Set theoretical forcing in quantum mechanics and AdS/CFT correspondence

We show unexpected connection of Set Theoretical Forcing with Quantum Mechanical lattice of projections over some separable Hilbert space. The basic ingredient of the construction is the rule of indistinguishability of Standard and some Nonstandard models of Peano Arithmetic. The ingeneric reals introduced by M. Ozawa will correspond to simultaneous measurement of incompatible observables. We also discuss some results concerning model theoretical analysis of Small Exotic Smooth Structures on topological 4-space. Forcing appears rather naturally in this context and the rule of indistinguishability is crucial again. As an unexpected application we are able to approach Maldacena Conjecture on $AdS/CFT$ correspondence in the case of AdS_5xS^5 and Super YM Conformal Field Theory in 4 dimensions. We conjecture that there is possibility of breaking Supersymetry via sources of gravity generated in 4 dimensions by exotic smooth structures on R^4 emerging in this context.Comment: 16 pages, 1 eps figure, LaTeX 2e. Presented at QCS02 held in Ustron, Poland on September 2002, to appear in Int. J. of Theor. Phy

Model theory and the AdS/CFT correspondence

We give arguments that exotic smooth structures on compact and noncompact 4-manifolds are essential for some approaches to quantum gravity. We rely on the recently developed model-theoretic approach to exotic smoothness in dimension four. It is possible to conjecture that exotic $R^4$'s play fundamental role in quantum gravity similarily as standard local 4-spacetime patches do for classical general relativity. Renormalization in gravity--field theory limit of AdS/CFT correspondence is reformulated in terms of exotic $R^4$'s. We show how doubly special relativity program can be related to some model-theoretic self-dual $R^4$'s. The relevance of the structures for the Maldacena conjecture is discussed, though explicit calculations refer to the would be noncompact smooth 4-invariants based on the intuitionistic logic.Comment: 17 pages, presented at the IPM String School and Workshop, Queshm Island, Iran, 05-14. 01. 2005. Some bibliographical corrections include

Model and Set-Theoretic Aspects of Exotic Smoothness Structures on R4\mathbb{R}^4

Model-theoretic aspects of exotic smoothness were studied long ago uncovering unexpected relations to noncommutative spaces and quantum theory. Some of these relations were worked out in detail in later work. An important point in the argumentation was the forcing construction of Cohen but without a direct application to exotic smoothness. In this article we assign the set-theoretic forcing on trees to Casson handles and characterize small exotic smooth $R^4$ from this point of view. Moreover, we show how models in some Grothendieck toposes can help describing such differential structures in dimension 4. These results can be used to obtain the deformation of the algebra of usual complex functions to the noncommutative algebra of operators on a Hilbert space. We also discuss the results in the context of the Epstein-Glaser renormalization in QFT.Comment: 32 pages, 2 figures, to appear in: At the Frontiers of Spacetime: Scalar-Tensor Theory, Bell's Inequality, Mach's Principle, Exotic Smoothness, ed. T. Asselmeyer-Maluga (Springer, 2016), in honor of Carl Brans's 80th birthda

Synthetic Approach to the Singularity Problem

We try to convince the reader that the categorical version of differential geometry, called Synthetic Differential Geometry (SDG), offers valuable tools which can be applied to work with some unsolved problems of general relativity. We do this with respect to the space-time singularity problem. The essential difference between the usual differential geometry and SDG is that the latter enriches the real line by introducing infinitesimal of various kinds. Owing to this geometry acquires a tool to penetrate "infinitesimally small" parts of a given manifold. However, to make use of this tool we must switch from the category of sets to some other suitable category. We try two topoi: the topos ${\cal G}$ of germ determined ideals and the so-called Basel topos ${\cal B}$. The category of manifolds is a subcategory of both of them. In ${\cal G}$, we construct a simple model of a contracting sphere. As the sphere shrinks, its curvature increases, but when the radius of the sphere reaches infinitesimal values, the curvature becomes infinitesimal and the singularity is avoided. The topos ${\cal B}$, unlike the topos ${\cal G}$, has invertible infinitesimal and infinitely large nonstandard natural numbers. This allows us to see what happens when a function "goes through a singularity". When changing from the category of sets to another topos, one must be ready to switch from classical logic to intuitionistic logic. This is a radical step, but the logic of the universe is not obliged to conform to the logic of our brains.Comment: 17 pages, no figure

How Logic Interacts with Geometry: Infinitesimal Curvature of Categorical Spaces

In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic logic of SDG enforces the existence of infinitesimal objects which essentially modify the local structure of spaces considered in SDG. We focus on an "infinitesimal version" of SDG, an infinitesimal $n$-dimensional formal manifold, and develop differential geometry on it. In particular, we show that the Riemann curvature tensor on infinitesimal level is itself infinitesimal. We construct a heuristic model $S^3 \times \mathbb{R} \subset \mathbb{R}^4$ and study it from two perspectives: the perspective of the category SET and that of the so-called topos $\mathcal{G}$ of germ-determined ideals. We show that the fact that in this model the curvature tensor is infinitesimal (in $\mathcal{G}$-perspective) eliminates the existing singularity. A surprising effect is that the hybrid geometry based on the existence of the infinitesimal and the SET levels generates an exotic smooth structure on $\mathbb{R}^4$. We briefly discuss the obtained results and indicate their possible applications.Comment: 18 page

Towards superconformal and quasi-modular representation of exotic smooth R^4 from superstring theory II

This is the second part of the work where quasi-modular forms emerge from small exotic smooth $\mathbb{R}^4$'s grouped in a fixed radial family. SU(2) Seiberg-Witten theory when formulated on exotic $\mathbb{R}^4$ from the radial family, in special foliated topological limit can be described as SU(2) Seiberg-Witten theory on flat standard $\mathbb{R}^4$ with the gravitational corrections derived from coupling to ${\cal N}=2$ supergravity. Formally, quasi-modular expressions which follow the Connes-Moscovici construction of the universal Godbillon-Vey class of the codimension-1 foliation, are related to topological correlation functions of superstring theory compactified on special Callabi-Yau manifolds. These string correlation functions, in turn, generate Seiberg-Witten prepotential and the couplings of Seiberg-Witten theory to ${\cal N}=2$ supergravity sector. Exotic 4-spaces are conjectured to serve as a link between supersymmetric and non-supersymmetric Yang-Mills theories in dimension 4.Comment: 12 page

Abelian gerbes, generalized geometries and foliations of small exotic R^4

In the paper we prove the existence of the strict but relative relation between small exotic $\mathbb{R}^{4}$ for a fixed radial family of DeMichelis-Freedman type, and cobordism classes of codimension one foliations of $S^{3}$ distinguished by the Godbillon-Vey invariant, $GV\in H^{3}(S^{3},\mathbb{R})$ (represented by a 3-form). This invariant can be integrated to get the Godbillon-Vey number. For a fixed radial family, we will show that the isotopy classes (invariance w.r.t. small diffeomorphisms or coordinate transformations) of all members in this family are distinguished by the Godbillon-Vey number of the foliation which is equal to the square of the radius of the radial family. The special case of integer Godbillon-Vey invariants $GV\in H^{3}(S^{3},\mathbb{Z})$ is also discussed and is connected to flat $PSL(2,\mathbb{R})-$bundles. Next we relate these distinguished small exotic smooth $\mathbb{R}^{4}$'s to twisted generalized geometries of Hitchin on $TS^{3}\oplus T^{\star}S^{3}$ and abelian gerbes on $S^{3}$. In particular the change of the smoothness on $\mathbb{R}^{4}$ corresponds to the twisting of the generalized geometry by the abelian gerbe. We formulate the localization principle for exotic 4-regions in spacetime and show that the existence of these domains causes the quantization of electric charge, the effect usually ascribed to the existence of magnetic monopoles.Comment: 54 pages, 11 Figures, WS RMP style, complete revision with many background material, an error in the argumentation was fixed (many thanks to L. Taylor), the proof about the relation between foliations and exotic R^4 was completed, an construction of a foliation from a Casson handle was adde

Gerbes on orbifolds and exotic smooth R^4

By using the relation between foliations and exotic R^4, orbifold $K$-theory deformed by a gerbe can be interpreted as coming from the change in the smoothness of R^4. We give various interpretations of integral 3-rd cohomology classes on S^3 and discuss the difference between large and small exotic R^4. Then we show that $K$-theories deformed by gerbes of the Leray orbifold of S^3 are in one-to-one correspondence with some exotic smooth R^4's. The equivalence can be understood in the sense that stable isomorphisms classes of bundle gerbes on S^{3} whose codimension-1 foliations generates the foliations of the boundary of the Akbulut cork, correspond uniquely to these exotic R^{4}'s. Given the orbifold $SU(2)\times SU(2)\rightrightarrows SU(2)$ where SU(2) acts on itself by conjugation, the deformations of the equivariant $K$-theory on this orbifold by the elements of $H_{SU(2)}^{3}(SU(2),\mathbb{Z})$, correspond to the changes of suitable exotic smooth structures on R^4.Comment: now 22 pages, the construction of the foliation in the appendix was added, no figures, adjustment of the changes in arXiv:0904.1276, subm. to Comm. Math. Phy

Magnetic monopoles, squashed 3-spheres and gravitational instantons from exotic R^4

We show that, in some limit, gravitational instantons correspond to exotic smooth geometry on R^4. The geometry of this exotic R^4 represents magnetic monopoles of Polyakov-'t Hooft type and the BPS condition allows for the generation of the charges from the gravitational sources of exotic R^4. Higgs field is, as usual, present in the monopole configurations. We indicate some possible scenarios in cosmology, particle physics and condensed matter where pure SU(2) Yang-Mills theory on the exotic R^4 aquires non-zero mass for its gauge boson when the smoothness is changed to the standard R^4 and, then, it is described by Yang-Mills Higgs theory. The derivation of the results is based on the quasi-modularity of the expressions and the geometry of foliations in the limit.Comment: 16 page

Higgs potential and confinement in Yang-Mills theory on exotic R^4

We show that pure SU(2) Yang-Mills theory formulated on certain exotic R^4 from the radial family shows confinement. The condensation of magnetic monopoles and the qualitative form of the Higgs potential are derived from the exotic R^4, e. A relation between the Higgs potential and inflation is discussed. Then we obtain a formula for the Higgs mass and discuss a particular smoothness structure so that the Higgs mass agrees with the experimental value. The singularity in the effective dual U(1) potential has its cause by the exotic 4-geometry and agrees with the singularity in the maximal abelian gauge scenario. We will describe the Yang-Mills theory on e in some limit as the abelian-projected effective gauge theory on the standard R^4. Similar results can be derived for SU(3) Yang-Mills theory on an exotic R^4 provided dual diagonal effective gauge bosons propagate in the exotic 4-geometry.Comment: 12 page