General boundary conditions ("branes") for the Poisson sigma model are studied. They turn out to be labeled by coisotropic submanifolds of the given Poisson manifold. The role played by these boundary conditions both at the classical and at the perturbative quantum level is discussed. It turns out to be related at the classical level to the category of Poisson manifolds with dual pairs as morphisms and at the perturbative quantum level to the category of associative algebras (deforming algebras of functions on Poisson manifolds) with bimodules as morphisms. Possibly singular Poisson manifolds arising from reduction enter naturally into the picture and, in particular, the construction yields (under certain assumptions) their deformation quantization.
In this paper, we attach an $L_\infty$-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), Lê-Oh (locally conformal symplectic case). As a new special case, we attach an $L_\infty$-algebra to any coisotropic submanifold in a contact manifold. The $L_\infty$-algebra of a coisotropic submanifold $S$ governs the (formal) deformation problem of $S$.
We propose a reduction procedure that leads to a reduced star product on the reduced phase space of a `First Class'-constrained system, where no symmetries, group actions or the like are present. For the case that the coisotropic submanifold has codimension 1, we establish a constructive method to compute the reduced star product explicitly. Concluding examples show that this method depends crucially on the constraint function singled out to describe the constrained submanifold and not only on this submanifold itself, and that two different constraint functions for the same constraint submanif…
In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of {\it closed} one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of {\it strong…
We describe the differential graded Lie algebras governing Poisson deformations of a holomorphic Poisson manifold and coisotropic embedded deformations of a coisotropic holomorphic submanifold. In both cases, under some mild additional assumption, we show that the infinitesimal first order deformations induced by the anchor map are unobstructed. Applications include the analog of Kodaira stability theorem for coisotropic deformation and a generalization of McLean-Voisin's theorem about the local moduli space of lagrangian submanifold. Finally it is shown that our construction is homotopy equiv…
In this paper, we study deformations of coisotropic submanifolds in a locally conformal symplectic manifold. Firstly, we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. Secondly, we prove that the formal deformation problem is governed by an $L_\infty$-structure which is a $\frak b$-deformation of strong homotopy Lie algebroids introduced in Oh and Park (2005) in the symplectic context. Then we study deformations of locally conformal symplectic…
We prove a relative version of Kontsevich's formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevich's theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L-infinity-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevich's theorem for supermanifolds, which we a…
We complete the study of the Poisson-Sigma model over Poisson-Lie groups. Firstly, we solve the models with targets $G$ and $G^*$ (the dual group of the Poisson-Lie group $G$) corresponding to a triangular $r$-matrix and show that the model over $G^*$ is always equivalent to BF-theory. Then, given an arbitrary $r$-matrix, we address the problem of finding D-branes preserving the duality between the models. We identify a broad class of dual branes which are subgroups of $G$ and $G^*$, but not necessarily Poisson-Lie subgroups. In particular, they are not coisotropic submanifolds in the general …
We study coisotropic A-branes in the sigma model on a four-torus by explicitly constructing examples. We find that morphisms between coisotropic branes can be equated with a fundamental representation of the noncommutatively deformed algebra of functions on the intersection. The noncommutativity parameter is expressed in terms of the bundles on the branes. We conjecture these findings hold in general. To check mirror symmetry, we verify that the dimensions of morphism spaces are equal to the corresponding dimensions of morphisms between mirror objects.
The BFV-formalism was introduced to handle classical systems, equipped with symmetries. It associates a differential graded Poisson algebra to any coisotropic submanifold $S$ of a Poisson manifold $(M,Π)$. However the assignment (coisotropic submanifold) $\leadsto$ (differential graded Poisson algebra) is not canonical, since in the construction several choices have to be made. One has to fix: 1. an embedding of the normal bundle $NS$ of $S$ into $M$, 2. a connection $\nabla$ on $NS$ and 3. a special element $Ω$. We show that different choices of the connection and $Ω$ -- but with the tubular …
We introduce a technique to realise brane wrapping and double dimensional reduction in the context of AKSZ topological sigma models and also in their target spaces, which are symplectic $L_n$-algebroids (i.e. QP-manifolds). Our procedure involves a novel coisotropic reduction combined with an AKSZ transgression that realises degree-shifting; the reduced QP-manifold depends on topological data of the `wrapped' cycle. We check our procedure against the known rules for fluxes under wrapping in the context of M-theory/type IIA duality, and we also find a new relation between Courant algebroids and…
We prove that non-coisotropic branes in the Poisson-Sigma model are allowed at the quantum level. When the brane is defined by second-class constraints, the perturbative quantization of the model yields the Kontsevich's star product associated to the Dirac bracket on the brane. We also discuss the quantization when both first and second-class constraints are present.
We analyse the problem of boundary conditions for the Poisson-Sigma model and extend previous results showing that non-coisotropic branes are allowed. We discuss the canonical reduction of a Poisson structure to a submanifold, leading to a Poisson algebra that generalizes Dirac's construction. The phase space of the model on the strip is related to the (generalized) Dirac bracket on the branes through a dual pair structure.
We study the Poisson sigma model which can be viewed as a topological string theory. Mainly we concentrate our attention on the Poisson sigma model over a group manifold G with a Poisson-Lie structure. In this case the flat connection conditions arise naturally. The boundary conditions (D-branes) are studied in this model. It turns out that the D-branes are labelled by the coisotropic subgroups of G. We give a description of the moduli space of classical solutions over Riemann surfaces both without and with boundaries. Finally we comment briefly on the duality properties of the model.
This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and pre-Poisson submanifolds, their appearance as branes of the PSM, quantization in terms of L-infinity and A-infinity algebras, and bimodule structures are recalled. As an application, an "almost" functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson-Lie groups.
We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle S^7 --> S^4 making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson--Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU_q(2); it determines a new deformation of the 4-sphere Sigma^4_q as the algebra of coinvariants in S_q^7. We show that the quantum vector bundle associated to the fundamental corepresentation of …
We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an L-infinity algebra, which we construct explicitly. Our machinery is based on Th. Voronov's derived bracket construction. In this paper we consider only geometric applications, including deformations of coisotropic submanifolds in Poisson manifolds, of twisted Poisson structures, and of complex structures within generalized complex geometry. These applications can not be, to our knowledge, obtained by other methods such as operad theory.
These are lecture notes for the course "Poisson geometry and deformation quantization" given by the author during the fall semester 2020 at the University of Zurich. The first chapter is an introduction to differential geometry, where we cover manifolds, tensor fields, integration on manifolds, Stokes' theorem, de Rham's theorem and Frobenius' theorem. The second chapter covers the most important notions of symplectic geometry such as Lagrangian submanifolds, Weinstein's tubular neighborhood theorem, Hamiltonian mechanics, moment maps and symplectic reduction. The third chapter gives an introd…
Motivated by the path integral analysis of boundary conditions in a 3-dimensional topological sigma-model, we suggest a definition of the 2-category associated with a holomorphic symplectic manifold X and study its properties. The simplest objects of this 2-category are holomorphic lagrangian submanifolds of X. We pay special attention to the case when X is the total space of the cotangent bundle of a complex manifold U or a deformation thereof. In the latter case the endomorphism category of the zero section is a monoidal category which is an A-infinity deformation of the 2-periodic derived c…
We study boundary conditions and defects in a three-dimensional topological sigma-model with a complex symplectic target space X (the Rozansky-Witten model). We show that boundary conditions correspond to complex Lagrangian submanifolds in X equipped with complex fibrations. The set of boundary conditions has the structure of a 2-category; morphisms in this 2-category are interpreted physically as one-dimensional defect lines separating parts of the boundary with different boundary conditions. This 2-category is a categorification of the Z/2-graded derived category of X; it is also related to …
