A minus sign that used to annoy me but now I know why it is there

We consider two well known constructions of link invariants. One uses skein theory: you resolve each crossing of the link as a linear combination of things that don't cross, until you eventually get a linear combination of links with no crossings, which you turn into a polynomial. The other uses quantum groups: you construct a functor from a topological category to some category of representations in such a way that (directed framed) links get sent to endomorphisms of the trivial representation, which are just rational functions. Certain instances of these two constructions give rise to essentially the same invariants, but when one carefully matches them there is a minus sign that seems out of place. We discuss exactly how the constructions match up in the case of the Jones polynomial, and where the minus sign comes from. On the quantum group side, one is led to use a non-standard ribbon element, which then allows one to consider a larger topological category.Comment: Expository paper. 16 pages. v2: Significant revision, including several new reference

Twisted Supersymmetric Gauge Theories and Orbifold Lattices

We examine the relation between twisted versions of the extended supersymmetric gauge theories and supersymmetric orbifold lattices. In particular, for the $\mathcal{N}=4$ SYM in $d=4$, we show that the continuum limit of orbifold lattice reproduces the twist introduced by Marcus, and the examples at lower dimensions are usually Blau-Thompson type. The orbifold lattice point group symmetry is a subgroup of the twisted Lorentz group, and the exact supersymmetry of the lattice is indeed the nilpotent scalar supersymmetry of the twisted versions. We also introduce twisting in terms of spin groups of finite point subgroups of $R$-symmetry and spacetime symmetry.Comment: 32 page

The eight Cayley-Dickson doubling product

The purpose of this paper is to identify all eight of the basic Cayley-Dickson doubling products. A Cayley-Dickson algebra \cda{N+1} of dimension $2^{N+1}$ consists of all ordered pairs of elements of a Cayley-Dickson algebra \cda{N} of dimension $2^N$ where the product $(a,b)(c,d)$ of elements of \cda{N+1} is defined in terms of a pair of second degree binomials $\left(f(a,b,c,d),g(a,b,c,d)\right)$ satisfying certain properties. The polynomial pair$(f,g)$ is called a `doubling product.' While \cda{0} may denote any ring, here it is taken to be the set $\mathbb{R}$ of real numbers. The binomials $f$ and $g$ should be devised such that \cda{1}=\mathbb{C} the complex numbers, \cda{2}=\mathbb{H} the quaternions, and \cda{3}=\mathbb{O} the octonions. Historically, various researchers have used different yet equivalent doubling products.Comment: 32 candidates for alternate Cayley-Dickson doubling products are winnowed down to 8 products. Author now finds that 4 of those 8 should also be discarded: each allows zero divisors at the eight dimensional stage. The 4 remaining products are denoted in the paper as P0,P3(the standard doubling product),P4,P7. Those four produce algebras isomorphic to the standard Cayley-Dickson algebra

Spectral networks and Fenchel-Nielsen coordinates

We explain that spectral networks are a unifying framework that incorporates both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate systems on moduli spaces of flat SL(2,C) connections, in the following sense. Given a spectral network W on a punctured Riemann surface C, we explain the process of "abelianization" which relates flat SL(2)-connections (with an additional structure called "W-framing") to flat C*-connections on a covering. For any W, abelianization gives a construction of a local Darboux coordinate system on the moduli space of W-framed flat connections. There are two special types of spectral network, combinatorially dual to ideal triangulations and pants decompositions; these two types of network lead to Fock-Goncharov and Fenchel-Nielsen coordinates respectively.Comment: 63 pages; v2: expository improvements, journal versio

Remarks on the symmetric powers of cusp forms on GL(2)

In this paper we prove the following conditional result: Let F be a number field, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assume that all the symmetric powers sym^m(pi) are modular, i.e., define automorphic forms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers are cuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is an abelian twist of the functorial product of pi with the symmetric square of a cusp form pi' on GL(2)/F.Comment: A sentence has been modified in the Introduction. It has nothing to do with the main result of the pape

B(s)→SB_{(s)}\to S transitions in the light cone sum rules with the chiral current

$B_{(s)}$ semi-leptonic decays to the light scalar meson, $B_{(s)}\to S l\bar{\nu}_l, S l \bar{l}\,\,(l=e,\mu,\tau)$, are investigated in the QCD light-cone sum rules (LCSR) with chiral current correlator. Having little knowledge of ingredients of the scalar mesons, we confine ourself to the two quark picture for them and work with the two possible Scenarios. The resulting sum rules for the form factors receive no contributions from the twist-3 distribution amplitudes (DA's), in comparison with the calculation of the conventional LCSR approach where the twist-3 parts play usually an important role. We specify the range of the squared momentum transfer $q^2$, in which the operator product expansion (OPE) for the correlators remains valid approximately. It is found that the form factors satisfy a relation consistent with the prediction of soft collinear effective theory (SCET). In the effective range we investigate behaviors of the form factors and differential decay widthes and compare our calculations with the observations from other approaches. The present findings can be beneficial to experimentally identify physical properties of the scalar mesons.Comment: 22 pages,16 figure