Exotic Smoothness Research Articles

Particulate Exotica

Recent advances in differential topology single out four-dimensions as being special, allowing for vast varieties of exotic smoothness (differential) structures, distinguished by their handlebody decompositions, even as the coarser algebraic topology is fixed. Should the spacetime we reside in takes up one of the more exotic choices, and there is no obvious reason why it shouldn’t, apparent pathologies would inevitably plague calculus-based physical theories assuming the standard vanilla structure, due to the non-existence of a diffeomorphism and the consequent lack of a suitable portal through which to transfer the complete information regarding the exotic physical dynamics into the vanilla theories. An obvious plausible consequence of this deficiency would be the uncertainty permeating our attempted description of the microscopic world. We tentatively argue here, that a re-inspection of the key ingredients of the phenomenological particle models, from the perspective of exotica, could possibly yield interesting insights. Our short and rudimentary discussion is qualitative and speculative, because the necessary mathematical tools have only just began to be developed.

Categorical Smoothness of 4-Manifolds from Quantum Symmetries and the Information Loss Paradox.

In this paper, we focus on some aspects of the relation of spacetime and quantum mechanics and the study counterparts (in Set) of the categorical local symmetries of smooth 4-manifolds. In the set-theoretic limit, there emerge some exotic smoothness structures on (hence the Riemannian nonvanishing curvature), which fit well with the quantum mechanical lattice of projections on infinite-dimensional Hilbert spaces. The method we follow is formalization localized on the open covers of the spacetime manifold. We discuss our findings in the context of the information paradox assigned to evaporating black holes. A black hole can evaporate entirely, but the smoothness structure of spacetime will be altered and, in this way, the missing information about the initial states of matter forming the black hole will be encoded. Thus, the possible global geometric remnant of black holes in spacetime is recognized as exotic 4-smoothness. The full-fledged verification of this proposal will presumably be possible within the scope of future quantum gravity theory research.

Does Our Universe Prefer Exotic Smoothness?

Various experimentally verified values of physical parameters indicate that the universe evolves close to the topological phase of exotic smoothness structures on R 4 and K3 surface. The structures determine the α parameter of the Starobinski model, the number of e-folds, the spectral tilt, the scalar-to-tensor ratio and the GUT and electroweak energy scales, as topologically supported quantities. Neglecting exotic R 4 and K3 leaves these free parameters undetermined. We present general physical and mathematical reasons for such preference of exotic smoothness. It appears that the spacetime should be formed on open domains of smooth K 3 # C P 2 ¯ at extra-large scales possibly exceeding our direct observational capacities. Such potent explanatory power of the formalism is not that surprising since there exist natural physical conditions, which we state explicitly, that allow for the unique determination of a spacetime within the exotic K3.

Quantum Mechanics, Formalization and the Cosmological Constant Problem

Based on formal arguments from Zermelo–Fraenkel set theory we develop the environment for explaining and resolving certain fundamental problems in physics. By these formal tools we show that any quantum system defined by an infinite dimensional Hilbert space of states interferes with the spacetime structure M. M and the quantum system both gain additional degrees of freedom, given by models of Zermelo–Fraenkel set theory. In particular, M develops the ground state where classical gravity vanishes. Quantum mechanics distinguishes set-theoretic random forcing such that M and gravitational degrees of freedom are parameterized by extended real line. The large scale smooth geometry compatible with the forcing extensions is one of exotic smoothness structures of $${\mathbb {R}}^4$$ . The amoeba forcing makes the old real line to have Lebesgue measure zero in the extended one. We apply the entire procedure to the cosmological constant problem, especially to discard the zero-modes contributions to the gravitational vacuum density. Moreover, there exists certain exotic smooth $${\mathbb {R}}^4$$ from which one determines the realistic, agreeing with observation, small value of the vacuum energy density.

Local External/Internal Symmetry of Smooth Manifolds and Lack of Tovariance in Physics

Category theory allows one to treat logic and set theory as internal to certain categories. What is internal to SET is 2-valued logic with classical Zermelo–Fraenkel set theory, while for general toposes it is typically intuitionistic logic and set theory. We extend symmetries of smooth manifolds with atlases defined in Set towards atlases with some of their local maps in a topos T . In the case of the Basel topos and R 4 , the local invariance with respect to the corresponding atlases implies exotic smoothness on R 4 . The smoothness structures do not refer directly to Casson handless or handle decompositions, which may be potentially useful for describing the so far merely putative exotic R 4 underlying an exotic S 4 (should it exist). The tovariance principle claims that (physical) theories should be invariant with respect to the choice of topos with natural numbers object and geometric morphisms changing the toposes. We show that the local T -invariance breaks tovariance even in the weaker sense.

A topological approach to neutrino masses by using exotic smoothness

In this paper, we will consider a cosmological model with two topological transitions of the space. The smooth 4-dimensional spacetime of the model admits topological transitions of its 3-dimensional slices. The whole approach is inspired by a class of exotic smoothness structures on [Formula: see text]. In particular, this class of smoothness structures induces two topological transitions. Then, we are able to calculate the energy scales as associated to these topological transitions. For the first transition we will get the value of the GUT scale and the energy of the second transition is at the electroweak scale. The topology of the exotic [Formula: see text] determines both, the energy of the scales by certain topological invariants, and the existence of the right-handed sterile neutrino. It is the input for the seesaw mechanism. Secondly, based on this model, we are able to calculate the neutrino masses which are in a very good agreement with experiments. Finally, we will speculate, again based on topology, why there are three generations of neutrinos and an asymmetry between neutrinos and anti-neutrinos.

Hyperbolic groups, 4-manifolds and Quantum Gravity

4-manifolds have special topological properties which can be used to get a different view on quantum mechanics. One important property (connected with exotic smoothness) is the natural appearance of 3-manifold wild embeddings (Alexanders horned sphere) which can be interpreted as quantum states. This relation can be confirmed by using the Turaev-Drinfeld quantization procedure. Every part of the wild embedding admits a hyperbolic geometry uncovering a deep connection between quantum mechanics and hyperbolic geometry. Then the corresponding symmetry is used to get a dimensional reduction from 4 to 2 for infinite curvatures. Physical consequences will be discussed. At the end we will obtain a spacetime representation of a quantum state of geometry by a non-singular fractal space (wild embedding) which is stable in the limit of infinite curvatures.

Group actions, corks and exotic smoothings of $${\mathbb R}^4$$ R 4

We provide the first information on diffeotopy groups of exotic smoothings of R^4: For each of uncountably many smoothings, there are uncountably many isotopy classes of self-diffeomorphisms. We realize these by various explicit group actions. There are also actions at infinity by nonfinitely generated groups, for which no nontrivial element extends over the whole manifold. In contrast, every diffeomorphism of the end of the universal R^4 extends. Our techniques apply to many other open 4-manifolds, and are related to cork theory. We show that under broad hypotheses, cork twisting is equivalent (up to blowups) to twisting on an exotic R^4, and give applications.

From Quantum to Cosmological Regime. The Role of Forcing and Exotic 4-Smoothness

Recently, a cosmological model based on smooth open 4-manifolds admitting non-standard smoothness structures was proposed. The manifolds are exotic versions of R 4 and S 3 × R . The model has been developed further and proven to be capable of obtaining some realistic cosmological parameters from these exotic smoothings. The important problem of the quantum origins of the exotic smoothness of space-time is addressed here. It is shown that the algebraic structure of the quantum-mechanical lattice of projections enforces exotic smoothness on R n . Since the only possibility for such a structure is exotic R 4 , it is found to be a reasonable explanation of the large-scale four-dimensionality of space-time. This is based on our recent research indicating the role of set-theoretic forcing in quantum mechanics. In particular, it is shown that a distributive lattice of projections implies the standard smooth structure on R 4 . Two examples of models valid for cosmology are discussed. The important result that the cosmological constant can be identified with the constant curvature of the embedding ( exotic R 4 ) → R 4 is referred. . The calculations are in good agreement with the observed small value of the dark energy density.

Minimal genera of open 4–manifolds

We study exotic smoothings of open 4-manifolds using the minimal genus function and its analog for end homology. While traditional techniques in open 4-manifold smoothing theory give no control of minimal genera, we make progress by using the adjunction inequality for Stein surfaces. Smoothings can be constructed with much more control of these genus functions than the compact setting seems to allow. As an application, we expand the range of 4-manifolds known to have exotic smoothings (up to diffeomorphism). For example, every 2-handlebody interior (possibly infinite or nonorientable) has an exotic smoothing, and "most" have infinitely, or sometimes uncountably many, distinguished by the genus function and admitting Stein structures when orientable. Manifolds with 3-homology are also accessible. We investigate topological submanifolds of smooth 4-manifolds. Every domain of holomorphy (Stein open subset) in the complex plane $C^2$ is topologically isotopic to uncountably many other diffeomorphism types of domains of holomorphy with the same genus functions, or with varying but controlled genus functions.

Exotic smoothings via large ℝ4’s in Stein surfaces

We study the relationship between exotic R^4's and Stein surfaces as it applies to smoothing theory on more general open 4-manifolds. In particular, we construct the first known examples of large exotic R^4's that embed in Stein surfaces. This relies on an extension of Casson's Embedding Theorem for locating Casson handles in closed 4-manifolds. Under sufficiently nice conditions, we show that using these R^4's as end-summands produces uncountably many diffeomorphism types while maintaining independent control over the genus-rank function and the Taylor invariant.

How to include fermions into general relativity by exotic smoothness

This paper is two-fold. At first we will discuss the generation of source terms in the Einstein-Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary: a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, denoted as hyperbolic knot complements in the following. We will discuss the fermionic properties of this class of 3-manifolds, i.e. we are able to identify a fermion with a hyperbolic knot complement. Secondly we will construct a large class of space-times, the exotic $\mathbb{R}^{4}$, containing this class of 3-manifolds naturally. We begin with a topological trivial space, the $\mathbb{R}^{4}$, and change only the differential structure to obtain many nontrivial 3-manifolds. It is known for a long time that exotic $\mathbb{R}^{4}$'s generate extra sources of gravity (Brans conjecture) but here we will analyze the structure of these source terms more carefully. Finally we will state that adding a hyperbolic knot complement will result in the appearance of a fermion as source term in the Einstein-Hilbert action.

Inflation and Topological Phase Transition Driven by Exotic Smoothness

We will discuss a model which describes the cause of inflation by a topological transition. The guiding principle is the choice of an exotic smoothness structure for the space-time. Here we consider a space-time with topologyS3×ℝ. In case of an exoticS3×ℝ, there is a change in the spatial topology from a 3-sphere to a homology 3-sphere which can carry a hyperbolic structure. From the physical point of view, we will discuss the path integral for the Einstein-Hilbert action with respect to a decomposition of the space-time. The inclusion of the boundary terms produces fermionic contributions to the partition function. The expectation value of an area (with respect to some surface) shows an exponential increase; that is, we obtain inflationary behavior. We will calculate the amount of this increase to be a topological invariant. Then we will describe this transition by an effective model, the Starobinski orR2model which is consistent with the current measurement of the Planck satellite. The spectral index and other observables are also calculated.

Smooth embeddings with Stein surface images

A simple characterization is given of open subsets of a complex surface that smoothly perturb to Stein open subsets. As applications, ℂ2 contains domains of holomorphy (Stein open subsets) that are exotic smoothings of ℝ4, and others homotopy equivalent to S2 but cut out by smooth, compact 3-manifolds. Pseudoconvex embeddings of Brieskorn spheres and other 3-manifolds into complex surfaces are constructed, as are pseudoconcave holomorphic fillings (with disagreeing contact and boundary orientations). Pseudoconcave complex structures on Milnor fibers are found. A byproduct of this construction is a simple polynomial expression for the signature of the (p, q, npq−1) Milnor fiber. Akbulut corks in complex surfaces can always be chosen to be pseudoconvex or pseudoconcave submanifolds. The main theorem is expressed via Stein handlebodies (possibly infinite), which are defined holomorphically in all dimensions by extending Stein theory to manifolds with non-compact boundary.

On the geometrization of matter by exotic smoothness

In this paper we discuss the question how matter may emerge from space. For that purpose we consider the smoothness structure of spacetime as underlying structure for a geometrical model of matter. For a large class of compact 4-manifolds, the elliptic surfaces, one is able to apply the knot surgery of Fintushel and Stern to change the smoothness structure. The influence of this surgery to the Einstein-Hilbert action is discussed. Using the Weierstrass representation, we are able to show that the knotted torus used in knot surgery is represented by a spinor fulfilling the Dirac equation and leading to a mass-less Dirac term in the Einstein-Hilbert action. For sufficient complicated links and knots, there are "connecting tubes" (graph manifolds, torus bundles) which introduce an action term of a gauge field. Both terms are genuinely geometrical and characterized by the mean curvature of the components. We also discuss the gauge group of the theory to be U(1)xSU(2)xSU(3).

Exotic smooth ℝ4 and quantum matter

We follow the point of view that superstring theory, as the theory of quantum gravity in the number of spacetime dimensions bigger than 4, serves as mathematics for both, 4 dimensional QG and exotic smoothness on open 4-manifolds. Extra-dimensions, supersymmetry or some other string techniques, belong to the mathematical toolkit suitable for the above purposes. Physics in dimension 4 is reached via exotic 4-geometries on ℝ4 . In the paper we discuss the techniques of exact superstring backgrounds, CFT and SU(2)κ WZW models, as suitable for the description of effects assigned to the magnetic field and its gravitational backreactions on exotic Euklidean ℝ4 which is the underlying smoothness for the 4-dimensional spacetime.

TOPOLOGICAL QUANTUM D-BRANES AND WILD EMBEDDINGS FROM EXOTIC SMOOTH ℝ4

This is the next step of uncovering the relation between string theory and exotic smooth ℝ4. Exotic smoothness of ℝ4 is correlated with D6-brane charges in IIA string theory. We construct wild embeddings of spheres and relate them to a class of topological quantum D p-branes as well to KK theory. These branes emerge when there are nontrivial NS–NS H-fluxes on S3 where the topological classes are determined by wild embeddings S2→S3. Then wild embeddings of higher dimensional p-complexes into Sn correspond to D p-branes. These wild embeddings as constructed by using gropes are basic objects to understand exotic smoothness as well Casson handles. Next we build C⋆-algebras corresponding to the embeddings. Finally we consider topological quantum D-branes as those which emerge from wild embeddings in question. We construct an action for these quantum D-branes and show that the classical limit agrees with the Born–Infeld action such that flat branes = usual embeddings.

EXOTIC SMOOTH ℝ4AND CERTAIN CONFIGURATIONS OF NS AND D BRANES IN STRING THEORY

In this paper, we show that in some important cases, four-dimensional data can be extracted from superstring theory such that (i) the data are Euclidean four-geometries embedded in standard ℝ4, (ii) these data depend on NS and D brane charges of some string backgrounds, (iii) it is of potential relevance to four-dimensional physics, (iv) the compactification and stabilization techniques are not in use, but are rather replaced. We analyze certain configurations of NS and D-branes in the context of SU(2) WZW model and find the correlations with different exotic smoothings of ℝ4. First, the dynamics of D-branes in SU(2) WZW model at finite k, i.e. the charges of the branes, refers to the exoticness of ambient ℝ4. Next, the correspondence between exotic smoothness on four-space, transversal to the world volume of NS5 branes in type IIA, and the number of these NS5 branes follows. Finally, the translation of ten-dimensional string backgrounds to Euclidean four-spaces embedded as open subsets in the standard ℝ4is achieved.

Exotic smoothness and quantum gravity

Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. Thus, the negative result of Duston (2009 arXiv:0911.4068) was a surprise. A closer look into the semi-classical approach uncovered the implicit assumption of a close connection between geometry and smoothness structure. But both structures, geometry and smoothness, are independent of each other. In this paper we calculate the ‘smoothness structure’ part of the path integral in quantum gravity assuming that the ‘sum over geometries’ is already given. For that purpose we use the knot surgery of Fintushel and Stern applied to the class E(n) of elliptic surfaces. We mainly focus our attention to the K3 surfaces E(2). Then we assume that every exotic smoothness structure of the K3 surface can be generated by knot or link surgery in the manner of Fintushel and Stern. The results are applied to the calculation of expectation values. Here we discuss the two observables, volume and Wilson loop, for the construction of an exotic 4-manifold using the knot 52 and the Whitehead link Wh. By using Mostow rigidity, we obtain a topological contribution to the expectation value of the volume. Furthermore, we obtain a justification of area quantization.

Background Independence in Quantum Gravity and Forcing Constructions

A general duality connecting the level of a formal theory and of a metatheory is proposed. Because of the role of natural numbers in a metatheory the existence of a dual theory is conjectured, in which the natural numbers become formal in the theory but in formalizing non-formal natural numbers taken from the dual metatheory these numbers become nonstandard. For any formal theory there may be in principle a dual theory. The dual shape of the lattice of projections over separable Hilbert space is exhibited. The crucial ingredient of this dual theory is set-theoretical (Cohen) forcing. Then, some attempts have been made to exhibit the dual shape of the exotic smooth (small) structures on topological ℝ4. There are striking similarities of both pictures. Nonstandard solutions of the equations of motion of 10-dimensional superstring of IIB type are described as a model theoretic extension of standard ones. Based on Brans conjecture one can predict the existence of sources of gravity in 4 dimensions. By their dual origins they carry quantum information as well. The generalized Maldacena conjecture appears naturally, which sheds light on a definition of background independent string theory.