7,775 research outputs found
Link polynomial calculus and the AENV conjecture
Using the recently proposed differential hierarchy (Z-expansion) technique,
we obtain a general expression for the HOMFLY polynomials in two arbitrary
symmetric representations of link families, including Whitehead and Borromean
links. Among other things, this allows us to check and confirm the recent
conjecture of arXiv:1304.5778 that the large representation limit (the same as
considered in the knot volume conjecture) of this quantity matches the
prediction from mirror symmetry consideration. We also provide, using the
evolution method, the HOMFLY polynomial in two arbitrary symmetric
representations for an arbitrary member of the one-parametric family of
2-component 3-strand links, which includes the Hopf and Whitehead links.Comment: 20 page
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
Cut-and-join structure and integrability for spin Hurwitz numbers
Spin Hurwitz numbers are related to characters of the Sergeev group, which
are the expansion coefficients of the Q Schur functions, depending on odd times
and on a subset of all Young diagrams. These characters involve two dual
subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur
functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators
W_\Delta with \Delta\in OP. The eigenvalues of these operators are the
generalized Sergeev characters, their algebra is isomorphic to the algebra of Q
Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the
generating function of spin Hurwitz numbers is a \tau-function of an integrable
hierarchy, that is, of the BKP type. At last, we discuss relations of the
Sergeev characters with matrix models.Comment: 22 page
A Hurwitz theory avatar of open-closed strings
We review and explain an infinite-dimensional counterpart of the Hurwitz
theory realization of algebraic open-closed string model a la Moore and
Lizaroiu, where the closed and open sectors are represented by conjugation
classes of permutations and the pairs of permutations, i.e. by the algebra of
Young diagrams and bipartite graphes respectively. An intriguing feature of
this Hurwitz string model is coexistence of two different multiplications,
reflecting the deep interrelation between the theory of symmetric and linear
groups S_\infty and GL(\infty).Comment: 11 page
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