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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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In this note we make an attempt to compare a cohomological theory of Hilbert spaces of ground states in the ${cal N}=(2,2)$ 2d Landau-Ginzburg theory in models describing link embeddings in ${mathbb{R}}^3$ to Khovanov and Khovanov-Rozansky homologies. To confirm the equivalence we exploit the invariance of Hilbert spaces of ground states for interfaces with respect to homotopy. In this attempt to study solitons and instantons in the Landau-Giznburg theory we apply asymptotic analysis also known in the literature as exact WKB method, spectral networks method, or resurgence. In particular, we associate instantons in LG model to specific WKB line configurations we call null-webs.
By studying the infra-red fixed point of an $mathcal{N}=(0,2)$ Landau-Ginzburg model, we find an example of modular invariant partition function beyond the ADE classification. This stems from the fact that a part of the left-moving sector is a new conformal field theory which is a variant of the parafermion model.
For each given $ngeq 2$, we construct a family of entire solutions $u_varepsilon (z,t)$, $varepsilon>0$, with helical symmetry to the 3-dimensional complex-valued Ginzburg-Landau equation begin{equation*} onumber Delta u+(1-|u|^2)u=0, quad (z,t) in mathbb{R}^2times mathbb{R} simeq mathbb{R}^3. end{equation*} These solutions are $2pi/varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, with asymptotic behavior as $varepsilonto 0$ $$ u_varepsilon (z,t) approx prod_{j=1}^n Wleft( z- varepsilon^{-1} f_j(varepsilon t) right), $$ where $W(z) =w(r) e^{itheta} $, $z= re^{itheta},$ is the standard degree $+1$ vortex solution of the planar Ginzburg-Landau equation $ Delta W+(1-|W|^2)W=0 text{ in } mathbb{R}^2 $ and $$ f_j(t) = frac { sqrt{n-1} e^{it}e^{2 i (j-1)pi/ n }}{ sqrt{|logvarepsilon|}}, quad j=1,ldots, n. $$ Existence of these solutions was previously conjectured, being ${bf f}(t) = (f_1(t),ldots, f_n(t))$ a rotating equilibrium point for the renormalized energy of vortex filaments there derived, $$ mathcal W_varepsilon ( {bf f} ) :=pi int_0^{2pi} Big ( , frac{|log varepsilon|} 2 sum_{k=1}^n|f_k(t)|^2-sum_{j eq k}log |f_j(t)-f_k(t)| , Big ) mathrm{d} t, $$ corresponding to that of a planar logarithmic $n$-body problem. These solutions satisfy $$ lim_{|z| to +infty } |u_varepsilon (z,t)| = 1 quad hbox{uniformly in $t$} $$ and have nontrivial dependence on $t$, thus negatively answering the Ginzburg-Landau analogue of the Gibbons conjecture for the Allen-Cahn equation, a question originally formulated by H. Brezis.
In this work we study the symmetry breaking conditions, given by a (anti)de Sitter-valued vector field, of a full (anti)de Sitter-invariant MacDowell-Mansouri inspired action. We show that under these conditions the action breaks down to General Relativity with a cosmological constant, the four dimensional topological invariants, as well as the Holst term. We obtain the equations of motion of this action, and analyze the symmetry breaking conditions.
We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via $Gamma$-convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.
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