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1. Sato–Tate Groups of Abelian Threefolds

   https://doi.org/10.1090/memo/1582  

   Fité, Francesc; Kedlaya, Kiran; Sutherland, Andrew (August 2025, Memoirs of the American Mathematical Society)

   Given an abelian variety over a number field, its Sato–Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the $L$-function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato–Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato–Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato–Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura’s theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations inGAP,SageMath, andMagma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated $L$-functions. 

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2. Quark masses and mixing in string-inspired models

   https://doi.org/10.1007/JHEP06(2025)175  

   Constantin, Andrei; Fraser-Taliente, Cristofero S; Harvey, Thomas R; Leung, Lucas_T Y; Lukas, Andre (June 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We study a class of supersymmetric Froggatt-Nielsen (FN) models with multiple U(1) symmetries and Standard Model (SM) singlets inspired by heterotic string compactifications on Calabi-Yau threefolds. The string-theoretic origin imposes a particular charge pattern on the SM fields and FN singlets, dividing the latter into perturbative and non-perturbative types. Employing systematic and heuristic search strategies, such as genetic algorithms, we identify charge assignments and singlet VEVs that replicate the observed mass and mixing hierarchies in the quark sector, and subsequently refine the Yukawa matrix coefficients to accurately match the observed values for the Higgs VEV, the quark and charged lepton masses and the CKM matrix. This bottom-up approach complements top-down string constructions and our results demonstrate that string FN models possess a sufficiently rich structure to account for flavour physics. On the other hand, the limited number of distinct viable charge patterns identified here indicates that flavour physics imposes tight constraints on string theory models, adding new constraints on particle spectra that are essential for achieving a realistic phenomenology. 

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3. On the string landscape without hypermultiplets

   https://doi.org/10.1007/JHEP04(2024)121  

   Baykara, Zihni Kaan; Hamada, Yuta; Tarazi, Houri-Christina; Vafa, Cumrun (April 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> In this work we study interesting corners of the quantum gravity landscape with 8 supercharges pushing the boundaries of our current understanding. Calabi-Yau threefolds compactifications of F/M/type II theories to 6, 5 and 4 dimensions are the most prominent examples of this class, and these always lead to a universal hypermultiplet coming from the volume/string coupling constant. We find new examples of asymmetric orbifold constructions which have no hypermultiplets in 4 or 5 dimensions and no neutral hypers in 6d. We argue that these theories can also be obtained by going to strong coupling/small volume regions of geometric constructions where a new Coulomb branch opens up and moving in this direction freezes the volume/string coupling constant. Interestingly we find that the Kodaira condition encountered in geometric limits of F-theory compactifications to 6 dimensions is violated in these corners of the landscape due to strong quantum corrections. We also construct a theory in 3 dimensions which if it were to arise by toroidal compactifications from 5d, it would have to come from pure$$ \mathcal{N} $$ $N$= 1 supergravity with no massless scalar fields. 

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4. Bounds on field range for slowly varying positive potentials

   https://doi.org/10.1007/JHEP02(2024)175  

   van_de_Heisteeg, Damian; Vafa, Cumrun; Wiesner, Max; Wu, David H (February 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> In the context of quantum gravitational systems, we place bounds on regions in field space with slowly varying positive potentials. Using the fact that$$ V<{\Lambda}_s^2 $$ $V<{\Lambda }_s^2$, where Λ_s(ϕ) is the species scale, and the emergent string conjecture, we show this places a bound on the maximum diameter of such regions in field space: ∆ϕ≤alog(1/V) +bin Planck units, wherea≤$$ \sqrt{\left(d-1\right)\left(d-2\right)} $$ $\sqrt{\left(d-1\right)\left(d-2\right)}$, andbis an 𝒪(1) number and expected to be negative. The coefficient of the logarithmic term has previously been derived using TCC, providing further confirmation. For type II string flux compactifications on Calabi-Yau threefolds, using the recent results on the moduli dependence of the species scale, we can check the above relation and determine the constantb, which we verify is 𝒪(1) and negative in all the examples we studied. 

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5. Some remarks about deformation theory and formality conjecture

   https://doi.org/10.1007/s11565-024-00500-0  

   Chen, Huachen; Pertusi, Laura; Zhao, Xiaolei (February 2024, ANNALI DELL'UNIVERSITA' DI FERRARA)

   Abstract Using the algebraic criterion proved by Bandiera, Manetti and Meazzini, we show the formality conjecture for universally gluable objects with linearly reductive automorphism groups in the bounded derived category of a K3 surface. As an application, we prove the formality conjecture for polystable objects in the Kuznetsov components of Gushel–Mukai threefolds and quartic double solids. 

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