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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
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Geometry of Poisson-Lie T-duality
Svoboda, Josef ; Jurčo, Branislav (advisor) ; Deser, Andreas (referee)
In this thesis we study geometry of Poisson-Lie T-duality. We develop the language of Lie and Courant algebroids and study generalized metrics on them. Then we use Dirac structures and generalized isometries to formulate a general version of Poisson-Lie T-duality, a non-abelian version of T-duality, known from string theory.
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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.
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Finite dimensional BV formalism
Skácel, Ondřej ; Jurčo, Branislav (advisor) ; Pulmann, Ján (referee)
We study the BV formalism in both the infinite- and finite-dimensional case. We outline the use in QFT and provide explicit calculations for the Yang- Mills theories. We summarize the flat finite-dimensional case and show an equiv- alence bewteen two definitions of the effective observable. An overview of graded geometry is included. 1
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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)
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S-matrix and homological perturbation lemma
Pulmann, Ján ; Jurčo, Branislav (advisor) ; Doubek, Martin (referee)
Loop homotopy Lie algebras, which appear in closed string field theory, are a generalization of homotopy Lie algebras. For a loop homotopy Lie algebra, we transfer its structure on its homology and prove that the transferred structure is again a loop homotopy algebra. Moreover, we show that the homological perturbation lemma can be regarded as a path integral, integrating out the degrees of freedom which are not in the homology. The transferred action then can be interpreted as an effective action in the Batalin-Vilkovisky formalism. A review of necessary results from Batalin- Vilkovisky formalism and homotopy algebras is included as well. Powered by TCPDF (www.tcpdf.org)
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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.
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Operads and field theory
Pulmann, Ján ; Jurčo, Branislav (advisor) ; Křižka, Libor (referee)
Operads and their variants, modular and cyclic operads, naturally describe compositions of objects of various types. We provide an accessible introduction to the theory of operads, the formalism for modular operads from [1] and modern application of modular operads to physics, due to Barannikov [2]. Through examples, we introduce Batalin-Vilkovisky formalism as a tool for cohomological integration of path integral in quantum field theories. A master equation, consistency condition for action, follows from this formalism. Solutions to master equation also describe algebras over Feynman transform of a modular operad. We explore the master equation defined in terms of modular operad and review an application to closed string field theory. [1] Martin Doubek, Branislav Jurco, and Korbinian Muenster. Modular operads and the quantum open-closed homotopy algebra. 2013. arXiv: 1308.3223 [math-AT]. [2] Serguei Barannikov. "Modular operads and Batalin-Vilkovisky geometry". In: International Mathematics Research Notices 2007 (2007), rnm075.
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