Ing. Ivo Petr, Ph.D.

Článek

10.1093/ptep/ptae054

Katedra aplikované matematiky

Poisson-Lie T-duality/plurality was recently generalized to Jacobi-Lie T-plurality formulated in terms of Double Field Theory and based on Leibniz algebras given by structure coefficients $f_{ab}{}^{c}, f_{c}{}^{ab},$ and $Z_a, Z^a$. We investigate three- and four-dimensional sigma models corresponding to six-dimensional Leibniz algebras with $f_b{}^{ba} \neq 0$, $Z^a=0$. We show that these algebras are plural one to another and, moreover, to an algebra with $f_b{}^{ba}= 0$, $Z^a=0$. These pluralities are used for construction of Jacobi--Lie models. It was conjectured that plural models should satisfy generalized supergravity equations. We have found examples of models satisfying ``true'' generalized supergravity equations where no trivialization to usual supergravity equations s is possible. On the other hand, we show that there are also models corresponding to algebras with $f_b{}^{ba}\neq 0$, $Z^a=0$ where the Killing vector appearing in generalized supergravity equations either vanishes or can be removed by suitable gauge transformation. Such models then satisfy usual supergravity equations, i.e. vanishing beta-function equations.

Článek

10.1140/epjc/s10052-021-09254-x

Katedra aplikované matematiky

In this paper we investigate Poisson–Lie transformation of dilaton and vector field J appearing in generalized supergravity equations. While the formulas appearing in literature work well for isometric sigma models, we present examples for which generalized supergravity equations are not preserved. Therefore, we suggest modification of these formulas.

Článek

10.1007/JHEP04(2020)068

Katedra aplikované matematiky

Poisson--Lie T-duality and plurality are important solution generating techniques in string theory and (generalized) supergravity. Since duality/plurality does not preserve conformal invariance, the usual beta function equations are replaced by Generalized Supergravity Equations containing vector $\mathcal{J}$. In this paper we apply Poisson--Lie T-plurality on Bianchi cosmologies. We present a formula for the vector $\mathcal{J}$ as well as transformation rule for dilaton, and show that plural backgrounds together with this dilaton and $\mathcal{J}$ satisfy the Generalized Supergravity Equations. The procedure is valid also for non-local dilaton and non-constant $\mathcal{J}$. We also show that $Div\,\Theta$ of the non-commutative structure $\Theta$ used for non-Abelian T-duality or integrable deformations does not give correct $\mathcal{J}$ for Poisson--Lie T-plurality.

Článek

10.1140/epjc/s10052-020-08446-1

Katedra aplikované matematiky

In previous papers we have presented many purely bosonic solutions of Generalized Supergravity Equations obtained by Poisson-Lie T-duality and plurality of flat and Bianchi cosmologies. In this paper we focus on their compactifications and identify solutions that can be interpreted as T-folds. To recognize T-folds we adopt the language of Double Field Theory and discuss how Poisson-Lie T-duality/plurality fits into this framework. As a special case we confirm that all non-Abelian T-duals can be compactified as T-folds.

Článek

10.1007/JHEP04(2019)157

Katedra aplikované matematiky

We investigate (non-)Abelian T-duality from the perspective of Poisson-Lie T-plurality. We show that sigma models related by duality/plurality are given not only by Manin triples obtained from decompositions of Drinfel’d double, but also by their particular embeddings, i.e. maps that relate bases of these decompositions. This allows us to get richer set of dual or plural sigma models than previously thought. That’s why we ask how T-duality is defined and what should be the “canonical” duality or plurality transformation.

Článek

10.1140/epjc/s10052-019-7356-5

Katedra aplikované matematiky

We investigate a special class of Poisson--Lie T-plurality transformations of Bianchi cosmologies invariant with respect to non-semisimple Bianchi groups. For six-dimensional semi-Abelian Manin triples $\mathfrak{b}\bowtie\mathfrak{a}$ containing Bianchi algebras $\mathfrak{b}$ we identify general forms of Poisson--Lie identities and dualities. We show that these can be decomposed into simple factors, namely automorphisms of Manin triples, B-shifts, $\beta$-shifts, and ``full'' or ``factorized'' dualities. Further, we study effects of these transformations and utilize the decompositions to obtain new backgrounds which, supported by corresponding dilatons, satisfy Generalized Supergravity Equations.

Článek

10.1088/1361-6382/aaaeed

Katedra aplikované matematiky

By addition of non-zero, but torsionless B -field, we expand the classification of (non-)Abelian T-duals of the flat background in four dimensions with respect to 1, 2, 3 and 4D subgroups of the Poincaré group. We discuss the influence of the additional B -field on the process of dualization, and identify essential parts of the torsionless B -field that cannot in general be eliminated by coordinate or gauge transformation of the dual background. These effects are demonstrated using particular examples. Due to their physical importance, we focus on duals whose metrics represent plane-parallel (pp-)waves. Besides the previously found metrics, we find new pp-waves depending on parameters originating from the torsionless B -field. These pp-waves are brought into their standard forms in Brinkmann and Rosen coordinates.

Článek

10.1088/1361-6382/aa7908

Katedra aplikované matematiky

We give the classification of T-duals of the flat background in four dimensions with respect to one-, two-, and three-dimensional subgroups of the Poincaré group using non-Abelian T-duality with spectators. As duals we find backgrounds for sigma models in the form of plane-parallel waves or diagonalizable curved metrics often with torsion. Among others, we find exactly solvable time-dependent isotropic pp-wave, singular pp-waves, or generalized plane wave (K-model).

Článek

10.1088/0264-9381/32/3/035005

Katedra aplikované matematiky

We give a classification of non-Abelian T-duals ('T' standing for topological or toroidal) of the flat metric in D = 4 dimensions with respect to the four-dimensional continuous subgroups of the Poincaré group. After dualizing the flat background, we identify the majority of dual models as conformal sigma models in plane-parallel wave backgrounds, most of them having torsion. We give their form in Brinkmann coordinates. We find, besides the plane-parallel waves, several diagonalizable curved metrics with nontrivial scalar curvature and torsion. Using the non-Abelian T-duality, we find general solutions of the classical field equations for all the sigma models in terms of dʼAlembert solutions of the wave equation.