Distribution Theory and Spin Geometry Methods in General Relativity and String Theories

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Açik Ö.

TUBITAK Project, 2018 - 2020

  • Project Type: TUBITAK Project
  • Begin Date: August 2018
  • End Date: August 2020

Project Abstract

In this project, topics such as extended particles, string and M-theories, supergravity and higher spin fields which are among the leading research interests of current theoretical physics will be studied. Distribution theory, gravitational charges, spin geometry, hidden symmetries and related methods will be used in the research that will be done in the project. The main aims of the project are; i) finding a relevant mathematical language for the distribution theory in curved space-time and using it in the analysis of the dynamics of closed and charged immersions, ii) analysis of the motion of gravitationally charged basic or composite systems and the effects of gravitational charges on the external and internal geometries of immersion and stability of the motion, iii) obtaining hidden symmetries on special manifolds arising from the compactifications of string and M-theories and finding results about the classification problem of string and M-theory backgrounds by constructing superalgebra structures on these special manifolds, iv) Finding the solutions of the Rarita-Schwinger equation which arise in supergravity theories and describe the motion of spin 3/2 particles by using spin raising and lowering operators. 

Relativistic dynamics of extended particles can be used to model physical events in the scope of astrophysics, fundamental particle physics and string theories and for this reason it is involed among the current fields of interest of theoretical physics. Classification of consistent backgrounds of string and M-theories and finding solutions of supergravity Rarita-Schwinger equation are also studied extensively in the recent literature. In this project, by using distributional differential geometry techniques in General Relativity and other background geometries, which is a less known topic in our country, extended particles, boundary layers and shock waves will be studied systematically and open problems in the literature will be analyzed. Usage of gravitational charges in the dynamics of closed and charged immersions will be dealed in the scope of this project for the first time. Spin geometry tools that appear in the compactification methods which provide to obtain four-dimensional space-times from higher dimensional theories will be used for the first time in the literature for the aim of obtaining background hidden symmetries of higher dimensional theories. In this way, symmetry structures of these backgrounds will be introduced by constructing extended superalgebras and new conclusions about the classification of string theory and M-theory backgrounds, which is an open problem in the literature, will be reached. In the project, the method of obtaining hidden symmetries of higher dimensional manifolds by using hidden symmetries of lower dimensional manifolds will be used in this concept for the first time. The method that will be used in finding the solutions of the Rarita-Schwinger equation which is satisfied by spin 3/2 particles that appear in supergravity theories will be developed for the first time in the scope of this project. Spin raising and lowering operators which are constructed in the literature previously for the aim of establishing relations between the solutions of the field equations satisfied by the lower spin particles will be generalized for the first time to higher spin particles. In that way, starting with the known solutions of the Dirac equation satisfied by spin 1/2 particles and the Maxwell equations satisfied by spin 1 particles, an original method which does not exist in the literature and intended for finding the solutions of the Rarita-Schwinger equation satisfied by spin 3/2 particles will be developed. Contributions to the topics included in the project from Turkey are in tiny amounts in comparison with the world literature. By the studies that will be done in the project, original contributions will be made to the world literature and a step will be taken devoted to increase the activities of Turkey in this field. 

The basic methods that are used in the project are the theory of distributions, gravitational charges, spin geometry and hidden symmetries. The mathematics of the theory of distributions will be developed in correspondence with the subjects of study; the motions of the particles that are either fundamental or composite but gravitationally charged will be investigated as well as the stability of these motions. For the case of single immersions where the metric is piecewise-continious and the stress tensor is discontinious, the motion under the presence of gravitation and other external fields will be resolved by using Einstein’s equations and other field equations. The motion of a condansate foliating a space-time with a smooth metric will be solved by equating the divergence of the associated stress tensor to zero which is consistent with Einstein’s equations. The manifolds which occur during the compactification process of string theories and M-theory are Calabi-Yau manifolds that are Ricci-flat Kähler manifolds and the manifolds that have G2 and Spin(7) holonomy groups. The methods of spin geometry will be extended to these particular manifolds. Some special spinors that are defined in the framework of spin geometry, hidden symmetries generated from them and the extended superalgebra structures will be used in understanding the properties of string theory and M-theory backgrounds. A method related to the lifting of Killing-Yano forms and conformal Killing-Yano forms to higher dimensions will be used for obtaining the hidden symmetries of string and M-theory backgrounds. Spin raising and lowering operators will be built for higher 

spin particles and symmetry operators will be deduced for higher spin field equations and their solutions. 

As outputs of the research that will be done during the realisation of the project, it is aimed that new and original results that does not exist in the literature will be obtained related to the theory of distributions and motion of extended particles, the classification of string theory and M-theory backgrounds and the solutions of Rarita-Schwinger equations. On the other hand, by means of working on these subjects in our country that are ranging from elementary particle physics to nuclear physics and from astrophysics to cosmology, the project has effective importance towards increasing the number of students and researchers that are willing to make theoretical or experimental works in these subjects and also increasing the contribution to the world literature from Turkey. Besides, two school organizations related to subjects of the project are planned for educating MSc and PhD students and a workshop including the participation of scientists working in these subjects is also planned. The scholarship students in the project will find opportunities for doing research and writing their MSc thesis on current topics in world literature. Accordingly, new researchers from our country will be educated in leading subjects of the world literature and it is provided that the contributions to these subjects from our country will increase. The results of the project will generate new problems in quantum gravity theories, string theories and M-theory and related topics and will lead to the developments of national and international new projects.