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Classical anomalies
Haman, Pavel ; Iorio, Alfredo (advisor) ; Jizba, Petr (referee)
Classical anomalies defined as due to centrally extended algebras of Noether charges are discussed. Specific emphasis is given to the (anomalous) conformal symmetry in two dimensions, governed by the Virasoro algebra of the Noether charges. Liouville field theory as an example of a classically anomalous theory is studied, in flat and curved spacetimes. We find that the presence of a central charge leads to an energy-momentum tensor with non- tensorial conformal transformations in flat spacetime and a non-vanishing trace in curved spacetime. A set of improvements of Liouville theory, leading to the Weyl invariance, is given. By explicit calculations it is shown how the conservation of the improved energy-momentum tensor is lost, while the improvements restore the tracelessness and the relation between the chosen improvement and the corresponding subset of preserved diffeomorphisms is given. The non-tensorial transformation rule of the improved energy-momen- tum tensor in a curved spacetime is related back to the central charge of the Virasoro algebra.
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Conformal symmetry and vortices in graphene
Kůs, Pavel ; Iorio, Alfredo (advisor) ; Jizba, Petr (referee)
This study provides an introductory insight into the complex field of graphene and its relativistic-like behaviour. The thesis is opened by an overview to this topic and draws special attention to interesting non-topological vortex solutions of the Liouville equation found by P. A. Horváthy and J.-C. Yéra, which emerge in a context of the Chern-Simons theory [1], [2] and have been put into context of graphene [3], [4]. We introduce the massless Dirac field theory, well describing electronic properties of graphene in the low energy limit, and point to the fact that the action of the massless Dirac field is invariant under Weyl transformations, which has far-reaching consequences. When the graphene membrane is suitably deformed, we assume that the correct description is that of a Dirac field on a curved spacetime. In particular, an important case is that of conformally flat 2+1-dimensional spacetimes. These are obtained when the spatial part of the metric describes a surface of constant intrinsic curvature [3]. In other words, the conformal factor of such spatial metrics has to satisfy the Liouville equation, an important equation of mathematical physics. In this work, we have identified the kind of surfaces to which the Horváthy- Yéra conformal factors, above recalled, correspond, and have provided...
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