Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology

Dmitry Galakhov; Gregory W. Moore

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Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology

Dmitry Galakhov, Gregory W. Moore

High Energy Physics - Theory · Submitted 14 Jul 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of $SU(2)$ magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.57 · human participants 0.01 · controlled comparison 0.08 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.93 · mentions code or data availability 0.02.

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We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of $SU(2)$ magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$.	contribution 1.00	0.93	0.19	asserted 1.00
s1	It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry.	method 0.91	0.69	0.09	asserted 1.00
s2	We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology.	result 0.82	0.94	0.10	hedged possibility 0.73
s3	We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of $SU(2)$ magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.	result 0.97	0.86	0.68	asserted 1.00

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Original abstract

We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of $SU(2)$ magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.

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12 arXiv papers cite this in 33 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 · 24 sentences without a confident label

Uses its method or tool 2 papers

Invariance under all these moves they can be easily checked along the lines of [ref] .

Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology? · Link invariant · 0.88
In the current work, we would like to take the above viewpoint while following the approach of [ref] (see also [ref] ) to construct conformal blocks which are interpreted as wavefunctions of a 2d Landau-Ginzburg theory specified by a superpotential $𝒲$.

Liouville conformal blocks and Stokes phenomena · Introduction · 0.94

Compares with it 1 paper

Calculating detours and taking into account the hovering solutions in vacua $(-,-)$ and $(+,+)$ we derive the following expressions for the interfaces (to compare to [ref] one should switch $(q,t)$-gradings to opposite ones):

Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology? · Braiding, fusion/defusion interfaces · 0.93

Cites it as background 3 papers

The Janus interface allows one to parallel transport the theory from theory $p_0$ to theory $p_1$ along some path $℘$, so that for the partition functions we have: [equation] where $M_ij(p_1,p_0;℘)$ is a supersymmetric index of the theory on cylinder $[0,1]×S^1$ with boundary conditions defined by indices $i$ and $j$. $M_ij(p_1,p_0;℘)$ provides a connection on the bundle of partition functions $Z⃗$ over the parameter space.[In principle, one could construct a differential form of this connection analogously to the tt$^*$-connection [ref] (see also [ref] ).

Gauge/Bethe correspondence from quiver BPS algebras · From Janus interfaces to twisted R-matrices · 0.82
Many of the Landau-Ginzburg models that are of interest in low dimensional topology [ref] have non-trivial twisted masses.

On the Algebra of the Infrared with Twisted Masses · Introduction · 0.80
Similar categorical constructions appear in the recent physics literature on the web-based formalism; cf. [ref] , [ref] , [ref] .

A sheaf-theoretic model for SL(2,C) Floer homology · Categorification · 0.88

What it cites · 51 sentences without a confident label

Builds on 2 papers

To do this we consider a slight generalization of the famous CFIV index [ref] .

A New Supersymmetric Index · Fermion Number Grading And $tt^*$-geometry · 0.91
This requires an excursion - interesting in its own right - into $tt^*$ equations [ref] for a generalization of the CFIV index [ref] to the case of LG interfaces.

On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version) · Introduction And Conclusion · 0.80

Uses the method or tool of 1 paper

As explained in [ref] , counting rigid $σζ$-instantons produces a map

Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions · Invariance Under Reidemeister Moves · 0.80

Compares with 7 papers

[Notice that due to our choice of fermion number convention [ref] we have a minus sign in front of $_τ$ in comparison to [ref] .

Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions · Zeta Instantons And Curved Webs · 0.88
* At this stage we can compare the link complexes following form LG interfaces with the well-known mathematical construction due to Khovanov [ref] (and other generalizations of the hypercube model, see e.g. [ref] ).

Matrix model and dimensions at hypercube vertices · The Braiding Interface · 0.98
* At this stage we can compare the link complexes following form LG interfaces with the well-known mathematical construction due to Khovanov [ref] (and other generalizations of the hypercube model, see e.g. [ref] ).

Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial · The Braiding Interface · 0.98

Cites as background 11 papers

In this section we review some essential points from [ref] .

Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions · Lightning review · 0.99
In this section we review some essential points from [ref] .

An Introduction To The Web-Based Formalism · Lightning review · 0.99
The first interpretation of knot homology in terms of BPS states of which we are aware is [ref] . ]

Khovanov-Rozansky Homology and Topological Strings · Introduction And Conclusion · 0.94

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