arXiv subjects

Torsten Asselmeyer-Maluga

Publications and source records attributed to Torsten Asselmeyer-Maluga.

At least 19 recordsLinked to original sources

arXiv2024

A geometric phase approach to quark confinement from stochastic gauge-geometry flows

We apply a stochastic version of the geometric (Ricci) flow, complemented with the stochastic flow of the gauge Yang--Mills sector, in order to seed the chromo-magnetic and chromo-electric vortices that source the area-law for QCD confinement. The area-law is the key signature of quark confinement in Yang--Mills gauge theories with a non-trivial center symmetry. In particular, chromo-magnetic vortices enclosed within the chromo-electric Wilson loops instantiate the area-law asymptotic behaviour of the Wilson loop vacuum expectation values. The stochastic gauge-geometry flow is responsible for the topology changes that induce the appearance of the vortices. When vortices vanish, due to topology changes in the manifolds associated to the hadronic ground states, the evaluation of the Wilson loop yields a dependence on the length of the path, hence reproducing the perimeter law of the hadronic (Higgs) phase of real QCD. Confinement, instead, is naturally achieved within this context as a by-product of the topology change of the manifold over which the dynamics of the Yang--Mills fields is defined. It is then provided by the Aharonov--Bohm effect induced by the concatenation of the compact chromo-electric and chromo-magnetic fluxes originated by the topology changes. The stochastic gauge-geometry flow naturally accomplishes a treatment of the emergence of the vortices and the generation of turbulence effects. Braiding and knotting, resulting from topology changes, namely stochastic fluctuations, stabilize the chromo-magnetic vortices. Finally, we observe that dimensional transmutation for the Yang-Mills fields can be derived from the scaling property of the geometric part of the stochastic flow. Specifically, a relation that involves the infrared equilibrium limit of the Planck constant can be derived that yields the correct order of magnitude for $\Lambda_{\rm QCD}$.

hep-th↗

arXiv2022

Big Bang and Topology

In this paper we discuss the initial state of the universe at the Big Bang. By using ideas of Freedman in the proof of the disk embedding theorem for 4-manifolds, we describe the corresponding spacetime as gravitational instanton. The spatial space is a fractal space (wild embedded 3-sphere). Then we construct the quantum state from this fractal space. This quantum state is part of the string algebra of Ocneanu. There is a link to the Jones polynomial and to Witten's topological field theory. Using this link, we are able to determine the physical theory (action) to be the Chern-Simons functional. The gauge fixing of this action determines the foliation of the spacetime and as well the smoothness properties. Finally, we determine the quantum symmetry of the quantum state to be the enveloped Lie algebra $U_{q}(sl_{2}(\mathbb{C}))$ where $q$ is the 4th root of unity.

gr-qc↗

arXiv2021

3D Topological Quantum Computing

In this paper we will present some ideas to use 3D topology for quantum computing extending ideas from a previous paper. Topological quantum computing used \textquotedblleft knotted\textquotedblright{} quantum states of topological phases of matter, called anyons. But anyons are connected with surface topology. But surfaces have (usually) abelian fundamental groups and therefore one needs non-abelian anyons to use it for quantum computing. But usual materials are 3D objects which can admit more complicated topologies. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere (see arXiv:2102.04452 for previous work). The whole system is designed as knotted superconductor where every crossing is a Josephson junction and the qubit is realized as flux qubit. We discuss the properties of this systems in particular the fluxion quantization by using the A-polynomial of the knot. Furthermore we showed that 2-qubit operations can be realized by linked (knotted) superconductors again coupled via a Josephson junction.

quant-ph↗

arXiv2021

Topological Quantum Computing and 3-Manifolds

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston's geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al.

quant-ph↗

arXiv2020

Dark Matter as gravitational solitons in the weak field limit

In this paper, we will describe the idea that dark matter partly consists of gravitational solitons (gravisolitons). The corresponding solution is valid for weak gravitational fields (weak field limit) with respect to a background metric. The stability of this soliton is connected with the existence of a special foliation and amazingly with the smoothness properties of spacetime. Gravisolitons have many properties of dark matter, such as no interaction with light but act on matter via gravitation. In this paper, we showed that the gravitational lensing effect of gravisolitons agreed with the lensing effect of usual matter. Furthermore, we obtained the same equation of state $w=0$ as matter.

gr-qc↗

arXiv2019

Braids, 3-manifolds, elementary particles: number theory and symmetry in particle physics

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston's geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement $C(K)=S^{3}\setminus(K\times D^{2})$ of a knot $K$ carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which~will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of $S^{3}$ branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk $D^{3}$ branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld--Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson--Thompson model).

physics.gen-ph↗

arXiv2018

Quantum computing and the brain: quantum nets, dessins d'enfants and neural networks

In this paper, we will discuss a formal link between neural networks and quantum computing. For that purpose we will present a simple model for the description of the neural network by forming sub-graphs of the whole network with the same or a similar state. We will describe the interaction between these areas by closed loops, the feedback loops. The change of the graph is given by the deformations of the loops. This fact can be mathematically formalized by the fundamental group of the graph. Furthermore the neuron has two basic states $|0\rangle$ (ground state) and $|1\rangle$ (excited state). The whole state of an area of neurons is the linear combination of the two basic state with complex coefficients representing the signals (with 3 Parameters: amplitude, frequency and phase) along the neurons. Then it can be shown that the set of all signals forms a manifold (character variety) and all properties of the network must be encoded in this manifold. In the paper, we will discuss how to interpret learning and intuition in this model. Using the Morgan-Shalen compactification, the limit for signals with large amplitude can be analyzed by using quasi-Fuchsian groups as represented by dessins d'enfants (graphs to analyze Riemannian surfaces). As shown by Planat and collaborators, these dessins d'enfants are a direct bridge to (topological) quantum computing with permutation groups. The normalization of the signal reduces to the group $SU(2)$ and the whole model to a quantum network. Then we have a direct connection to quantum circuits. This network can be transformed into operations on tensor networks. Formally we will obtain a link between machine learning and Quantum computing.

quant-ph↗

arXiv2018

A topological model for inflation

In this paper we will discuss a new model for inflation based on topological ideas. For that purpose we will consider the change of the topology of the spatial component seen as compact 3-manifold. We analyzed the topology change by using Morse theory and handle body decomposition of manifolds. For the general case of a topology change of a $n-$manifold, we are forced to introduce a scalar field with quadratic potential or double well potential. Unfortunately these cases are ruled out by the CMB results of the Planck misssion. In case of 3-manifolds there is another possibility which uses deep results in differential topology of 4-manifolds. With the help of these results we will show that in case of a fixed homology of the 3-manifolds one will obtain a scalar field potential which is conformally equivalent to the Starobinsky model. The free parameter of the Starobinsky model can be expressed by the topological invariants of the 3-manifold. Furthermore we are able to express the number of e-folds as well as the energy and length scale by the Chern-Simons invariant of the final 3-manifold. We will apply these result to a specific model which was used by us to discuss the appearance of the cosmological constant with an experimentally confirmed value.

physics.gen-ph↗

arXiv2018

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.

gr-qc↗

arXiv2016

Smooth quantum gravity: Exotic smoothness and Quantum gravity

Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic $\mathbb{R}^{4}$, and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called \emph{smooth quantum gravity} by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard $\mathbb{R}^{4}$ to the exotic $R^{4}$ is a quantization of geometry.

gr-qc↗

arXiv2014

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.

hep-th↗

arXiv2013

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.

math-ph↗

arXiv2012

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.

math-ph↗

arXiv2012

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).

gr-qc↗

arXiv2012

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

We show that superconformal ${\cal N}=4,2$ algebras are well-suited to represent some invariant constructions characterizing exotic $\mathbb{R}^4$ relative to a given radial family. We examine the case of ${\cal N}=4, \hat{c}=4$ (at $r=1$ level) superconformal algebra which is realized on flat $\mathbb{R}^4$ and curved $S^3\times \mathbb{R}$. While the first realization corresponds naturally to standard smooth $\mathbb{R}^4$ the second describes the algebraic end of some small exotic smooth $\mathbb{R}^4$'s from the radial family of DeMichelis-Freedman and represents the linear dilaton background $SU(2)_k\times \mathbb{R}_Q$ of superstring theory. From the modular properties of the characters of the algebras one derives Witten-Reshetikhin-Turaev and Chern-Simons invariants of homology 3-spheres. These invariants are represented rather by false, quasi-modular, Ramanujan mock-type functions. Given the homology 3-spheres one determines exotic smooth structures of Freedman on $S^3\times \mathbb{R}$. In this way the fake ends are related to the SCA ${\cal N}=4$ characters. The case of the ends of small exotic $\mathbb{R}^4$'s is more complicated. One estimates the complexity of exotic $\mathbb{R}^4$ by the minimal complexity of some separating from the infinity 3-dimensional submanifold. These separating manifolds can be chosen, in some exotic $\mathbb{R}^4$'s, to be homology 3-spheres. The invariants of such homology 3-spheres are, again, obtained from the characters of SCA, ${\cal N}=4$. Next we take into account the modification of the algebra of modular forms due to the noncommutativity of the codimension-one foliations of the homology 3-spheres. Then, the modification of modular forms is represented by the Connes-Moscovici construction ...

hep-th↗

arXiv2012

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.

hep-th↗

arXiv2012

The modification of the energy spectrum of charged particles by exotic open 4-smoothness via superstring theory

In this paper we present a model where the modified Landau-like levels of charged particles in a magnetic field are determined due to the modified smoothness of $\mathbb{R}^{4}$ as underlying structure of the Minkowski spacetime. Then the standard smoothness of $\mathbb{R}^{4}$ is shifted to the exotic $\mathbb{R}_{k}^{4}$, $k=2p$, $p=1,2...$. This is achieved by superstring theory using gravitational backreaction induced from a strong, almost constant magnetic field on standard $\mathbb{R}^{4}$. The exact string background containing flat $\mathbb{R}^{4}$ is replaced consistently by the curved geometry of $SU(2)_{k}\times\mathbb{R}$ as part of the modified exact backgrounds. This corresponds to the change of smoothness on $\mathbb{R}^{4}$ from the standard $\mathbb{R}^{4}$ to some exotic $\mathbb{R}_{k}^{4}$. The calculations of the spectra are using the CFT marginal deformations and Wess-Zumino-Witten (WZW) models. The marginal deformations capture the effects of the magnetic field as well as its gravitational backreactions. The spectra depend on even level $k$ of WZW on SU(2). At the same time the WZ term as element of $H^{3}(SU(2),\mathbb{R})$ determines also the exotic smooth $\mathbb{R}_{k}^{4}$. As the consequence we obtain a non-zero mass-gap emerges in the spectrum induced from the presence of an exotic $\mathbb{R}_{k}^{4}$.

hep-th↗

arXiv2012

On the origin of inflation

In this paper we discuss a space-time having the topology of S^{3}xR but with different smoothness structure. This space-time is not a global hyperbolic space-time. Especially we obtain a time line with a topology change of the space from the 3-sphere to a homology 3-sphere and back but without a topology-change of the space-time. Among the infinite possible smoothness structures of this space-time, we choose a homology 3-sphere with hyperbolic geometry admitting a homogenous metric. Then the topology change can be described by a time-dependent curvature parameter k changing from k=+1 to k=-1 and back. The solution of the Friedman equation for dust matter (p=0) after inserting this function shows an exponential growing which is typical for inflation. In contrast to other inflation models, this process stops after a finite time.

gr-qc↗