Pavol Severa

Integration of exact Courant algebroids

In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [32] inverts our integration.

Quasi-Hamiltonian Groupoids and Multiplicative Manin Pairs

We reformulate notions from the theory of quasi-Poisson g-manifolds in terms of graded Poisson geometry and graded Poisson-Lie groups and prove that quasi-Poisson g-manifolds integrate to quasi-Hamiltonian g-groupoids. We then interpret this result within the theory of Dirac morphisms and multiplicative Manin pairs, to connect our work with more traditional approaches, and also to put it into a wider context suggesting possible generalizations.

Quantization of Poisson Families and of Twisted Poisson Structures

We describe a deformation quantization of a modification of Poisson geometry by a closed 3-form. Under suitable conditions, it gives rise to a stack of algebras. The basic object used for this aim is a kind of families of Poisson structures given by a Maurer–Cartan equation; they are easily quantized using Kontsevichs formality theorem. We conclude with a section on quantization of complex manifolds.

On the Origin of the BV Operator on Odd Symplectic Supermanifolds

Differential forms on an odd symplectic manifold form a bicomplex: one differential is the wedge product with the symplectic form and the other is de Rham differential. In the corresponding spectral sequence the next differential turns out to be the Batalin–Vilkoviski operator.

Formality of the Chain Operad of Framed Little Disks

We extend Tamarkin’s formality of the little disk operad to the framed little disk operad.

Poisson–Lie T-Duality and Courant Algebroids

This note explains Poisson-Lie T-duality from the point of view of Courant algebroids. It contains basically nothing new: all the material is already contained in my letters [9] to Alan Weinstein written in 1998-99, which circulated in the “Poisson community” (including, among others, Anton Alekseev, Paul Bressler, Yvette Kosmann-Schwarzbach and Ping Xu) for some time. During the 16 years since the letters were written, the basic technical tools (e.g. reduction of Courant algebroids [2]) were rediscovered and works linking (Abelian) T-duality and Courant algebroids appeared, notably the paper by Cavalcanti and Gualtieri [3]. I still decided to write my account and include details missing in [9]. Perhaps the most important reason is that I introduced exact Courant algebroids while trying to understand Poisson-Lie T-duality, and I believe that this duality, first introduced in [6], which generalized the usual Abelian T-duality, is essential for understanding of both Courant algebroids and of the world of T-dualities. This note summarizes the first four letters of [9]. In particular, in doesn’t deal with differential graded symplectic geometry and its link with Courant algebroids, which is discussed in the remaining letters. While it’s certainly relevant for PoissonLie T-duality, I decided to exclude it to keep the focus on one thing, and also because I already wrote about it in [10].

Poisson Actions up to Homotopy and their Quantization

Symmetries of Poisson manifolds are in general quantized just to symmetries up to homotopy of the quantized algebra of functions. It is therefore interesting to study symmetries up to homotopy of Poisson manifolds. We notice that they are equivalent to Poisson principal bundles and describe their quantization to symmetries up to homotopy of the quantized algebras of functions, using the formality theorem of Kontsevich.

Poisson-Lie T-duality as a boundary phenomenon of Chern-Simons theory

We give a “holographic” explanation of Poisson-Lie T-duality in terms of Chern-Simons theory (or, more generally, in terms of Courant σ-models) with appropriate boundary conditions.A bstractWe give a “holographic” explanation of Poisson-Lie T-duality in terms of Chern-Simons theory (or, more generally, in terms of Courant σ-models) with appropriate boundary conditions.

(Non-)Abelian Kramers-Wannier Duality And Topological Field Theory

We show for any oriented surface, possibly with a boundary, how to generalize Kramers-Wannier duality to the world of quantum groups. The generalization is motivated by quantization of Poisson-Lie T-duality from the string theory. Cohomologies with quantum coefficients are defined for surfaces and their meaning is revealed. They are functorial with respect to some glueing operations and connected with q-invariants of 3-folds.

Symplectic and Poisson Geometry of the Moduli Spaces of Flat Connections Over Quilted Surfaces

In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective. The moduli spaces we construct in this way include a number of important examples, including Poisson Lie groups and their Homogeneous spaces, moduli spaces for meromorphic connections over Riemann surfaces (following the work of Philip Boalch), and various symplectic groupoids. Realizing these examples as moduli spaces for quilted surfaces provides new insights into their geometry.