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C. Klimcik

Publications and source records attributed to C. Klimcik.

At least 19 recordsLinked to original sources
arXiv2018

Affine Poisson and affine quasi-Poisson T-duality

We generalize the Poisson-Lie T-duality by making use of the structure of the affine Poisson group which is the concept introduced some time ago in Poisson geometry as a generalization of the Poisson-Lie group. We also introduce a new notion of an affine quasi-Poisson group and show that it gives rise to a still more general T-duality framework. We establish for a class of examples that this new T-duality is compatible with the renormalization group flow.

hep-th↗
arXiv2016

Poisson-Lie T-duals of the bi-Yang-Baxter models

We prove the conjecture of Sfetsos, Siampos and Thompson that suitable analytic continuations of the Poisson-Lie T-duals of the bi-Yang-Baxter sigma models coincide with the recently introduced generalized lambda models. We then generalize this result by showing that the analytic continuation of a generic sigma model of "universal WZW-type" introduced by Tseytlin in 1993 is nothing but the Poisson-Lie T-dual of a generic Poisson-Lie symmetric sigma model introduced by Klimcik and Severa in 1995.

hep-th↗
arXiv2012

E pur si muove!

We comment a debate on quantum physics in Slovak TV in which several physicists advocated a point of view that "an electron does not move around the nucleus".

physics.pop-ph↗
arXiv2012

Totally classical Calogero model

We show that the standard Calogero Lax matrix can be interpreted as a function on the fuzzy sphere and the Avan-Talon r-matrix as a function on the direct product of two fuzzy spheres. We calculate the limiting Lax function and r-function when the fuzzy sphere tends to the ordinary sphere and we show that they define an integrable model interpreted as a large N Calogero model by Bordemann, Hoppe and Theisen.

math-ph↗
arXiv2012

The Ruijsenaars self-duality map as a mapping class symplectomorphism

This is a brief review of the main results of our paper arXiv:1101.1759 that contains a complete global treatment of the compactified trigonometric Ruijsenaars-Schneider system by quasi-Hamiltonian reduction. Confirming previous conjectures of Gorsky and collaborators, we have rigorously established the interpretation of the system in terms of flat SU(n) connections on the one-holed torus and demonstrated that its self-duality symplectomorphism represents the natural action of the standard mapping class generator S on the phase space. The pertinent quasi-Hamiltonian reduced phase space turned out to be symplectomorphic to the complex projective space equipped with a multiple of the Fubini-Study symplectic form and two toric moment maps playing the roles of particle-positions and action-variables that are exchanged by the duality map. Open problems and possible directions for future work are also discussed.

math-ph↗
arXiv2012

On the spectra of the quantized action-variables of the compactified Ruijsenaars-Schneider system

A simple derivation of the spectra of the action-variables of the quantized compactified Ruijsenaars-Schneider system is presented. The spectra are obtained by combining Kahler quantization with the identification of the classical action-variables as a standard toric moment map on the complex projective space. The result is consistent with the Schrodinger quantization of the system worked out previously by van Diejen and Vinet.

math-ph↗
arXiv2012

Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction

The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider III(b) system from a quasi-Hamiltonian reduction of the internally fused double SU(n) x SU(n). In particular, the reduced spectral functions depending respectively on the first and second SU(n) factor of the double engender two toric moment maps on the III(b) phase space CP(n-1) that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z) mapping class group of the torus with one boundary component is shown to act on the quasi-Hamiltonian double by automorphisms and, upon reduction, the standard generator S of the mapping class group is proved to descend to the Ruijsenaars self-duality symplectomorphism that exchanges the toric moment maps. We give also two new presentations of this duality map: one as the composition of two Delzant symplectomorphisms and the other as the composition of three Dehn twist symplectomorphisms realized by Goldman twist flows. Through the well-known relation between quasi-Hamiltonian manifolds and moduli spaces, our results rigorously establish the validity of the interpretation [going back to Gorsky and Nekrasov] of the III(b) system in terms of flat SU(n) connections on the one-holed torus.

math-ph↗
arXiv2010

Poisson-Lie interpretation of trigonometric Ruijsenaars duality

A geometric interpretation of the duality between two real forms of the complex trigonometric Ruijsenaars-Schneider system is presented. The phase spaces of the systems in duality are viewed as two different models of the same reduced phase space arising from a suitable symplectic reduction of the standard Heisenberg double of U(n). The collections of commuting Hamiltonians of the systems in duality are shown to descend from two families of `free' Hamiltonians on the double which are dual to each other in a Poisson-Lie sense. Our results give rise to a major simplification of Ruijsenaars' proof of the crucial symplectomorphism property of the duality map.

math-ph↗
arXiv2009

On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models

We consider two families of commuting Hamiltonians on the cotangent bundle of the group GL(n,C), and show that upon an appropriate single symplectic reduction they descend to the spectral invariants of the hyperbolic Sutherland and of the rational Ruijsenaars-Schneider Lax matrices, respectively. The duality symplectomorphism between these two integrable models, that was constructed by Ruijsenaars using direct methods, can be then interpreted geometrically simply as a gauge transformation connecting two cross sections of the orbits of the reduction group.

math-ph↗
arXiv2009

One loop renormalizability of the Poisson-Lie sigma models

We present the proof of the one loop renormalizability in the strict field theoretic sense of the Poisson-Lie sigma models. The result is valid for any Drinfeld double and it relies solely on the Poisson-Lie structure encoded in the target manifold.

hep-th↗
arXiv2009

Poisson-Lie generalization of the Kazhdan-Kostant-Sternberg reduction

The trigonometric Ruijsenaars-Schneider model is derived by symplectic reduction of Poisson-Lie symmetric free motion on the group U(n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U(n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars-Schneider Lax matrix upon reduction.

math-ph↗
arXiv2008

On integrability of the Yang-Baxter $\si$-model

We prove the integrability of the Yang-Baxter $\si$-model which is the Poisson-Lie deformation of the principal chiral model. We find also an explicit one-to-one map transforming every solution of the principal chiral model into a solution of the deformed model. With the help of this map, the standard procedure of the dressing of the principal chiral solutions can be directly transferred into the deformed Yang-Baxter context.

hep-th↗
arXiv2005

Nested T-duality

We identify the obstructions for T-dualizing the boundary WZW model and make explicit how they depend on the geometry of branes. In particular, the obstructions disappear for certain brane configurations associated to non-regular elements of the Cartan torus. It is shown in this case that the boundary WZW model is "nested" in the twisted boundary WZW model as the dynamical subsystem of the latter.

hep-th↗
arXiv2004

q-deformation of $z\to {az+b\over cz+d}$

We construct the action of the quantum double of $\uq$ on the standard Podleś sphere and interpret it as the quantum projective formula generalizing to the q-deformed setting the action of the Lorentz group of global conformal transformations on the ordinary Riemann sphere.

math-ph↗
arXiv2003

On Hopf anomaly

This is a very brief but selfcontained review of the concept of quantum group symmetries and their anomalies. Remarkably, general constructions can be very simply illustrated on the standard harmonic oscillator which is shown to possess a non-commutative and non-cocommutative anomalous quantum group symmetry.

hep-th↗
arXiv2003

One-loop renormalizability of all 2d dimensional Poisson-Lie sigma-models

We perform a systematic study of the one-loop renormalizability of all Poisson-Lie T-dualizable $\si$-models with two-dimensional targets. We show that whatever Drinfeld double and whatever matrix of coupling constants we consider the corresponding $\si$-model is always one-loop renormalizable in the strict field theoretical sense. Moreover, in all cases, the RG flow in the space of the coupling constants is compatible with the Poisson-Lie T-duality.

hep-th↗