scholarly journals Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers

2017 ◽  
Vol 2019 (18) ◽  
pp. 5674-5702 ◽  
Author(s):  
Hee-Joong Chung ◽  
Dohyeong Kim ◽  
Minhyong Kim ◽  
Georgios Pappas ◽  
Jeehoon Park ◽  
...  
Keyword(s):  
Path Integral ◽  
Knot Theory ◽  
Integral Formula ◽  
Simons Theory ◽  
Duality Theorems ◽  
Linking Numbers ◽  
Seifert Surfaces ◽  
Chern Simons ◽  
Chern Simons Theory

AbstractFollowing the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of $n$-th power residue symbols. This formalism leads to a precise arithmetic analogue of a “path-integral formula” for linking numbers.

scholarly journals CHERN–SIMONS THEORY AND THE QUANTUM RACAH FORMULA

2013 ◽  
Vol 25 (03) ◽  
pp. 1350004 ◽  
Author(s):  
SEBASTIAN DE HARO ◽  
ATLE HAHN
Keyword(s):  
Path Integral ◽  
Special Class ◽  
Wilson Loop ◽  
Knot Theory ◽  
Analytic Approach ◽  
Simons Theory ◽  
Compact Structure ◽  
Simply Connected ◽  
Chern Simons ◽  
Chern Simons Theory

We generalize several results on Chern–Simons models on Σ × S1in the so-called "torus gauge" which were obtained in [A. Hahn, An analytic approach to Turaev's shadow invariant, J. Knot Theory Ramifications17(11) (2008) 1327–1385] (= arXiv:math-ph/0507040) to the case of general (simply-connected simple compact) structure groups and general link colorings. In particular, we give a non-perturbative evaluation of the Wilson loop observables corresponding to a special class of simple but non-trivial links and show that their values are given by Turaev's shadow invariant. As a byproduct, we obtain a heuristic path integral derivation of the quantum Racah formula.

A THREE-DIMENSIONAL COVARIANT APPROACH TO MONODROMY (SKEIN RELATIONS) IN CHERN-SIMONS THEORY

1990 ◽  
Vol 05 (32) ◽  
pp. 2747-2751 ◽  
Author(s):  
B. BRODA
Keyword(s):  
Integral Representation ◽  
Path Integral ◽  
Wilson Loop ◽  
Three Dimensional ◽  
Simons Theory ◽  
Monodromy Operator ◽  
Covariant Approach ◽  
Path Integral Representation ◽  
Chern Simons ◽  
Chern Simons Theory

A genuinely three-dimensional covariant approach to the monodromy operator (skein relations) in the context of Chern-Simons theory is proposed. A holomorphic path-integral representation for the holonomy operator (Wilson loop) and for the non-abelian Stokes theorem is used.

scholarly journals PERTURBATIVE CHERN-SIMONS THEORY

1995 ◽  
Vol 04 (04) ◽  
pp. 503-547 ◽  
Author(s):  
DROR BAR-NATAN
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Integral Formula ◽  
Simons Theory ◽  
Gauge Field Theory ◽  
Complete Agreement ◽  
Knot Invariants ◽  
Arf Invariant ◽  
Chern Simons ◽  
Chern Simons Theory

We present the perturbation theory of the Chern-Simons gauge field theory and prove that to second order it indeed gives knot invariants. We identify these invariants and show that in fact we get a previously unknown integral formula for the Arf invariant of a knot, in complete agreement with earlier non-perturbative results of Witten. We outline our expectations for the behavior of the theory beyond two loops.

Hamiltonian, path integral, and BRST formulations of the Chern–Simons theory under appropriate gauge-fixing

2008 ◽  
Vol 86 (2) ◽  
pp. 401-407 ◽  
Author(s):  
U Kulshreshtha ◽  
D S Kulshreshtha
Keyword(s):  
Path Integral ◽  
Hamiltonian Path ◽  
Simons Theory ◽  
Gauge Fixing ◽  
Chern Simons ◽  
Chern Simons Theory

The Hamiltonian, path integral, and BRST formulations of the Chern–Simons theory in two-space one-time dimensions are investigated under appropriate gauge-fixing conditions.PACS Nos.: 11.10.Ef, 11.10.Kk, 12.20.Ds

GEOMETRY OF QUANTUM LOOPS AND KNOT THEORY

1989 ◽  
Vol 04 (24) ◽  
pp. 2409-2416 ◽  
Author(s):  
M.A. AWADA
Keyword(s):  
Perturbation Theory ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Three Dimensions ◽  
Simons Theory ◽  
Loop Equation ◽  
Chern Simons ◽  
Linear Quantum ◽  
Chern Simons Theory ◽  
Skein Relation

Starting from the linear quantum loop equation of non-abelian Chern-Simons theory in three dimensions, we prove that it yields precisely and to all loop orders in perturbation theory the exact skein relation satisfied by the Jones polynomial in knot theory.

A RIGOROUS CONSTRUCTION OF ABELIAN CHERN-SIMONS PATH INTEGRALS USING WHITE NOISE ANALYSIS

1996 ◽  
Vol 08 (03) ◽  
pp. 445-456 ◽  
Author(s):  
PETER LEUKERT ◽  
JÖRG SCHÄFER
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Path Integrals ◽  
Simons Theory ◽  
White Noise Analysis ◽  
Feynman Path ◽  
Feynman Path Integrals ◽  
Linking Numbers ◽  
Chern Simons ◽  
Chern Simons Theory

Using recent results in white noise analysis we rigorously construct Feynman path integrals for abelian Chern-Simons theory. We define Wilson loop variables and show that their expectation is a topological invariant, namely computable in terms of pairwise linking numbers as conjectured by E. Witten.

Erratum: “Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers”

2019 ◽  
Vol 2019 (18) ◽  
pp. 5854-5857
Author(s):  
Hee-Joong Chung ◽  
Dohyeong Kim ◽  
Minhyong Kim ◽  
George Pappas ◽  
Jeehoon Park ◽  
...  
Keyword(s):  
The Other ◽  
Simons Theory ◽  
Height Pairing ◽  
Main Error ◽  
Linking Numbers ◽  
Definition Of ◽  
Mathematics Research ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We wish to point out errors in the paper “Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers”, International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1–29. The main error concerns the symmetry of the “ramified case” of the height pairing, which relies on the vanishing of the Bockstein map in Proposition 3.5. The surjectivity claimed in the 1st line of the proof of Proposition 3.5 is incorrect. The specific results that are affected are Proposition 3.5; Lemmas 3.6, 3.7, 3.8, and 3.9; and Corollary 3.11. The definition of the $(S,n)$-height pairing following Lemma 3.9 is also invalid, since the symmetry of the pairing was required for it to be well defined. The results of Section 3 before Proposition 3.5 as well as those of the other Sections are unaffected. Proposition 3.10 is correct, but the proof is unclear and has some sign errors. So we include here a correction. As in the paper, let $I$ be an ideal such that $I^n$ is principal in ${\mathcal{O}}_{F,S}$. Write $I^n=(f^{-1})$. Then the Kummer cocycles $k_n(f)$ will be in $Z^1(U, {{\mathbb{Z}}/{n}{\mathbb{Z}}})$. For any $a\in F$, denote by $a_S$ its image in $\prod _{v\in S} F_v$. Thus, we get an element $$\begin{equation*}[f]_{S,n}:=[(k_n(f), k_{n^2}(f_S), 0)] \in Z^1(U, {{{\mathbb{Z}}}/{n}{{\mathbb{Z}}}} \times_S{\mathbb{Z}}/n^2{\mathbb{Z}}),\end{equation*}$$which is well defined in cohomology independently of the choice of roots used to define the Kummer cocycles. (We have also trivialized both $\mu _{n^2}$ and $\mu _n$.)

Sign in / Sign up

Export Citation Format

Share Document