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Refined Chern-Simons Theory in Genus Two
High Energy Physics - Theory · Submitted 10 Apr 2015
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the defining relations of the mapping class groups beyond genus 1.
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The abstract, sentence by sentence
Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117, opening a question if it extends to any genus. Our construction is built upon a q,t-deformation of the square of q-6j symbol of U_q(sl_2), which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Reshetikhin-Turaev (a.k.a. | background 1.00 | 0.04 | 0.01 | unclear 0.90 |
| s1 | Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. | background 1.00 | 0.05 | 0.03 | asserted 1.00 |
| s2 | The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the defining relations of the mapping class groups beyond genus 1. | contribution 0.90 | 0.40 | 0.60 | asserted 0.99 |
| s3 | For this we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. | contribution 0.92 | 0.95 | 0.19 | asserted 1.00 |
| s4 | This is a direct generalization of the original genus 1 construction of arXiv:1105.5117, opening a question if it extends to any genus. | future work 0.59 | 0.61 | 0.08 | asserted 0.72 |
| s5 | Our construction is built upon a q,t-deformation of the square of q-6j symbol of U_q(sl_2), which we define using the Macdonald version of Fourier duality. | method 0.99 | 0.90 | 0.06 | asserted 1.00 |
| s6 | This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. | result 0.73 | 0.84 | 0.11 | asserted 0.96 |
| s7 | In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology. | result 0.89 | 0.72 | 0.11 | hedged possibility 0.82 |
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The paper
Original abstract
Reshetikhin-Turaev (a.k.a. Chern-Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation -- refinement -- of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117, opening a question if it extends to any genus. Our construction is built upon a q,t-deformation of the square of q-6j symbol of U_q(sl_2), which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.
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References
How the literature uses it
8 arXiv papers cite this in 14 sentences
Papers citing it · 8 sentences without a confident label
Compares with it 1 paper
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It reproduces the previous suggestions [ref] for symmetric and antisymmetric representations $[r]$ and $[1^s]$ – on which there seems to be a consensus in the literature (with the single exception of [ref] ).
Rectangular superpolynomials for the figure-eight knot · Conclusion · 0.87
Cites it as background 5 papers
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Comparing to the normal $SU(2)_K$ CS TQFT, the Hilbert space is unchanged but the matrix elements of generators $S$ and $T$ depends on two parameters $$ and $$ [ref] , [equation] with $=e^2πi/K+2β$, $=^β=e^2πiβ/K+2β$, $K$ is the CS level and $S_00$ is a normalization constant. $i$ and $j$ run over non negative integers and are the Dynkin label of $SU(2)$ irreducible representations. $P_j(x_1,x_2;,)$ is the $SU(2)$ Macdonald polynomial of the spin-$j/2$ representation and $g_j$ is the quadratic norm of the Macdonald polynomials $P_j(x_1,x_2;,)$ under a natural orthogonality condition.
Argyres-Douglas Theories, Modularity of Minimal Models and Refined Chern-Simons · Refined CS representation · 0.84
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A quest for a TQFT describing Khovanov invariants [ref] , a categorification of Jones polynomials – Wilson loop averages in the 3d Chern-Simons theory [ref] , – or more desirably Khovanov-Rozansky invariants [ref] , or even more generally superpolynomials [ref] , delivering categorification to HOMFLY-PT invariants, has led to a wide and profound development in the both physics and mathematics, see e.g. [ref] .[The literature on generalizations and physical constructions of knot/link invariants is extremely rich.
Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology? · Introduction And Discussion · 1.00
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A generalization of DAHA for the skein algebra on higher-genus surface was initiated in [ref] .
Generalized Double Affine Hecke Algebra for Double Torus · Introduction · 0.85
What it cites · 28 sentences without a confident label
Builds on 1 paper
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[equation] This is the same procedure that has been used in [ref] , only now in genus 2 setting.
Knot Homology from Refined Chern-Simons Theory · Knot invariants · 0.94
Uses the method or tool of 1 paper
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In these cases the matrix elements of the TQFT representation can be actually expressed in a simple closed form, which is straightforward to prove using either of the methods of [ref] or [ref] .
TQFT computations and experiments · Unrefined TQFT representations for $g = 1,2$ · 0.87
Compares with 3 papers
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The resulting partition function agrees with the partition function of the refined topological string on $T^⋆ M$ [ref] .
Knot Homology from Refined Chern-Simons Theory · Discussion · 0.81
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It is enough to look at the simplest examples of twist knots ($4_1,6_1,8_1,…$) in s. 6 below to see that they are very different from those of [ref] .
Super-A-polynomials for Twist Knots · Introduction · 0.84
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It is enough to look at the simplest examples of twist knots ($4_1,6_1,8_1,…$) in s. 6 below to see that they are very different from those of [ref] .
The Superpolynomial for Knot Homologies · Introduction · 0.84
Cites as background 9 papers
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As explained in [ref] , to compute the TQFT knot invariants, in addition to the representation of the mapping class group one also needs the knot operators $O_j(K)$, that the TQFT functor associates to the bordisms inserting $K$ colored by representation $j$.
Knot Homology from Refined Chern-Simons Theory · Knot invariants · 0.84
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Let us briefly review here the argument of [ref] .
Colored knot amplitudes and Hall-Littlewood polynomials · Refined Chern-Simons Invariants of Pretzel knots · 0.89
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As discussed before in [ref] , knot operators in refined Chern-Simons theory on a torus generate an algebra which is isomorphic to the spherical DAHA, also known as the elliptic Hall algebra.
Refined knot invariants and Hilbert schemes · Discussion · 0.86
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