Surface framed braids

In this paper we introduce the framed pure braid group on $n$ strands of an oriented surface, a topological generalisation of the pure braid group $P_n$. We give different equivalents definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of $P_n$, usually called surface pure braid groups. The notion of surface framed braid groups is also introduced.Comment: 24 pages ; 7 figure

Framed BPS States

We consider a class of line operators in d=4, N=2 supersymmetric field theories which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call "framed BPS states." These include halo bound states similar to those of d=4, N=2 supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the Kontsevich-Soibelman wall-crossing formula for the ordinary BPS particles, by reducing it to the semiprimitive wall-crossing formula. After reducing on S1, the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the "Darboux coordinates" on the moduli space M of the theory. Moreover, we introduce a "protected spin character" which keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of protected spin characters admit a multiplication which defines a deformation of the algebra of functions on M. As an illustration of these ideas, we consider the six-dimensional (2,0) field theory of A1 type compactified on a Riemann surface C. Here we show (extending previous results) that line operators are classified by certain laminations on a suitably decorated version of C, and we compute the spectrum of framed BPS states in several explicit examples. Finally we indicate some interesting connections to the theory of cluster algebras.Comment: 123 pages, 52 figures; v2: minor correction

Framed Hitchin Pairs

We provide a construction of the moduli spaces of framed Hitchin pairs and their master spaces. These objects have come to interest as algebraic versions of solutions of certain coupled vortex equations by work of Lin and Stupariu. Our method unifies and generalizes constructions of several similar moduli spaces. Here are some points which are also of interest in other similar situations: - Our construction does not require the symmetricity condition that the map ^2G x E -> E be zero, usually appearing in the context of Higgs bundles. - We carry out a detailed analysis of the polynomial stability parameter without referring to GIT. This sheds some light on intrinsic properties of such parameter dependent stability concepts. - The construction corrects an inaccuracy in our previous construction of the compactification of the Hitchin space and generalizes it.Comment: To appear in the Revue roumaine de math'ematiques pures et appliq'ee

p-adic framed braids

In this paper we define the $p$-adic framed braid group ${\mathcal F}_{\infty,n}$, arising as the inverse limit of the modular framed braids and we give topological generators for ${\mathcal F}_{\infty, n}$. We also give geometric interpretations for the $p$-adic framed braids. We then construct a $p$-adic Yokonuma-Hecke algebra ${\rm Y}_{\infty,n}(u)$ as the inverse limit of a family of classical Yokonuma-Hecke algebras. These are quotients of the modular framed braid groups over a quadratic relation. We also give topological generators for ${\rm Y}_{\infty,n}(u)$. Finally, we construct on this new algebra a linear trace that supports the Markov property.Comment: 35 pages, 14 figures, LaTex documen

Framed Deformation of Galois Representation

We studied framed deformations of two dimensional Galois representation of which the residue representation restrict to decomposition groups are scalars, and established a modular lifting theorem for certain cases. We then proved a family version of the result, and used it to determine the structure of deformation rings over characteristic zero fields. These can be applied to the study of exceptional zero of p-adic L-function

Framed symplectic sheaves on surfaces

A framed symplectic sheaf on a smooth projective surface $X$ is a torsion-free sheaf $E$ together with a trivialization on a divisor $D\subseteq X$ and a morphism $\Lambda^{2}E\rightarrow\mathcal{O}_{X}$ satisfying some additional conditions. We construct a moduli space for framed symplectic sheaves on a surface, and present a detailed study for $X=\mathbb{P}_{\mathbb{C}}^{2}$. In this case, the moduli space is irreducible and admits an ADHM-type description and a birational proper map into the space of framed symplectic ideal instantons.Comment: 40p. Comments are welcome. Minor changes, Typos correcte

Framed knots at large N

We study the framing dependence of the Wilson loop observable of U(N) Chern-Simons gauge theory at large N. Using proposed geometrical large N dual, this leads to a direct computation of certain topological string amplitudes in a closed form. This yields new formulae for intersection numbers of cohomology classes on moduli of Riemann surfaces with punctures (including all the amplitudes of pure topological gravity in two dimensions). The reinterpretation of these computations in terms of BPS degeneracies of domain walls leads to novel integrality predictions for these amplitudes. Moreover we find evidence that large N dualities are more naturally formulated in the context of U(N) gauge theories rather than SU(N).Comment: 26 pages, harvma

The space of framed chord diagrams as a Hopf module

This note is dedicated to the study of a Hopf module structures on the space of framed chord diagrams and framed graphs. We also introduce a framed version of the chromatic polynomial and propose two methods to construct framed weight systems.Comment: updated version, 18 page

Framed sheaves on projective stacks

Given a normal projective irreducible stack $\mathscr X$ over an algebraically closed field of characteristic zero we consider framed sheaves on $\mathscr X$, i.e., pairs $(\mathcal E,\phi_{\mathcal E})$, where $\mathcal E$ is a coherent sheaf on $\mathscr X$ and $\phi_{\mathcal E}$ is a morphism from $\mathcal E$ to a fixed coherent sheaf $\mathcal F$. After introducing a suitable notion of (semi)stability, we construct a projective scheme, which is a moduli space for semistable framed sheaves with fixed Hilbert polynomial, and an open subset of it, which is a fine moduli space for stable framed sheaves. If $\mathscr X$ is a projective irreducible orbifold of dimension two and $\mathcal F$ a locally free sheaf on a smooth divisor $\mathscr D\subset \mathscr X$ satisfying certain conditions, we consider $(\mathscr{D}, \mathcal{F})$-framed sheaves, i.e., framed sheaves $(\mathcal E,\phi_{\mathcal E})$ with $\mathcal E$ a torsion-free sheaf which is locally free in a neighborhood of $\mathscr D$, and ${\phi_{\mathcal{E}}}_{| \mathscr{D}}$ an isomorphism. These pairs are $\mu$-stable for a suitable choice of a parameter entering the (semi)stability condition, and of the polarization of $\mathscr X$. This implies the existence of a fine moduli space parameterizing isomorphism classes of $(\mathscr{D}, \mathcal{F})$-framed sheaves on $\mathscr{X}$ with fixed Hilbert polynomial, which is a quasi-projective scheme. In an appendix we develop the example of stacky Hirzebruch surfaces. This is the first paper of a project aimed to provide an algebro-geometric approach to the study of gauge theories on a wide class of 4-dimensional Riemannian manifolds by means of framed sheaves on "stacky" compactifications of them. In particular, in a subsequent paper we will use these results to study gauge theories on ALE spaces of type $A_k$.Comment: v1: 62 pages. Comments welcome. v2: 64 pages, typos corrected. Appendix D now contains a formula for the dimension of the moduli spaces of framed sheaves on stacky Hirzebruch surfaces. v3: references added. v4: Typos corrected, references added; minor, inconsequential mistakes in Appendix D correcte

Three-Dimensional 2-Framed TQFTs and Surgery

The notion of 2-framed three-manifolds is defined. The category of 2-framed cobordisms is described, and used to define a 2-framed three-dimensional TQFT. Using skeletonization and special features of this category, a small set of data and relations is given that suffice to construct a 2-framed three-dimensional TQFT. These data and relations are expressed in the language of surgery.Comment: 19 pages, 9 figures using epsfi