On the information carried by programs about the objects they compute

In computability theory and computable analysis, finite programs can compute infinite objects. Presenting a computable object via any program for it, provides at least as much information as presenting the object itself, written on an infinite tape. What additional information do programs provide? We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object. Hence we identify the exact relationship between Markov-computability and Type-2-computability. We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets

The real projective spaces in homotopy type theory

Homotopy type theory is a version of Martin-L\"of type theory taking advantage of its homotopical models. In particular, we can use and construct objects of homotopy theory and reason about them using higher inductive types. In this article, we construct the real projective spaces, key players in homotopy theory, as certain higher inductive types in homotopy type theory. The classical definition of RP(n), as the quotient space identifying antipodal points of the n-sphere, does not translate directly to homotopy type theory. Instead, we define RP(n) by induction on n simultaneously with its tautological bundle of 2-element sets. As the base case, we take RP(-1) to be the empty type. In the inductive step, we take RP(n+1) to be the mapping cone of the projection map of the tautological bundle of RP(n), and we use its universal property and the univalence axiom to define the tautological bundle on RP(n+1). By showing that the total space of the tautological bundle of RP(n) is the n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an (n+1)-cell attached to it. The infinite dimensional real projective space, defined as the sequential colimit of the RP(n) with the canonical inclusion maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises as the subtype of the universe consisting of 2-element types. Indeed, the infinite dimensional projective space classifies the 0-sphere bundles, which one can think of as synthetic line bundles. These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of type theoretic and homotopy theoretic ideas.Comment: 8 pages, to appear in proceedings of LICS 201

5d quivers and their AdS(6) duals

We consider an infinite class of 5d supersymmetric gauge theories involving products of symplectic and unitary groups that arise from D4-branes at orbifold singularities in Type I' string theory. The theories are argued to be dual to warped AdS(6)x S4/Zn backgrounds in massive Type IIA supergravity. In particular, this demonstrates the existence of supersymmetric 5d fixed points of quiver type. We analyze the spectrum of gauge fields and charged states in the supergravity dual, and find a precise agreement with the symmetries and charged operators in the quiver theories. We also comment on other brane objects in the supergravity dual and their interpretation in the field theories.Comment: 29 pages, 15 figure

Do massive compact objects without event horizon exist in infinite derivative gravity

Einstein’s general theory of relativity is plagued by cosmological and black-hole type singularities Recently, it has been shown that infinite derivative, ghost free, gravity can yield nonsingular cosmological and mini-black hole solutions. In particular, the theory possesses a mass-gap determined by the scale of new physics. This paper provides a plausible argument, not a no-go theorem, based on the Area-law of gravitational entropy that within infinite derivative, ghost free, gravity nonsingular compact objects in the static limit need not have horizons

F-Theory on Spin(7) Manifolds: Weak-Coupling Limit

F-theory on appropriately fibered Spin(7) holonomy manifolds is defined to arise as the dual of M-theory on the same space in the limit of a shrinking fiber. A class of Spin(7) orbifolds can be constructed as quotients of elliptically fibered Calabi-Yau fourfolds by an anti-holomorphic involution. The F-theory dual then exhibits one macroscopic dimension that has the topology of an interval. In this work we study the weak-coupling limit of a subclass of such constructions and identify the objects that arise in this limit. On the Type IIB side we find space-time filling O7-planes as well as O5-planes and orbifold five-planes with a (-1)^{F_L} factor localised on the interval boundaries. These orbifold planes are referred to as X5-planes and are S-dual to a D5-O5 system. For other involutions exotic O3-planes and X3-planes on top of a six-dimensional orbifold singularity can appear. We show that the objects present preserve a mutual supersymmetry of four supercharges in the bulk of the interval and two supercharges on the boundary. It follows that in the infinite-interval and weak-coupling limit full four-dimensional N=1 supersymmetry is restored, which on the Type IIA side corresponds to an enhancement of supersymmetry by winding modes in the vanishing interval limit.Comment: 23 page