|
|
Algebras over operads and properads
Peksová, Lada ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Operads are objects that model operations with several inputs and one output. We define such structures in the context of graphs, namely oriented trees. Then we generalize operads to properads and modular operads by taking general graphs with, or without, orientation. Further we construct the cobar complex of operads and properads and illustrate the construction on the examples of the associative operad Ass and the Frobenius properad Frob. Algebras over the cobar complex of operads correspond to certain homotopy algebras, for our example of Ass it is A1. We find its Maurer-Cartan equation and convert it from coderivations to derivations. Similarly we find the Maurer-Cartan equation for cobar complex of Frobenius properad. Powered by TCPDF (www.tcpdf.org)
|
|
|
Generalized metric and gravity
Vrábel, Juraj ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Based on the knowledge from differential geometry, the generalized geometry is introduced. As a consequence of the symmetries in this new geometry, a B-field, known from the string theory, inherently emerges. Generalized metric based on ordinary metric tensor and the B-field will be established as well. This allows to construct connection in the framework of generalized geometry and develop a Riemannian generalized geometry. From this point, it is a straightforward way to the replacement of an ordinary scalar curvature by the generalized one in Einstein-Hilbert action. Obtained action closely resembles the supergravity action, especially the bosonic part.
|
|
|
AKSZ formalism and applications
Bialas, Filip ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Generalization of manifolds to the case of both commuting and anticommut- ing variables - Z-graded manifolds are described in this thesis. The language of categories and algebraic geometry is used for defining them and generalizing a few geometrical concepts such as vector fields, differential forms, and symplectic geometry. In the rest of the text, AKSZ construction is described. This construc- tion unifies a few topological field theories by constructing an action functional which is a solution to the classical BV master equation. We will describe one such theory (Poisson sigma model) using AKSZ formalism in greater detail. 1
|
|
|
Courant algebroid connections and string effective actions
Jurčo, B. ; Vysoký, Jan
Courant algebroids are a natural generalization of quadratic Lie algebras, appearing in various contexts in mathematical physics. A connection on a Courant algebroid gives an analogue of a covariant derivative compatible with a given fiber-wise metric. Imposing further conditions resembling standard Levi-Civita connections, one obtains a class of connections whose curvature tensor in certain cases gives a new geometrical description of equations of motion of low energy effective action of string theory. Two examples are given. One is the so called symplectic gravity, the second one is an application to the the so called heterotic reduction. All necessary definitions, propositions and theorems are given in a detailed and self-contained way.
|
|
|
Algebras over operads and properads
Peksová, Lada ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Operads are objects that model operations with several inputs and one output. We define such structures in the context of graphs, namely oriented trees. Then we generalize operads to properads and modular operads by taking general graphs with, or without, orientation. Further we construct the cobar complex of operads and properads and illustrate the construction on the examples of the associative operad Ass and the Frobenius properad Frob. Algebras over the cobar complex of operads correspond to certain homotopy algebras, for our example of Ass it is A1. We find its Maurer-Cartan equation and convert it from coderivations to derivations. Similarly we find the Maurer-Cartan equation for cobar complex of Frobenius properad. Powered by TCPDF (www.tcpdf.org)
|
|
|
Generalized metric and gravity
Vrábel, Juraj ; Jurčo, Branislav (advisor) ; Vysoký, Jan (referee)
Based on the knowledge from differential geometry, the generalized geometry is introduced. As a consequence of the symmetries in this new geometry, a B-field, known from the string theory, inherently emerges. Generalized metric based on ordinary metric tensor and the B-field will be established as well. This allows to construct connection in the framework of generalized geometry and develop a Riemannian generalized geometry. From this point, it is a straightforward way to the replacement of an ordinary scalar curvature by the generalized one in Einstein-Hilbert action. Obtained action closely resembles the supergravity action, especially the bosonic part.
|
guest ::
RSS feed.