Adapting Game Engines to Curved Spaces

Curved spaces are very un-intuitive to our eyes trained on Euclidean geometry. Games provide an interesting way to explore these strange worlds. Games are written with the help of modeling tools and game engines based on Euclidean geometry. This paper addresses the problem of adapting 3D game engines to the rules of curved spaces. We consider the conversion of Euclidean objects, geometric calculations, transformation pipeline, lighting and physical simulation. Finally, we identify where existing game engines should be modified.

Exact multi-instanton solutions to self-dual Yang–Mills equation on curved spaces

We find exact multi-instanton solutions to the self-dual Yang–Mills equation on a large class of curved spaces with [Formula: see text] isometry, generalizing the results previously found on [Formula: see text]. The solutions are featured with explicit multi-centered expressions and topological properties. As examples, we demonstrate the approach on several different curved spaces, including the Einstein static universe and [Formula: see text], and show that the exact multi-instanton solutions exist on these curved backgrounds.

Geometrizing $$ T\overline{T} $$

The $$ T\overline{T} $$ T T ¯ deformation can be formulated as a dynamical change of coordinates. We establish and generalize this relation to curved spaces by coupling the undeformed theory to 2d gravity. For curved space the dynamical change of coordinates is supplemented by a dynamical Weyl transformation. We also sharpen the holographic correspondence to cutoff AdS3 in multiple ways. First, we show that the action of the annular region between the cutoff surface and the boundary of AdS3 is given precisely by the $$ T\overline{T} $$ T T ¯ operator integrated over either the cutoff surface or the asymptotic boundary. Then we derive dynamical coordinate and Weyl transformations directly from the bulk. Finally, we reproduce the flow equation for the deformed stress tensor from the cutoff geometry.

On negative energies, strings, branes, and braneworlds: A review of novel approaches

On the way towards quantum gravity and the unification of interaction, several ideas have been rejected and avenues avoided because they were perceived as physically unviable. But in the literature there are works in which it was found the contrary, namely that those rejected topics make sense after all. Such topics, reviewed in this paper, are negative energies occurring in higher derivative theories and ultrahyperbolic spaces, ordering ambiguity of operators in curved spaces, the vast landscape of possible compactifications of extra dimensions in string theory, and quantization of a 3-brane in braneworld scenarios.