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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)

   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. Computational Mirror Symmetry

   https://doi.org/10.1007/JHEP01(2024)184  

   Demirtas, Mehmet ; Kim, Manki ; McAllister, Liam ; Moritz, Jakob ; Rios-Tascon, Andres ( January 2024 , Journal of High Energy Physics)

   We present an efficient algorithm for computing the prepotential in compactifications of type II string theory on mirror pairs of Calabi-Yau threefolds in toric varieties. Applying this method, we exhibit the first systematic computation of genus-zero Gopakumar-Vafa invariants in compact threefolds with many moduli, including examples with up to 491 vector multiplets.

    

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3. Symmetries of Calabi-Yau prepotentials with isomorphic flops

   https://doi.org/10.1007/JHEP02(2023)175  

   Lukas, Andre ; Ruehle, Fabian ( February 2023 , Journal of High Energy Physics)

   A bstract Calabi-Yau threefolds with infinitely many flops to isomorphic manifolds have an extended Kähler cone made up from an infinite number of individual Kähler cones. These cones are related by reflection symmetries across flop walls. We study the implications of this cone structure for mirror symmetry, by considering the instanton part of the prepotential in Calabi-Yau threefolds. We show that such isomorphic flops across facets of the Kähler cone boundary give rise to symmetry groups isomorphic to Coxeter groups. In the dual Mori cone, non-flopping curve classes that are identified under these groups have the same Gopakumar-Vafa invariants. This leads to instanton prepotentials invariant under Coxeter groups, which we make manifest by introducing appropriate invariant functions. For some cases, these functions can be expressed in terms of theta functions whose appearance can be linked to an elliptic fibration structure of the Calabi-Yau manifold. 

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4. The Weak Gravity Conjecture and BPS Particles

   https://doi.org/10.1002/prop.202100125  

   Alim, Murad ; Heidenreich, Ben ; Rudelius, Tom ( September 2021 , Fortschritte der Physik)

   Motivated by the Weak Gravity Conjecture, we uncover an intricate interplay between black holes, BPS particle counting, and Calabi‐Yau geometry in five dimensions. In particular, we point out that extremal BPS black holes exist only in certain directions in the charge lattice, and we argue that these directions fill out a cone that is dual to the cone of effective divisors of the Calabi‐Yau threefold. The tower and sublattice versions of the Weak Gravity Conjecture require an infinite tower of BPS particles in these directions, and therefore imply purely geometric conjectures requiring the existence of infinite towers of holomorphic curves in every direction within the dual of the cone of effective divisors. We verify these geometric conjectures in a number of examples by computing Gopakumar‐Vafa invariants.

    

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5. Small cosmological constants in string theory

   https://doi.org/10.1007/JHEP12(2021)136  

   Demirtas, Mehmet ; Kim, Manki ; McAllister, Liam ; Moritz, Jakob ; Rios-Tascon, Andres ( December 2021 , Journal of High Energy Physics)

   A bstract We construct supersymmetric AdS 4 vacua of type IIB string theory in compactifications on orientifolds of Calabi-Yau threefold hypersurfaces. We first find explicit orientifolds and quantized fluxes for which the superpotential takes the form proposed by Kachru, Kallosh, Linde, and Trivedi. Given very mild assumptions on the numerical values of the Pfaffians, these compactifications admit vacua in which all moduli are stabilized at weak string coupling. By computing high-degree Gopakumar-Vafa invariants we give strong evidence that the α ′ expansion is likewise well-controlled. We find extremely small cosmological constants, with magnitude < 10 − 123 in Planck units. The compactifications are large, but not exponentially so, and hence these vacua manifest hierarchical scale-separation, with the AdS length exceeding the Kaluza-Klein length by a factor of a googol. 

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