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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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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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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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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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Některá témata diferenciání a zobecněné geometrie s aplikacemi ve fyzice.
Červeň, Tomáš ; Jurčo, Branislav (advisor) ; Valach, Fridrich (referee)
Differential geometry plays a prominent role in many areas of modern theoretical physics, ranging from electromagnetism and Yang-Mills to Einstein's general theory of relativity. It became a standard tool to investigate both the local and global properties of classical and quantum physical systems. Generalized geometry, as it emerged from string theory and has been first formulated mathematically by Hitchin and his students, proves to be a valuable tool for further study of quantum field theory and strings. We rigorously introduce all essential concepts of differential geometry, which are then crucial to build a basic framework of the generalized tangent bundle. 1
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Geometrie symplektických gradovaných variet a jejich morfismů
Zika, Martin ; Jurčo, Branislav (advisor) ; Bugden, Mark (referee)
Graded manifolds naturally arise in the context of Batalin-Vilkovisky quantization as one introduces fields of non-trivial ghost degrees. We study the structures tied to the dy- namics and gauge symmetry of AKSZ models involving the classical master action or the antibracket on symplectic differential non-negatively graded manifolds (NQP manifolds) in the language of sheaves of graded-commutative algebras. We review the one-to-one correspondence between isomorphism classes of Courant algebroids and NQP manifolds of degree 2. Applying the construction of Lagrangian correspondences in the spirit of Weinstein's symplectic category, we extend the one-to-one correspondence on objects to an equivalence of categories. 1
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Nocommutative structures in quantum field theory
Peksová, Lada ; Jurčo, Branislav (advisor) ; Sachs, Ivo (referee) ; Golovko, Roman (referee)
In this thesis, structures defined via modular operads and properads are generalized to their non-commutative analogs. We define the connected sum for modular operads. This way we are able to construct the graded commutative product on the algebra over Feynman transform of the modular operad. This forms a Batalin-Vilkovisky algebra with symmetry given by the modular operad. We transfer this structure to the cohomology via the Homological perturbation lemma. In particular, we consider these constructions for Quantum closed and Quantum open modular operad. As a parallel project we introduce associative analog of Frobenius properad, called Open Frobenius properad. We construct the cobar complex over it and in the spirit of Barannikov interpret algebras over cobar complex as homological differential operators. Furthermore we present the IBA∞-algebras as analog of well-known IBL∞-algebras. 1
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Higher gauge theory
Mrozek, Jan ; Jurčo, Branislav (advisor) ; Bugden, Mark (referee)
This thesis gives a short introduction into the higher gauge algebras. We first in- troduce the BRST formalism in the context of ordinary gauge theories and show the properties that allow us to use it in the context of higher gauge theories. We define the 2-groups and show the correspondence between 2-groups and crossed modules. We then give a brief introduction into the theory of L∞-algebras - we give account of the graded manifolds and Q-manifolds. We give a short account of Homotopy Maurer-Cartan theory and show that it reduces to the BF theory in case of 4-dimensional manifold and 2-term L∞-algebra. 1
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Zobecněná komplexní geometrie
Zika, Martin ; Jurčo, Branislav (advisor) ; Bugden, Mark (referee)
In an attempt to unify the underlying geometry of Hamilton's equations with the language of complex geometry, we motivate the research of generalized com- plex geometry. We construct the structure of a Courant algebroid on the di- rect sum of the tangent and cotangent bundle TM ⊕ T∗ M as we research the Courant bracket. The key notion of an involutive fibre-wise isotropic subbun- dle, a Dirac structure, is introduced and serves to specify a generalized complex structure. Generalized complex submanifolds are mentioned as well as the process of Dirac redution. Generalized complex geometry and the natural mechanisms in the Courant algebroid setting are then utilised as an interpretational tool in mathematical physics and related areas. We study a reduction of the symplectic structure of a harmonic oscillator, reflect on the nature of the Dirac bracket in string theory and relate a solution of a PDE to a generalized complex submanifold through the Monge-Amp`ere equations. 1
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