arXiv subjects
Publications and source records attributed to Pavol Ševera.
This note is a detailed review of the geometry behind the Batalin-Vilkovisky formalism and how it fits into the framework of the quantum odd symplectic category and the odd quantization functor.
Let $Σ$ be a compact oriented 2-manifold (possibly with boundary), and let $\mathcal G_Σ$ be the linear span of free homotopy classes of closed oriented curves on $Σ$ equipped with the Goldman Lie bracket $[\cdot, \cdot]_\mathrm{Goldman}$ defined in terms of intersections of curves. A theorem of Goldman gives rise to a Lie homomorphism $Φ^\mathrm{even}$ from $(\mathcal G_Σ, [\cdot, \cdot]_\text{Goldman})$ to functions on the moduli space of flat connections $\mathcal{M}_Σ(G)$ for $G=U(N), GL(N)$, equipped with the Atiyah-Bott Poisson bracket. The space $\mathcal{G}_Σ$ also carries the Turaev Lie cobracket $δ_\mathrm{Turaev}$ defined in terms of self-intersections of curves. In this paper, we address the following natural question: which geometric structure on moduli spaces of flat connections corresponds to the Turaev cobracket? We give a constructive answer to this question in the following context: for $G$ a Lie supergroup with an odd invariant scalar product on its Lie superalgebra, and for nonempty $\partialΣ$, we show that the moduli space of flat connections $\mathcal{M}_Σ(G)$ carries a natural Batalin-Vilkovisky (BV) structure, given by an explicit combinatorial Fock-Rosly formula. Furthermore, for the queer Lie supergroup $G=Q(N)$, we define a BV-morphism $Φ^\mathrm{odd}\colon \wedge \mathcal{G}_Σ \to \mathrm{Fun}(\mathcal{M}_Σ(Q(N)))$ which replaces the Goldman map, and which captures the information both on the Goldman bracket and on the Turaev cobracket. The map $Φ^\mathrm{odd}$ is constructed using the "odd trace" function on $Q(N)$.
Polylogarithmic functions (polylogs) in $n$ variables can be viewed as elements of $(U\mathfrak{p}_{m})^*$, the dual of the universal enveloping algebra of the Lie algebra $\mathfrak{p}_{m}$ of infinitesimal spherical pure braids with $m=n+3$ strands. Polylogs with $m=4,5$ are used in the theory relating double shuffle relations and Drinfeld associators \cite{furusho_double_2011}. We give explicit formulas for elements of $(U\mathfrak{p}_{m})^*$ representing polylogs, and compute the left ideal $J_{m} \subset U\mathfrak{p}_{m}$ given by their joint kernel. We introduce Lie subalgebras $\mathfrak{k}_{m}=\mathfrak{p}_{m} \cap J_{m}$, and we compute them for $m=4, 5$.
A simple way of obtaining robust estimates of the "center" (or the "location") and of the "scatter" of a dataset is to use the maximum likelihood estimate with a class of heavy-tailed distributions, regardless of the "true" distribution generating the data. We observe that the maximum likelihood problem for the Cauchy distributions, which have particularly heavy tails, is geodesically convex and therefore efficiently solvable (Cauchy distributions are parametrized by the upper half plane, i.e. by the hyperbolic plane). Moreover, it has an appealing geometrical meaning: the datapoints, living on the boundary of the hyperbolic plane, are attracting the parameter by unit forces, and we search the point where these forces are in equilibrium. This picture generalizes to several classes of multivariate distributions with heavy tails, including, in particular, the multivariate Cauchy distributions. The hyperbolic plane gets replaced by symmetric spaces of noncompact type. Geodesic convexity gives us an efficient numerical solution of the maximum likelihood problem for these distribution classes. This can then be used for robust estimates of location and spread, thanks to the heavy tails of these distributions.
We describe a method for quantization of Poisson Hopf algebras in $\mathbb Q$-linear symmetric monoidal categories. It is compatible with tensor products and can also be used to produce braided Hopf algebras. The main idea comes from the fact that nerves of groups are symmetric simplicial sets. Nerves of Hopf algebras then turn out to be braided rather than symmetric and nerves of Poisson Hopf algebras to be infinitesimally braided. The problem is thus solved via the standard machinery of Drinfeld associators.
We introduce left central and right central functions and left and right leaves in quasi-Poisson geometry, generalizing central (or Casimir) functions and symplectic leaves from Poisson geometry. They lead to a new type of (quasi-)Poisson reduction, which is both simpler and more general than known quasi-Hamiltonian reductions. We study these notions in detail for moduli spaces of flat connections on surfaces, where the quasi-Poisson structure is given by an intersection pairing on homology.
We find a Chern-Simons propagator on the ball with the chiral boundary condition. We use it to study perturbatively Chern-Simons boundary conditions related to 2-dim $σ$-models and to Poisson-Lie T-duality. In particular, we find their renormalization group flow, given by the generalized Ricci tensor. Finally we briefly discuss what happens when the Chern-Simons theory is replaced by a Courant $σ$-model or possibly by a more general AKSZ model.
We consider a TFT on the product of a manifold with an interval, together with a topological and a non-topological boundary condition imposed at the two respective ends. The resulting (in general higher gauge) field theory is non-topological, with different choices of the topological conditions leading to field theories dual to each other. In particular, we recover the electric-magnetic duality, the Poisson-Lie T-duality, and we obtain new higher analogues thereof.
We consider the problem of integration of L_\infty-algebroids (differential graded manifolds) to L_\infty-groupoids. We first construct a "big" Kan simplicial manifold (Fréchet or Banach) whose points are solutions of a (generalized) Maurer-Cartan equation. The main analytic trick in our work is an integral transformation sending the solutions of the Maurer-Cartan equation to closed differential forms. Following ideas of Ezra Getzler we then impose a gauge condition which cuts out a finite-dimensional simplicial submanifold. This "smaller" simplicial manifold is (the nerve of) a local Lie k-groupoid. The gauge condition can be imposed only locally in the base of the L_\infty-algebroid; the resulting local k-groupoids glue up to a coherent homotopy, i.e. we get a homotopy coherent diagram from the nerve of a good cover of the base to the (simplicial) category of local Lie k-groupoids. Finally we show that a m-symplectic differential graded manifold integrates to a local m-symplectic Lie k-groupoid; globally these assemble to form an A_\infty-functor. As a particular case for m=2 we obtain integration of Courant algebroids.
We reexamine the notions of generalized Ricci tensor and scalar curvature on a general Courant algebroid, reformulate them using objects natural w.r.t. pull-backs and reductions, and obtain them from the variation of a natural action functional. This allows us to prove, in a very general setup, the compatibility of the Poisson-Lie T-duality with the renormalization group flow and with string background equations. We thus extend the known results to a much wider class of dualities, including the cases with gauging (so called dressing cosets, or equivariant Poisson-Lie T-duality). As an illustration, we use the formalism to provide new classes of solutions of modified supergravity equations on symmetric spaces.
We describe a simple procedure for constructing a Lax pair for suitable 2-dimensional $σ$-models appearing in Poisson-Lie T-duality
These letters, written in 1998-2000, contain various basic results about Courant algebroids (CAs), such as classification of exact and transitive CAs, reduction of CAs, description in terms of symplectic dg manifolds, a canonical generating Dirac operator, and a relation with Poisson-Lie T-duality.
We describe a new method of quantization of Lie bialgebras, based on a construction of Hopf algebras out of a cocommutative coalgebra and a braided comonoidal functor.
We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson-Lie T-duality is compatible with the 1-loop renormalization group.
We give a "holographic" explanation of Poisson-Lie T-duality in terms of Chern-Simons theory (or, more generally, in terms of Courant sigma-models) with appropriate boundary conditions.
We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson Lie groups are nilpotent radicals of parabolic subgroups. We also provide examples of moment maps in this new context coming from moduli spaces of flat connections on surfaces.
In this note, we revisit the quantization of Lie bialgebras described by the second author, placing it in the more general framework of the quantization of moduli spaces developed in our previous work. In particular, we show that embeddings of quilted surfaces (which are compatible with the choice of skeleton) induce morphisms between the corresponding quantized moduli spaces of flat connections. As an application, we describe quantizations of both the variety of Lagrangian subalgebras and the de-Cocini Procesi wonderful compactification, which are compatible with the action of the (quantized) Poisson Lie group.
Let G be a Lie group endowed with a bi-invariant pseudo-Riemannian metric. Then the moduli space of flat connections on a principal G-bundle, P\to Σ, over a compact oriented surface, Σ, carries a Poisson structure. If we trivialize P over a finite number of points on the boundary of Σ, then the moduli space carries a quasi-Poisson structure instead. Our first result is to describe this quasi-Poisson structure in terms of an intersection form on the fundamental groupoid of the surface, generalizing results of Massuyeau and Turaev. Our second result is to extend this framework to quilted surfaces, i.e. surfaces where the structure group varies from region to region and a reduction (or relation) of structure occurs along the borders of the regions, extending results of the second author. We describe the Poisson structure on the moduli space for a quilted surface in terms of an operation on spin networks, i.e. graphs immersed in the surface which are endowed with some additional data on their edges and vertices. This extends the results of various authors.