Zeta functions of plane curve singularities, compactified Jacobians and Landau–Ginzburg models

Generally, the zeta-equivalence of two schemes over $\mathbb{C}$ (upon the passage to finitely generated $\mathbb{Z}$-subalgebras in $\mathbb{C}$) implies the coincidence of their virtual Hodge numbers (N. Katz). A stronger theory can be expected for isolated singularities. At least for plane curve singularities, proper zeta functions capture their topological types. Their topological invariance is due to the following (mostly) conjectural links:  Galkin--Stohrzeta functions  $\rightarrow$ motivic superpolynomials  $\rightarrow$ DAHA superpolynomials (known to be topological invariants). Also, the latter conjecturally coincide with the stable Khovanov--Rozansky polynomials and the physics superpolynomials (from $M$-theory).  The motivic superpolynomials of singularities $W(x,y)=0$ (at $x=0=y$ for polynomials $W$) are given entirely in terms of the corresponding compactified Jacobians.  Due to the (second) link to the DAHA superpolynomials, they can be expected partition functions in LGSM theory for W. The (first) link to zeta-functions provides the functional equation for them, where the corresponding Riemann Hypothesis is possibly related to phase transitions in LGSM, by analogy with the Lee--Yang theorem from spin chains. We will touch the physics aspects only a bit, focusing on the definition and properties of motivic superpolynomials, including the Riemann Hypothesis.