A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Survey on algebraic aspects of Turaev cobracket and related problems.
The paper describes algebraic operations on surface fundamental groups.
In a previous paper, we defined an operation that generalizes Turaev's cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper we consider the corresponding que…
Goldman and Turaev constructed a Lie bialgebra structure on the free -module generated by free homotopy classes of loops on a surface. Turaev conjectured that his cobracket is zero if and only if is a power of a simple class. Chas constructed examples that show Turaev's conjecture is, unfortunate…
The paper defines a function for knots in Seifert manifolds and connects it to Witten-Reshetikhin-Turaev invariants.
Geometric proof of curve characterization using loop-bundles.
Constructs non-commutative modular vector fields for Poisson manifolds.
We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple closed curves on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provid…
Proof of Goldman-Turaev Lie bialgebra structure using Knizhnik-Zamolodchikov connection.
Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6)(1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of loops on an orientable surface. Chas (Combinatorial Lie bialgebras of curves on surfaces, Topology 43 (2004) 543), b…
Given two free homotopy classes of loops on an oriented surface, it is natural to ask how to compute the minimum number of intersection points of loops in these two classes. We show that for the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of and i…
Study how Turaev-Viro invariants change with cabling operations.
New algorithm simplifies Turaev-Viro invariants computation.
String topology coproduct and Turaev cobracket computed for surfaces.
New invariants from quantum group theory for hyperbolic 3-manifolds.
Researchers compute torsion for homology cylinders, proving it a finite-type invariant.
Paper connects Lie bialgebra and KV problem on genus zero surfaces.
Study Turaev-Viro invariants of 3-manifolds with toroidal boundary.
Paper introduces an invariant for knots in non-orientable manifolds, akin to Turaev's comultiplication.
Method computes centers of Poisson and skein algebras for loops on surfaces.
We consider topological field theories that compute the Reidemeister-Milnor-Turaev torsion in three dimensions. These are the psl(1|1) and the U(1|1) Chern-Simons theories, coupled to a background complex flat gauge field. We use the 3d mirror symmetry to derive the Meng-Taubes theorem, which relates the torsion and th…
New combinatorial approach to Goldman-Turaev Lie bialgebra using cyclic word partitions.
Study cobrackets on flat GL(1, R)-bundles over surfaces, finding none for Turaev cobracket.
New structure found in loops on quasi-surfaces.
We prove for the Reidemeister-Turaev torsion of closed oriented three-manifolds some finiteness properties in the sense of Goussarov and Habiro, that is, with respect to some cut-and-paste operations which preserve the homology type of the manifolds. In general, those properties require the manifolds to come equipped w…
Decomposes Goldman-Turaev Lie bialgebra via cutting a surface.
A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string …
The vector space $\V$ generated by the conjugacy classes in the fundamental group of an orientable surface has a natural Lie cobracket $\mapδ{\V}{\V\times \V}$. For negatively curved surfaces, can be computed from a geodesic representative as a sum over transversal self-intersection points. In particular is zer…
Chern-Simons and Reshetikhin-Turaev theories are shown equivalent.
We describe a construction of fibrewise inner products on the cotangent bundle of the smooth free loop space of a Riemannian manifold. Using this inner product, we construct an operator over the loop space of a string manifold which is directly analogous to the Dirac operator of a spin manifold.
New method for calculating loop operations on surfaces.
In this note, we consider the problem of constructing knot invariants from Yang-Baxter operators associated to (unitary associative) algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to construct knot invariants, as modified by Murakami, invariably produ…
The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.
Motivated by recent developments in the AdS/CFT correspondence, we provide several alternative bulk descriptions of an arbitrary Wilson loop operator in Chern-Simons theory. Wilson loop operators in Chern-Simons theory can be given a description in terms of a configuration of branes or alternatively anti-branes in the …
We study Wilson-'t Hooft loop operators in a class of N=2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We t…
We obtain generalizations of some results of Turaev relating leading order terms of the Turaev torsion of closed, oriented, connected 3-manifolds to certain ``determinants'' derived from cohomology operations such as the alternate trilinear form on the first cohomology group given by cup product. These determinants unf…
Solves Goldman-Turaev formality problem for higher genus surfaces.
For loop groups (free and based), we compute the exact order of the curvature operator of the Levi-Civita connection depending on a Sobolev space parameter. This extends results of Freed and Maeda-Rosenberg-Torres.
Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be tur…
We study the supersymmetric Wilson loop as introduced by Caron-Huot, which attaches to lightlike polygons certain edge and vertex operators, whose shape is determined by supersymmetry constraints. We state explicit formulas for the vertex operators to all orders in the Graßmann expansion, thus filling a gap in the lite…
We construct a space of string diagrams, which are a type of fatgraph with some additional data, and show that there are string topology operations on the chains of the free loop space of a closed Riemannian manifold which are parameterized by the chains on the space of string diagrams. These operations are shown to re…
Study of operators on loop spaces using Fermionic calculus and stochastic methods.
We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.
Using geometric quantization, we represent curve operators in the TQFT of Witten-Reshetikhin-Turaev with jauge group SU_2 as Toeplitz operators with symbols corresponding to trace functions. As an application, we show that eigenvectors of these operators are concentrated near the level sets of these trace functions, an…
We express the index of the Dirac operator on symplectic quotients of a Hamiltonian loop group manifold with proper moment map in terms of fixed point data.
Stochastic representation for determinants derived from Brownian loop soups.
We show that for every spherical category $\C$ with invertible dimension, the Turaev-Viro TQFT admits a splitting into blocks which come from an HQFT, called the Turaev-Viro HQFT. The Turaev-Viro HQFT has the classifying space $B\grad$ as target space, where $\grad$ is a group obtained from the category $\C$. This cons…
Quantum model for knotted graphs from knot theory.