scholarly journals Twisted spectral correspondence and torus knots

2020 ◽  
Vol 29 (06) ◽  
pp. 2050040
Author(s):  
Wu-Yen Chuang ◽  
Duiliu-Emanuel Diaconescu ◽  
Ron Donagi ◽  
Satoshi Nawata ◽  
Tony Pantev
Keyword(s):  
Conjugacy Classes ◽  
Present Approach ◽  
Higgs Bundles ◽  
Torus Knots ◽  
Cohomological Invariants ◽  
Character Varieties ◽  
Spectral Correspondence ◽  
Marked Points ◽  
Chern Simons

Cohomological invariants of twisted wild character varieties as constructed by Boalch and Yamakawa are derived from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots. Generalizing the untwisted case, the present approach is based on a spectral correspondence for meromorphic Higgs bundles with fixed conjugacy classes at the marked points. This construction is carried out for twisted wild character varieties associated to [Formula: see text] torus knots, providing a colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

scholarly journals Hodge polynomials of the SL(2, ℂ)-character variety of an elliptic curve with two marked points

2014 ◽  
Vol 25 (14) ◽  
pp. 1450125 ◽  
Author(s):  
Marina Logares ◽  
Vicente Muñoz
Keyword(s):  
Elliptic Curve ◽  
Moduli Space ◽  
Betti Numbers ◽  
Conjugacy Classes ◽  
Character Variety ◽  
Higgs Bundles ◽  
Hodge Numbers ◽  
Hodge Polynomials ◽  
Marked Points

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, ℂ). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.

THE HOMFLY POLYNOMIAL FOR TORUS LINKS FROM CHERN–SIMONS GAUGE THEORY

1995 ◽  
Vol 10 (07) ◽  
pp. 1045-1089 ◽  
Author(s):  
J. M. F. LABASTIDA ◽  
M. MARIÑO
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Homfly Polynomial ◽  
Fundamental Representation ◽  
Torus Knots ◽  
Polynomial Invariants ◽  
Chern Simons ◽  
Knots And Links ◽  
Torus Links

Polynomial invariants corresponding to the fundamental representation of the gauge group SU(N) are computed for arbitrary torus knots and links in the framework of Chern–Simons gauge theory making use of knot operators. As a result, a formula for the HOMFLY polynomial for arbitrary torus links is presented.

Fundamental classes of 3-manifold groups representations in SL(4,R)

2017 ◽  
Vol 26 (07) ◽  
pp. 1750036
Author(s):  
Thilo Kuessner
Keyword(s):  
Connected Components ◽  
Fundamental Groups ◽  
Fundamental Class ◽  
Character Varieties ◽  
Chern Simons

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of [Formula: see text]-manifolds to [Formula: see text] that factor over [Formula: see text], in particular for those factoring over the isomorphism [Formula: see text]. We also discuss consequences for the number of connected components of [Formula: see text]-character varieties, and we show that there are knots with arbitrarily many components of vanishing Chern–Simons invariant in their [Formula: see text]-character varieties.

scholarly journals Character varieties of virtually nilpotent Kähler groups and G–Higgs bundles

2015 ◽  
Vol 65 (6) ◽  
pp. 2601-2612
Author(s):  
Indranil Biswas ◽  
Carlos Florentino
Keyword(s):  
Higgs Bundles ◽  
Character Varieties ◽  
Kähler Groups

scholarly journals Refined large N duality for knots

2020 ◽  
pp. 2041001
Author(s):  
Masaya Kameyama ◽  
Satoshi Nawata
Keyword(s):  
Moduli Spaces ◽  
Bound States ◽  
Simons Theory ◽  
Global Symmetry ◽  
Torus Knots ◽  
Torus Knot ◽  
Large N ◽  
Bps Invariants ◽  
Bps States ◽  
Chern Simons

We formulate large [Formula: see text] duality of [Formula: see text] refined Chern–Simons theory with a torus knot/link in [Formula: see text]. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the [Formula: see text]-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in [Formula: see text] to refined BPS invariants in the resolved conifold. Assuming that the extra [Formula: see text] global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

scholarly journals SOLITONS ON NONCOMMUTATIVE TORUS AS ELLIPTIC CALOGERO–GAUDIN MODELS, BRANES AND LAUGHLIN WAVE FUNCTIONS

2003 ◽  
Vol 18 (14) ◽  
pp. 2477-2500 ◽  
Author(s):  
BO-YU HOU ◽  
DAN-TAO PENG ◽  
KANG-JIE SHI ◽  
RUI-HONG YUE
Keyword(s):  
Quantum Hall ◽  
Moment Map ◽  
Simons Theory ◽  
Noncommutative Torus ◽  
The Moment ◽  
Gaudin Models ◽  
Laughlin Wave Function ◽  
Marked Points ◽  
Chern Simons ◽  
Chern Simons Theory

For the noncommutative torus [Formula: see text], in the case of the noncommutative parameter [Formula: see text], we construct the basis of Hilbert space ℋn in terms of θ functions of the positions zi of n solitons. The wrapping around the torus generates the algebra [Formula: see text], which is the Zn × Zn Heisenberg group on θ functions. We find the generators g of a local elliptic su (n), which transform covariantly by the global gauge transformation of [Formula: see text]. By acting on ℋn we establish the isomorphism of [Formula: see text] and g. We embed this g into the L-matrix of the elliptic Gaudin and Calogero–Moser models to give the dynamics. The moment map of this twisted cotangent [Formula: see text] bundle is matched to the D-equation with the Fayet–Illiopoulos source term, so the dynamics of the noncommutative solitons become that of the brane. The geometric configuration (k, u) of the spectral curve det |L(u) - k| = 0 describes the brane configuration, with the dynamical variables zi of the noncommutative solitons as the moduli T⊗ n/Sn. Furthermore, in the noncommutative Chern–Simons theory for the quantum Hall effect, the constrain equation with quasiparticle source is identified also with the moment map equation of the noncommutative [Formula: see text] cotangent bundle with marked points. The eigenfunction of the Gaudin differential L-operators as the Laughlin wave function is solved by Bethe ansatz.

scholarly journals Compact connected components in relative character varieties of punctured spheres

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Nicolas Tholozan ◽  
Jérémy Toulisse
Keyword(s):  
Lie Groups ◽  
Invariant Theory ◽  
Simple Closed Curve ◽  
Closed Curve ◽  
Connected Components ◽  
Higgs Bundles ◽  
Geometric Invariant ◽  
Projective Varieties ◽  
Elliptic Element ◽  
Character Varieties

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

scholarly journals The $SU(2)$-character varieties of torus knots

2015 ◽  
Vol 45 (2) ◽  
pp. 583-600
Author(s):  
Javier Martínez-Martínez ◽  
Vicente Muñoz
Keyword(s):  
Torus Knots ◽  
Character Varieties
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