Related papers: Spin networks and SL(2,C)-Character varieties

Let R be a commutative ring and let n,m be two positive integers. Let be the polynomial ring in m x n commuting independent variables R. The symmetric group on n letters acts diagonally on A(n,m). We give generators and relations of the…

A biperiodic planar network is a pair $(G,c)$ where $G$ is a graph embedded on the torus and $c$ is a function from the edges of $G$ to non-zero complex numbers. Associated to the discrete Laplacian on a biperiodic planar network is its…

For an important class of arithmetic Dedekind domains O including the ring of integers of not totally complex number fields, we describe explicitly the group of linear characters of SL_2(O). For this, we determine, for arbitrary Dedekind…

We diagonalize the $B$-element of monodromy matrix for noncompact open $SL(2,\mathbb{C})$ spin chain with boundary interaction. The monodromy matrix is defined in terms of $SL(2,\mathbb{C})$ $L$-operator and boundary $K$-matrix. The…

The action of $SL(2, {\bf Z})$ on the integer torus and its quotient by central symmetry and Artin's presentation of three strings braid group $B_{3}$, produces a presentation with parabolic generators $\pmatrix{1& -1\cr 0& 1\cr}$ and…

In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic…

Recall first the algebraic treatment of flows or tensions in a transportation network $N$, i.e. a connected antisymmetric 1-graph $G(X, U)$. Assume that, unusually, we take the values of flows (resp. tensions) in $\mathbb{C}$. So the…

This is a set of notes which reviews and addresses issues in the SL(2,R) conformal field theory, while working primarily in a basis of vertex operators of definite weight under the affine algebra. Following a review of the H3 coset model…

We investigate the spin structure function $g_2 (x, Q^2 )$ in the framework of the operator product expansion and the renormalization group. The twist-3 operators appearing in QCD are examined and their relations are studied. It is noted…

We propose a new version of the $AdS/CFT$ correspondence, where polymer structures similar to Loop Quantum Cosmology spin networks can be induced on an isotropic and homogeneous flat Randall-Sundrum II brane, corresponding to the…

The group $SL(n,{\bf Z})$ acts linearly on $\R^n$, preserving the integer lattice $\Z^{n} \subset \R^{n}$. The induced (left) action on the n-torus $\T^{n} = \R^{n}/\Z^{n}$ will be referred to as the ``standard action''. It has recently…

We show that an element w of a free group F on n generators defines a surjective word map of PSL(2,C)^n onto PSL(2,C) unless w belongs to the second derived subgroup of F. We also describe certain words maps that are surjective on SL(2,C) x…

Culler and Vogtmann defined a simplicial space $O(g)$ called outer space to study the outer automorphism group of the free group $F_g$. Using representation theoretic methods, we give an embedding of $O(g)$ into the analytification of…

The canonical components of SL_2-character varieties of arithmetic two bridge link groups are determined.

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations…

We define extended SL(2,R)/U(1) characters which include a sum over winding sectors. By embedding these characters into similarly extended characters of N=2 algebras, we show that they have nice modular transformation properties. We…

Let $g, n \geq 0$ and $\Sigma = \Sigma_{g, n}$ be a connected oriented surface of genus $g$ with $n$ punctures. The $\mathrm{SL}_2$-character variety of $\Sigma$ has a rigid relative automorphism group, whose elements fix each monodromies…

There are four groups $G$ fitting into a short exact sequence $ 1\rightarrow SL(2,5)\rightarrow G\rightarrow C_2\rightarrow 1, $ where $SL(2,5)$ is the special linear group of $(2\times 2)$-matrices with entries in the field of five…

We provide a novel Neural Network architecture that can: i) output R-matrix for a given quantum integrable spin chain, ii) search for an integrable Hamiltonian and the corresponding R-matrix under assumptions of certain symmetries or other…

We describe on any finitely generated group G the space of maps G->C which satisfy the parallelogram identity, f(xy)+f(xy^{-1})=2f(x)+2f(y). It is known (but not well-known) that these functions correspond to Zariski-tangent vectors at the…