are natural generalizations of Brane Brick Models, which capture the gauge theories on D1-branes probing toric Calabi-Yau 4-folds. The dynamics and triality of the gauge theories translate into simple transformations of the underlying CW complexes. We introduce various combinatorial tools for analyzing these theories and investigate their connections to toric Calabi-Yau manifolds, which arise as their master and moduli spaces. Invariance of the moduli space is indeed a powerful criterion for identifying theories in the same triality class. We also investigate the reducibility of these theories. 

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4. Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

   https://doi.org/10.1515/crelle-2023-0043  

   Argüz, Hülya; Bousseau, Pierrick (July 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Cluster varieties come in pairs: for any $\mathcal{𝒳}${\mathcal{X}}cluster variety there is an associated Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the $\mathcal{𝒳}${\mathcal{X}}cluster variety is a degeneration of the Fock–Goncharov dual $\mathcal{𝒜}${\mathcal{A}}cluster varietyand vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties. 

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5. Deep multi-task mining Calabi–Yau four-folds

   https://doi.org/10.1088/2632-2153/ac37f7  

   Erbin, Harold; Finotello, Riccardo; Schneider, Robin; Tamaazousti, Mohamed (November 2021, Machine Learning: Science and Technology)

   Abstract We continue earlier efforts in computing the dimensions of tangent space cohomologies of Calabi–Yau manifolds using deep learning. In this paper, we consider the dataset of all Calabi–Yau four-folds constructed as complete intersections in products of projective spaces. Employing neural networks inspired by state-of-the-art computer vision architectures, we improve earlier benchmarks and demonstrate that all four non-trivial Hodge numbers can be learned at the same time using a multi-task architecture. With 30% (80%) training ratio, we reach an accuracy of 100% for $h^{(1,1)}$and 97% for $h^{(2,1)}$(100% for both), 81% (96%) for $h^{(3,1)}$, and 49% (83%) for $h^{(2,2)}$. Assuming that the Euler number is known, as it is easy to compute, and taking into account the linear constraint arising from index computations, we get 100% total accuracy. 

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