Superpolynomials, zeta-functions and Riemann hypothesis for plane curve singularities

Ivan Cherednik

Generally, the zeta-equivalence of two schemes over $C$ (upon the passage to finitely generated $Z$-subalgebras in $C$) implies the coincidence of their virtual Hodge numbers (N.Katz). A much stronger theory can be expected for (local rings of) singularities. For plane curve singularities (at least), proper zeta-functions capture their topological type. Namely, motivic superpolynomials of plane curve singularities conjecturally coincide with the Galkin-Stohr $L$-functions and with the corresponding DAHA-superpolynomials (known to be topological invariants). In their turn, the latter conjecturally coincide with the stable Khovanov-Rozansky polynomials (proven for uncolored torus knots) and with physics superpolynomials. In these two theories, the roles of $q,t$ are quite involved. Motivically: $q$ is the cardinality of a finite field, $t$ is simply $T$ from the zeta-function. The superduality becomes the functional equation, but the Riemann Hypothesis mostly holds only for $q$ sufficiently close to $0$. In physics, properly extended L-functions of singularities$W(x,y)=0$ are expected some correlation functions for LGSMwith superpotentials $W(x,y)$; no Representation Theory(VOA, $W$-algebras, DAHA) is needed in the motivic approach.

http://scgp.stonybrook.edu/video_portal/video.php?id=3908

Simons- Workshop: Vertex Algebras and Gauge Theory