On Factorization of Generalized Macdonald Polynomials

A remarkable feature of Schur functions -- the common eigenfunctions of cut-and-join operators from $W_\infty$ -- is that they factorize at the peculiar two-parametric topological locus in the space of time-variables, what is known as the hook formula for quantum dimensions of representations of $U_q(SL_N)$ and plays a big role in various applications. This factorization survives at the level of Macdonald polynomials. We look for its further generalization to {\it generalized} Macdonald polynomials (GMP), associated in the same way with the toroidal Ding-Iohara-Miki algebras, which play the central role in modern studies in Seiberg-Witten-Nekrasov theory. In the simplest case of the first-coproduct eigenfunctions, where GMP depend on just two sets of time-variables, we discover a weak factorization -- on a codimension-one slice of the topological locus, what is already a very non-trivial property, calling for proof and better understanding.Comment: 8 page

The 2-leg vertex in K-theoretic DT theory

K-theoretic Donaldson-Thomas counts of curves in toric and many related threefolds can be computed in terms of a certain canonical 3-valent tensor, the K-theoretic equivariant vertex. In this paper we derive a formula for the vertex in the case when two out of three entries are nontrivial. We also discuss some applications of this result.Comment: 27 page

Multivalued current-phase relationship in a.c. Josephson effect for a three-dimensional Weyl semimetal WTe2_2

We experimentally study electron transport between two superconducting indium leads, coupled to a single WTe$_2$ crystal, which is a three-dimensional Weyl semimetal. We demonstrate Josephson current in long 5~$\mu$m In-WTe$_2$-In junctions, as confirmed by the observation of integer (1,2,3) and fractional (1/3, 1/2, 2/3) Shapiro steps under microwave irradiation. Demonstration of fractional a.c. Josephson effect indicates multivalued character of the current-phase relationship, which we connect with Weyl topological surface states contribution to Josephson current. In contrast to topological insulators and Dirac semimetals, we do not observe $4\pi$ periodicity in a.c. Josephson effect for WTe$_2$ at different frequencies and power, which might reflect chiral character of the Fermi arc surface states in Weyl semimetal.Comment: the text is seriously corrected. arXiv admin note: text overlap with arXiv:1801.0955