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1. Topological Strings on Non-commutative Resolutions

   https://doi.org/10.1007/s00220-023-04896-2  

   Katz, Sheldon; Klemm, Albrecht; Schimannek, Thorsten; Sharpe, Eric (February 2024, Communications in Mathematical Physics)

   Abstract In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi–Yau. Our main example will be a singular degeneration of the generic Calabi–Yau double cover of$${\mathbb {P}}^3$$ $P^3$and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau–Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi–Yau manifolds by one of the authors and clarify the associated enumerative geometry. 

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2. 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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3. Free quotients of favorable Calabi-Yau manifolds

   https://doi.org/10.1007/JHEP07(2022)116  

   Gray, James; Wang, Juntao (July 2022, Journal of High Energy Physics)

   A bstract Non-simply connected Calabi-Yau threefolds play a central role in the study of string compactifications. Such manifolds are usually described by quotienting a simply connected Calabi-Yau variety by a freely acting discrete symmetry. For the Calabi-Yau threefolds described as complete intersections in products of projective spaces, a classification of such symmetries descending from linear actions on the ambient spaces of the varieties has been given in [16]. However, which symmetries can be described in this manner depends upon the description that is being used to represent the manifold. In [24] new, favorable, descriptions were given of this data set of Calabi-Yau threefolds. In this paper, we perform a classification of cyclic symmetries that descend from linear actions on the ambient spaces of these new favorable descriptions. We present a list of 129 symmetries/non-simply connected Calabi-Yau threefolds. Of these, at least 33, and potentially many more, are topologically new varieties. 

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4. Warped quasi-asymptotically conical Calabi-Yau metrics

   Conlon, Ronan J.; Rochon, Frederic (August 2023, arXivorg)

   We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted Hölder spaces. 

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5. Orientifold Calabi-Yau threefolds with divisor involutions and string landscape

   https://doi.org/10.1007/JHEP03(2022)087  

   Altman, Ross; Carifio, Jonathan; Gao, Xin; Nelson, Brent D. (March 2022, Journal of High Energy Physics)

   A bstract We establish an orientifold Calabi-Yau threefold database for h 1 , 1 ( X ) ≤ 6 by considering non-trivial ℤ 2 divisor exchange involutions, using a toric Calabi-Yau database ( www.rossealtman.com/tcy ). We first determine the topology for each individual divisor (Hodge diamond), then identify and classify the proper involutions which are globally consistent across all disjoint phases of the Kähler cone for each unique geometry. Each of the proper involutions will result in an orientifold Calabi-Yau manifold. Then we clarify all possible fixed loci under the proper involution, thereby determining the locations of different types of O -planes. It is shown that under the proper involutions, one typically ends up with a system of O 3 /O 7-planes, and most of these will further admit naive Type IIB string vacua. The geometries with freely acting involutions are also determined. We further determine the splitting of the Hodge numbers into odd/even parity in the orbifold limit. The final result is a class of orientifold Calabi-Yau threefolds with non-trivial odd class cohomology ( $$ {h}_{-}^{1,1} $$ h − 1 , 1 ( X/σ * ) ≠ 0). 

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