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1. Weak gravity conjecture

   https://doi.org/10.1103/RevModPhys.95.035003  

   Harlow, Daniel; Heidenreich, Ben; Reece, Matthew; Rudelius, Tom (September 2023, Reviews of Modern Physics)

   The weak gravity conjecture holds that in a theory of quantum gravity any gauge force must mediate interactions stronger than gravity for some particles. This statement has surprisingly deep and extensive connections to many different areas of physics and mathematics. Several variations on the basic conjecture have been proposed, including statements that are much stronger but are nonetheless satisfied by all known consistent quantum gravity theories. These related conjectures and the evidence for their validity in the string theory landscape are reviewed. Also reviewed are a variety of arguments for these conjectures, which tend to fall into two categories: qualitative arguments that claim the conjecture is plausible based on general principles and quantitative arguments for various special cases or analogs of the conjecture. The implications of these conjectures for particle physics, cosmology, general relativity, and mathematics are also outlined. Finally, important directions for future research are highlighted. 

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2. Generic Beauville’s Conjecture

   https://doi.org/10.1017/fms.2024.21  

   Coskun, Izzet; Larson, Eric; Vogt, Isabel (January 2024, Forum of Mathematics, Sigma)

   Abstract Let$$\alpha \colon X \to Y$$be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under$$\alpha $$is semistable if the genus ofYis at least$$1$$and stable if the genus ofYis at least$$2$$. We prove this conjecture if the map$$\alpha $$is general in any component of the Hurwitz space of covers of an arbitrary smooth curveY. 

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3. The K(π,1)-conjecture implies the center conjecture for Artin groups

   https://doi.org/10.1016/j.jalgebra.2022.10.024  

   Jankiewicz, Kasia; Schreve, Kevin (February 2023, Journal of Algebra)
4. On Gupta’s Codensity Conjecture

   https://doi.org/10.1137/22M1514003  

   Cao, Yan; Chen, Guantao; Ding, Guoli; Jing, Guangming; Zang, Wenan (September 2023, SIAM Journal on Discrete Mathematics)
5. A CFT distance conjecture

   https://doi.org/10.1007/JHEP10(2021)070  

   Perlmutter, Eric; Rastelli, Leonardo; Vafa, Cumrun; Valenzuela, Irene (October 2021, Journal of High Energy Physics)

   A bstract We formulate a series of conjectures relating the geometry of conformal manifolds to the spectrum of local operators in conformal field theories in d > 2 spacetime dimensions. We focus on conformal manifolds with limiting points at infinite distance with respect to the Zamolodchikov metric. Our central conjecture is that all theories at infinite distance possess an emergent higher-spin symmetry, generated by an infinite tower of currents whose anomalous dimensions vanish exponentially in the distance. Stated geometrically, the diameter of a non-compact conformal manifold must diverge logarithmically in the higher-spin gap. In the holographic context our conjectures are related to the Distance Conjecture in the swampland program. Interpreted gravitationally, they imply that approaching infinite distance in moduli space at fixed AdS radius, a tower of higher-spin fields becomes massless at an exponential rate that is bounded from below in Planck units. We discuss further implications for conformal manifolds of superconformal field theories in three and four dimensions. 

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