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

Asselmeyer-Maluga, Torsten; Król, Jerzy

doi:10.48550/arxiv.0904.1276

Cited by 12 publications

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“…Finally, one arrives at (7) for flat R 4 in the background R 4 × W 6 . Thus, the inclusion of the magnetic field on R 4 results in the shift in the heterotic string background as follows:…”

Section: D Effects From String Backgroundsmentioning

confidence: 99%

“…One can relate these 4-exotics with some structures on S 3 (see e.g. [7,9,23]) provided it is placed at the boundary of a compact contractible 4-submanifold -the Akbulut cork. If so, one can prove that exotic smoothness of the R 4 is tightly related with codimension-one foliations of this S 3 , hence with the 3-rd real cohomology classes of S 3 .…”

Section: D Cft From the End Of Small Exotic Rmentioning

confidence: 99%

“…If so, one can prove that exotic smoothness of the R 4 is tightly related with codimension-one foliations of this S 3 , hence with the 3-rd real cohomology classes of S 3 . In this sense we classify exotic smooth R 4 's, from the so called radial family, by H 3 (S 3 , R) [7,10].…”

Section: D Cft From the End Of Small Exotic Rmentioning

confidence: 99%

“…This is because the integer classes in H 3 (S 3 , Z) are of special character. Topologically, this case refers to flat P SL(2, R)−bundles over the space (S 2 \ {k punctures}) × S 1 and due to the Heegard decomposition one obtains the relation [7]:…”

Section: D Cft From the End Of Small Exotic Rmentioning

confidence: 99%

“…Here we deal with small exotic R 4 's which emerge as a result of the failure of the h-cobordism theorem in dimension 5 [2,3]. Even though exotic R 4 's are smooth 4-manifolds, a big mathematical problem, which constrains also applications to physics, is to find a suitable effective coordinate presentation such that one can do calculus respecting the exoticness of these manifolds (see, however, [1,[4][5][6][7][8][9][10][11][12][13][14][15][16][16][17][18]).…”

Section: Introductionmentioning

confidence: 99%

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Exotic smooth ℝ⁴and quantum matter

Król¹

2012

J. Phys.: Conf. Ser.

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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 R 4 . In the paper we discuss the techniques of exact superstring backgrounds, CFT and SU (2) k WZW models, as suitable for the description of effects assigned to the magnetic field and its gravitational backreactions on exotic Euklidean R 4 which is the underlying smoothness for the 4-dimensional spacetime. * Electronic address: iriking@wp.pl

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Section: D Effects From String Backgroundsmentioning

confidence: 99%

Section: D Cft From the End Of Small Exotic Rmentioning

confidence: 99%

Section: D Cft From the End Of Small Exotic Rmentioning

confidence: 99%

Section: D Cft From the End Of Small Exotic Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exotic smooth ℝ⁴and quantum matter

Król¹

2012

J. Phys.: Conf. Ser.

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

Król

2016

Fundamental Theories of Physics

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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.

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Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

Asselmeyer-Maluga

2016

Fundamental Theories of Physics

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Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic 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 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 R 4 to the exotic R 4 is a quantization of geometry.

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

Cited by 12 publications

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Exotic smooth ℝ⁴and quantum matter

Exotic smooth ℝ⁴and quantum matter

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

Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

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

Cited by 12 publications

References 0 publications

Exotic smooth ℝ4and quantum matter

Exotic smooth ℝ4and quantum matter

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

Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

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Product

Resources

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Exotic smooth ℝ⁴and quantum matter

Exotic smooth ℝ⁴and quantum matter