Knot Invariants and New Weight Systems from General 3D TFTs

Jian Qiu; Maxim Zabzine

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Knot Invariants and New Weight Systems from General 3D TFTs

Jian Qiu, Maxim Zabzine

High Energy Physics - Theory · Submitted 7 Jun 2010

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The authors' main claim We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory.

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We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the Ward identities of these theories imply that the expectation value of the Wilson loop is a pairing of two dual constructions of (co)cycles of certain extended graph complex (extended from Kontsevich's graph complex to accommodate the Wilson loop). We also prove that there is an isomorphism between the same complex and certain extended Chevalley-Eilenberg complex of Hamiltonian vector fields. This isomorphism allows us to generalize the Lie algebra weight system for knots to weight systems associated with any homological vector field and its representations. As an example we construct knot invariants using holomorphic vector bundle over hyperKahler manifolds.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory.	contribution 0.92	0.93	0.59	asserted 1.00
s1	We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the Ward identities of these theories imply that the expectation value of the Wilson loop is a pairing of two dual constructions of (co)cycles of certain extended graph complex (extended from Kontsevich's graph complex to accommodate the Wilson loop).	method 0.47	0.85	0.12	asserted 1.00
s2	We also prove that there is an isomorphism between the same complex and certain extended Chevalley-Eilenberg complex of Hamiltonian vector fields.	result 0.91	0.91	0.09	asserted 1.00
s3	This isomorphism allows us to generalize the Lie algebra weight system for knots to weight systems associated with any homological vector field and its representations.	result 0.48	0.90	0.36	asserted 0.99
s4	As an example we construct knot invariants using holomorphic vector bundle over hyperKahler manifolds.	contribution 0.59	0.93	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

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Original abstract

We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the Ward identities of these theories imply that the expectation value of the Wilson loop is a pairing of two dual constructions of (co)cycles of certain extended graph complex (extended from Kontsevich's graph complex to accommodate the Wilson loop). We also prove that there is an isomorphism between the same complex and certain extended Chevalley-Eilenberg complex of Hamiltonian vector fields. This isomorphism allows us to generalize the Lie algebra weight system for knots to weight systems associated with any homological vector field and its representations. As an example we construct knot invariants using holomorphic vector bundle over hyperKahler manifolds.

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References

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4 arXiv papers cite this in 12 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 9 sentences without a confident label

Contests it 1 paper

[ref] , and we shall rectify this remiss in this paper.

Knot Weight Systems from Graded Symplectic Geometry · Introduction · contest 0.82

Compares with it 1 paper

Note that in [ref] , the vertices are labeled starting from 0 instead of 1, so the sign factor there is $(-1)^j+1$.

Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications · Bits of graph theory · 0.93

Cites it as background 1 paper

Quantizations of the AKSZ TFTs have been analyzed by many authors, for example, [ref] [ref] [ref] [ref] [ref] [ref] , but it is not completely successful to analyze quantization of general AKSZ theories.

Lectures on AKSZ Sigma Models for Physicists · Comments and Future Outlook · 0.82

What it cites · 31 sentences without a confident label

Uses the method or tool of 2 papers

[ref] to cure this.

Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes · Extended Graph Complex · 0.91
The perturbation theory can be carried out using the super Feynman rules [ref] .

Chern-Simons Perturbation Theory · Knot Invariants from Chern-Simons Theory · 0.96

Cites as background 7 papers

We first list a few useful identities (see [ref] for more details)

Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes · Extended CE Complex · 0.80
[ref] and [ref] respectively and is highly non-trivial.

Hyper-Kahler Geometry and Invariants of Three-Manifolds · AKSZ Topological Field Theory · 0.88
In this section we quickly review the construction of knot invariants and 3-manifold invariants from Chern-Simons theory initiated by Witten [ref] , for a nice review see [ref] .

Chern-Simons Gauge Theory: Ten Years After · Knot Invariants from Chern-Simons Theory · 1.00

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