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1. On the degeneration of asymptotically conical Calabi–Yau metrics

   https://doi.org/10.1007/s00208-021-02229-z  

   Collins, T. (July 2021, Mathematische Annalen)

   null (Ed.)

   We study the degenerations of asymptotically conical Ricci-flat Kahler metrics as the Kahler class degenerates to a semi-positive class. We show that under appropriate assumptions, the Ricci-flat Kahler metrics converge to a incomplete smooth Ricci-flat Kahler metric away from a compact subvariety. As a consequence, we construct singular Calabi–Yau metrics with asymptotically conical behaviour at infinity on certain quasi-projective varieties and we show that the metric geometry of these singular metrics are homeomorphic to the topology of the singular variety. Finally, we will apply our results to study several classes of examples of geometric transitions between Calabi–Yau manifolds. 

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2. Brane brick models for the Sasaki-Einstein 7-manifolds Yp,k(ℂℙ1 × ℂℙ1) and Yp,k(ℂℙ2)

   https://doi.org/10.1007/JHEP03(2023)050  

   Franco, Sebastián; Ghim, Dongwook; Seong, Rak-Kyeong (March 2023, Journal of High Energy Physics)

   A bstract The 2 d (0 , 2) supersymmetric gauge theories corresponding to the classes of Y p,k (ℂℙ 1 × ℂℙ 1 ) and Y p,k (ℂℙ 2 ) manifolds are identified. The complex cones over these Sasaki-Einstein 7-manifolds are non-compact toric Calabi-Yau 4-folds. These infinite families of geometries are the largest ones for Sasaki-Einstein 7-manifolds whose metrics, toric diagrams, and volume functions are known explicitly. This work therefore presents the largest list of 2 d (0 , 2) supersymmetric gauge theories corresponding to Calabi-Yau 4-folds with known metrics. 

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3. Metric flows with neural networks

   https://doi.org/10.1088/2632-2153/ad8533  

   Halverson, James; Ruehle, Fabian (October 2024, Machine Learning: Science and Technology)

   Abstract We develop a general theory of flows in the space of Riemannian metrics induced by neural network (NN) gradient descent. This is motivated in part by recent advances in approximating Calabi–Yau metrics with NNs and is enabled by recent advances in understanding flows in the space of NNs. We derive the corresponding metric flow equations, which are governed by a metric neural tangent kernel (NTK), a complicated, non-local object that evolves in time. However, many architectures admit an infinite-width limit in which the kernel becomes fixed and the dynamics simplify. Additional assumptions can induce locality in the flow, which allows for the realization of Perelman’s formulation of Ricci flow that was used to resolve the 3d Poincaré conjecture. We demonstrate that such fixed kernel regimes lead to poor learning of numerical Calabi–Yau metrics, as is expected since the associated NNs do not learn features. Conversely, we demonstrate that well-learned numerical metrics at finite-width exhibit an evolving metric-NTK, associated with feature learning. Our theory of NN metric flows therefore explains why NNs are better at learning Calabi–Yau metrics than fixed kernel methods, such as the Ricci flow. 

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4. 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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5. Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning

   https://doi.org/10.1007/JHEP05(2021)013  

   Anderson, Lara B.; Gerdes, Mathis; Gray, James; Krippendorf, Sven; Raghuram, Nikhil; Ruehle, Fabian (May 2021, Journal of High Energy Physics)

   null (Ed.)

   A bstract We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ 4 . 

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