Strings in Spacetime Cotangent Bundle and T-duality

A simple geometric description of T-duality is given by identifying the cotangent bundles of the original and the dual manifold. Strings propagate naturally in the cotangent bundle and the original and the dual string phase spaces are obtained by different projections. Buscher's transformation follows readily and it is literally projective. As an application of the formalism, we prove that the duality is a symplectomorphism of the string phase spaces.Comment: 10 pages, LaTeX (1 reference added

Non-Abelian Momentum-Winding Exchange

A non-Abelian analogue of the Abelian T-duality momentum-winding exchange is described. The non-Abelian T-duality relates $\sigma$-models living on the cosets of a Drinfeld double with respect to its isotropic subgroups. The role of the Abelian momentum-winding lattice is in general played by the fundamental group of the Drinfeld double.Comment: 12 pages, LaTe

Poisson-Lie T-duality and loop groups of Drinfeld doubles

A duality invariant first order action is constructed on the loop group of a Drinfeld double. It gives at the same time the description of both of the pair of \sigma-models related by Poisson-Lie T-duality. Remarkably, the action contains a WZW-term on the Drinfeld double not only for conformally invariant \si-models. The resulting actions of the models from the dual pair differ just by a total derivative corresponding to an ambiguity in specifying a two-form whose exterior derivative is the WZW three-form. This total derivative is nothing but the Semenov-Tian-Shansky symplectic form on the Drinfeld double and it gives directly a generating function of the canonical transformation relating the \si-models from the dual pair

Dual non-abelian duality and the Drinfeld double

The standard notion of the non-abelian duality in string theory is generalized to the class of \sigma-models admitting `non-commutative conserved charges'. Such \sigma-models can be associated with every Lie bialgebra ({\cal G},\tilde{\cal G}) and they posses an isometry group iff the commutant [\tilde{\cal G},\tilde{\cal G}] is not equal to \tilde{\cal G}. Within the enlarged class of the backgrounds the non-abelian duality {\it is} a duality transformation in the proper sense of the word. It exchanges the roles of {\cal G} and \tilde{\cal G} and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. We give explicit formulas for the non-abelian duality transformation for any ({\cal G}, \tilde{\cal G}). The non-abelian analogue of the abelian modular space O(d,d;Z) consists of all maximally isotropic decompositions of the corresponding Drinfeld double.The standard notion of the non-abelian duality in string theory is generalized to the class of $\sigma$-models admitting `non-commutative conserved charges'. Such $\sigma$-models can be associated with every Lie bialgebra $({\cal G},\tilde{\cal G})$ and they posses an isometry group iff the commutant $[\tilde{\cal G},\tilde{\cal G}]$ is not equal to $\tilde{\cal G}$. Within the enlarged class of the backgrounds the non-abelian duality {\it is} a duality transformation in the proper sense of the word. It exchanges the roles of ${\cal G}$ and $\tilde{\cal G}$ and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. We give explicit formulas for the non-abelian duality transformation for any $({\cal G},\tilde{\cal G})$. The non-abelian analogue of the abelian modular space $O(d,d;Z)$ consists of all maximally isotropic decompositions of the corresponding Drinfeld double.The standard notion of the non-Abelian duality in string theory is generalized to the class of σ-models admitting a Poisson-Lie-like 3ymmetry. Such σ-models can be associated with every Lie bialgebra ( g , G ). Within the enlarged class of the backgrouds the non-Abelian duality is a duality transformation in the pacer sense of the word. It exchanges the roles of G and G and it can be interpreted as a symplectomorphism of the phase spaces of the mutually dual theories. We give explicit formulas for the non-Abelian duality transformation for any ( g , G ). The non-Abelian analogue of the Abelian modular space O ( d , d ; Z ) consists of all maximally isotropic decompositions of the corresponding Drinfeld double

L∞L_\infty-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism

We review in detail the Batalin-Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an $L_\infty$-algebra and how quasi-isomorphisms between $L_\infty$-algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern-Simons theories and give some useful shortcuts in usually rather involved computations.Comment: v3: 131 pages, minor improvements, published versio

Six-Dimensional (1,0) Superconformal Models and Higher Gauge Theory

We analyze the gauge structure of a recently proposed superconformal field theory in six dimensions. We find that this structure amounts to a weak Courant-Dorfman algebra, which, in turn, can be interpreted as a strong homotopy Lie algebra. This suggests that the superconformal field theory is closely related to higher gauge theory, describing the parallel transport of extended objects. Indeed we find that, under certain restrictions, the field content and gauge transformations reduce to those of higher gauge theory. We also present a number of interesting examples of admissible gauge structures such as the structure Lie 2-algebra of an abelian gerbe, differential crossed modules, the 3-algebras of M2-brane models and string Lie 2-algebras.Comment: 31+1 pages, presentation slightly improved, version published in JM

Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a "generalized" Rota-Baxter identity of weight zero. Such operators are called generalized Rota-Baxter operators. We will show that generalized Rota-Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator "twisted Rota-Baxter operators" which is a natural generalization of generalized Rota-Baxter operators. It is known that classical Rota-Baxter operators are closely related with dendriform algebras. We will show that twisted Rota-Baxter operators induce NS-algebras which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer-Cartan equation up to homotopy. We will show the twisted Rota-Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota-Baxter condition arises.Comment: 18 pages. Final versio