scholarly journals ON THE GENERALIZED VOLUME CONJECTURE AND REGULATOR

2008 ◽  
Vol 10 (supp01) ◽  
pp. 1023-1032 ◽  
Author(s):  
WEIPING LI ◽  
QINGXUE WANG
Keyword(s):  
Line Bundle ◽  
Chern Class ◽  
Quantization Condition ◽  
Character Variety ◽  
Characteristic Class ◽  
Volume Conjecture ◽  
Smooth Part ◽  
First Chern Class ◽  
Chern Simons ◽  
Secondary Characteristic

In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL2(ℂ) character variety of the hyperbolic knot in S3. Furthermore, we prove that the corresponding ℂ*-valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the smooth part of the character variety, where the flat line bundle is the pullback of the universal Heisenberg line bundle over ℂ* × ℂ*. Based on this result, we give a reformulation of Gukov's generalized volume conjecture from a motivic perspective.

scholarly journals The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

2021 ◽  
Vol 9 ◽  
Author(s):  
Patrick Graf ◽  
Martin Schwald
Keyword(s):  
Chern Class ◽  
Cohomology Group ◽  
Canonical Bundle ◽  
Deformation Space ◽  
Projective Varieties ◽  
Quotient Singularities ◽  
Finite Quotient ◽  
First Chern Class ◽  
Small Deformations

Abstract Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

The First Chern Class of the Verlinde Bundles

2015 ◽  
pp. 87-111 ◽  
Author(s):  
Alina Marian ◽  
Dragos Oprea ◽  
Rahul Pandharipande
Keyword(s):  
Chern Class ◽  
First Chern Class

Topological invariance of the Hall conductance and quantization

2015 ◽  
Vol 29 (24) ◽  
pp. 1550135
Author(s):  
Paul Bracken
Keyword(s):  
Chern Class ◽  
Fiber Bundle ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Hall Conductance ◽  
Integer Multiple ◽  
Principal Fiber Bundle ◽  
Topological Invariance ◽  
First Chern Class

It is shown that the Kubo equation for the Hall conductance can be expressed as an integral which implies quantization of the Hall conductance. The integral can be interpreted as the first Chern class of a [Formula: see text] principal fiber bundle on a two-dimensional torus. This accounts for the conductance given as an integer multiple of [Formula: see text]. The formalism can be extended to deduce the fractional conductivity as well.

ON SOME SYMPLECTIC ASPECTS OF KNOT FRAMINGS

2006 ◽  
Vol 15 (07) ◽  
pp. 883-912 ◽  
Author(s):  
ALBERTO BESANA ◽  
MAURO SPERA
Keyword(s):  
Integral Formula ◽  
Geometric Quantization ◽  
Quantum Hall ◽  
Quantization Condition ◽  
Fractional Quantum ◽  
Fractional Quantum Hall ◽  
Linking Number ◽  
Witten Theory ◽  
Plane Projection ◽  
Chern Simons

The present article delves into some symplectic features arising in basic knot theory. An interpretation of the writhing number of a knot (with reference to a plane projection thereof) is provided in terms of a phase function analogous to those encountered in geometrical optics, its variation upon switching a crossing being akin to the passage through a caustic, yielding a knot theoretical analogue of Maslov's theory, via classical fluidodynamical helicity. The Maslov cycle is given by knots having exactly one double point, among those having a fixed plane shadow and lying on a semi-cone issued therefrom, which turn out to build up a Lagrangian submanifold of Brylinski's symplectic manifold of (mildly) singular knots. A Morse family (generating function) for this submanifold is determined and can be taken to be the Abelian Chern–Simons action plus a source term (knot insertion) appearing in the Jones–Witten theory. The relevance of the Bohr–Sommerfeld conditions arising in geometric quantization are investigated and a relationship with the Gauss linking number integral formula is also established, together with a novel derivation of the so-called Feynman–Onsager quantization condition. Furthermore, an additional Chern–Simons interpretation of the writhe of a braid is discussed and interpreted symplectically, also making contact with the Goldin–Menikoff–Sharp approach to vortices and anyons. Finally, a geometrical setting for the ground state wave functions arising in the theory of the Fractional Quantum Hall Effect is established.

scholarly journals Chern–Simons line bundle on Teichmüller space

2014 ◽  
Vol 18 (1) ◽  
pp. 327-377 ◽  
Author(s):  
Colin Guillarmou ◽  
Sergiu Moroianu
Keyword(s):  
Line Bundle ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Chern Simons

scholarly journals Remarks on Determinant Line Bundles, Chern-Simons Forms and Invariants

2002 ◽  
Vol 91 (1) ◽  
pp. 5 ◽  
Author(s):  
Johan L. Dupont ◽  
Flemming Lindblad Johansen
Keyword(s):  
Line Bundle ◽  
Line Bundles ◽  
Manifolds With Boundary ◽  
Principal Bundles ◽  
Determinant Line Bundles ◽  
Chern Simons ◽  
Determinant Line Bundle

We study generalized determinant line bundles for families of principal bundles and connections. We explore the connections of this line bundle and give conditions for the uniqueness of such. Furthermore we construct for families of bundles and connections over manifolds with boundary, a generalized Chern-Simons invariant as a section of a determinant line bundle.

scholarly journals Families of K-3 Surfaces

1972 ◽  
Vol 48 ◽  
pp. 1-17 ◽  
Author(s):  
Alan L. Mayer
Keyword(s):  
Complex Manifold ◽  
Chern Class ◽  
Compact Complex Manifold ◽  
Canonical Sheaf ◽  
First Chern Class

Let V be a 2-dimensional compact complex manifold. V is called a K-3 surface if : a) the irregularity q = dim H1(V, θ) of V vanishes and b) the first Chern class c1 of V vanishes. The canonical sheaf (of holo-morphic 2-forms) K of such a surface is trivial, since q = 0 implies that the Chern class map cx : Pic (V) → H2(V, Z) is injective : thus V has a nowhere zero holomorphic 2-form.

Sign in / Sign up

Export Citation Format

Share Document