We investigate the bound on the Lyapunov exponents by a charged particle in Kerr-Newman-de Sitter black holes using analytic and numerical methods. We determine whether the Lyapunov exponent can exceed the bound by an electrically charged particle with an angular momentum. Our tests are applied to the de Sitter spacetime by the positive cosmological constant such as Reissner-Nordström-de Sitter, Kerr-de Sitter, and Kerr-Newman-de Sitter black holes. In particular, we consider Nariai and ultracold limits on these black holes for our tests. From our analysis results, there remain violations on the bound under the positive cosmological constant, and electric charge and angular momentum of the particle significantly impact the Lyapunov exponent.

We investigate the conjectured bound on the Lyapunov exponent for a charged particle with angular motion in the Kerr-Newman-AdS black hole. The Lyapunov exponent is calculated based on the effective Lagrangian. We show that the negative cosmological constant reduces the chaotic behavior of the particle, namely, it decreases the Lyapunov exponent. Hence, the bound is more effective in the AdS spacetime than in the flat spacetime. Nevertheless, we find that the bound can be violated when the angular momenta of the black hole are turned on. Moreover, we show that in an extremal black hole, the bound is more easily violated compared to that in a nonextremal black hole.

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We consider the special roles of the zero loci of the Weierstrass invariants g 2 ðτðzÞÞ and g 3 ðτðzÞÞ in F theory on an elliptic fibration over P 1 or a further fibration thereof. They are defined as the zero loci of the coefficient functions fðzÞ and gðzÞ of a Weierstrass equation. They are thought of as complex codimension-1 objects and correspond to the two kinds of critical points of a dessin d'enfant of Grothendieck. The P 1 base is divided into several cell regions bounded by some domain walls extending from these planes and D-branes, on which the imaginary part of the J function vanishes. This amounts to drawing a dessin with a canonical triangulation. We show that the dessin provides a new way of keeping track of mutual nonlocalness among 7-branes without employing unphysical branch cuts or their base point. With the dessin, we can see that weak-and strongcoupling regions coexist and are located across an S wall from each other. We also present a simple method for computing a monodromy matrix for an arbitrary path by tracing the walls it goes through.