5,918 research outputs found
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
On skew tau-functions in higher spin theory
Recent studies of higher spin theory in three dimensions concentrate on
Wilson loops in Chern-Simons theory, which in the classical limit reduce to
peculiar corner matrix elements between the highest and lowest weight states in
a given representation of SL(N). Despite these "skew" tau-functions can seem
very different from conventional ones, which are the matrix elements between
the two highest weight states, they also satisfy the Toda recursion between
different fundamental representations. Moreover, in the most popular examples
they possess simple representations in terms of matrix models and Schur
functions. We provide a brief introduction to this new interesting field,
which, after quantization, can serve as an additional bridge between knot and
integrability theories.Comment: 36 page
S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
A recent claim that the S-duality between 4d SUSY gauge theories, which is
AGT related to the modular transformations of 2d conformal blocks, is no more
than an ordinary Fourier transform at the perturbative level, is further traced
down to the commutation relation [P,Q]=-i\hbar between the check-operator
monodromies of the exponential resolvent operator in the underlying
Dotsenko-Fateev matrix models and beta-ensembles. To this end, we treat the
conformal blocks as eigenfunctions of the monodromy check operators, what is
especially simple in the case of one-point toric block. The kernel of the
modular transformation is then defined as the intertwiner of the two
monodromies, and can be obtained straightforwardly, even when the eigenfunction
interpretation of the blocks themselves is technically tedious. In this way, we
provide an elementary derivation of the old expression for the modular kernel
for the one-point toric conformal block.Comment: 15 page
Knot invariants from Virasoro related representation and pretzel knots
We remind the method to calculate colored Jones polynomials for the plat
representations of knot diagrams from the knowledge of modular transformation
(monodromies) of Virasoro conformal blocks with insertions of degenerate
fields. As an illustration we use a rich family of pretzel knots, lying on a
surface of arbitrary genus g, which was recently analyzed by the evolution
method. Further generalizations can be to generic Virasoro modular
transformations, provided by integral kernels, which can lead to the Hikami
invariants.Comment: 29 page
Dualities in persistent (co)homology
We consider sequences of absolute and relative homology and cohomology groups
that arise naturally for a filtered cell complex. We establish algebraic
relationships between their persistence modules, and show that they contain
equivalent information. We explain how one can use the existing algorithm for
persistent homology to process any of the four modules, and relate it to a
recently introduced persistent cohomology algorithm. We present experimental
evidence for the practical efficiency of the latter algorithm.Comment: 16 pages, 3 figures, submitted to the Inverse Problems special issue
on Topological Data Analysi
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure
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