8,200 research outputs found
The algebra of cell-zeta values
In this paper, we introduce cell-forms on , which are
top-dimensional differential forms diverging along the boundary of exactly one
cell (connected component) of the real moduli space
. We show that the cell-forms generate the
top-dimensional cohomology group of , so that there is a
natural duality between cells and cell-forms. In the heart of the paper, we
determine an explicit basis for the subspace of differential forms which
converge along a given cell . The elements of this basis are called
insertion forms, their integrals over are real numbers, called cell-zeta
values, which generate a -algebra called the cell-zeta algebra. By
a result of F. Brown, the cell-zeta algebra is equal to the algebra of
multizeta values. The cell-zeta values satisfy a family of simple quadratic
relations coming from the geometry of moduli spaces, which leads to a natural
definition of a formal version of the cell-zeta algebra, conjecturally
isomorphic to the formal multizeta algebra defined by the much-studied double
shuffle relations
Universal Lyndon Words
A word over an alphabet is a Lyndon word if there exists an
order defined on for which is lexicographically smaller than all
of its conjugates (other than itself). We introduce and study \emph{universal
Lyndon words}, which are words over an -letter alphabet that have length
and such that all the conjugates are Lyndon words. We show that universal
Lyndon words exist for every and exhibit combinatorial and structural
properties of these words. We then define particular prefix codes, which we
call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in
bijection with the set of the shortest unrepeated prefixes of the conjugates of
a universal Lyndon word. This allows us to give an algorithm for constructing
all the universal Lyndon words.Comment: To appear in the proceedings of MFCS 201
Abelian Primitive Words
We investigate Abelian primitive words, which are words that are not Abelian
powers. We show that unlike classical primitive words, the set of Abelian
primitive words is not context-free. We can determine whether a word is Abelian
primitive in linear time. Also different from classical primitive words, we
find that a word may have more than one Abelian root. We also consider
enumeration problems and the relation to the theory of codes
Algebraic Relations Between Harmonic Sums and Associated Quantities
We derive the algebraic relations of alternating and non-alternating finite
harmonic sums up to the sums of depth~6. All relations for the sums up to
weight~6 are given in explicit form. These relations depend on the structure of
the index sets of the harmonic sums only, but not on their value. They are
therefore valid for all other mathematical objects which obey the same
multiplication relation or can be obtained as a special case thereof, as the
harmonic polylogarithms. We verify that the number of independent elements for
a given index set can be determined by counting the Lyndon words which are
associated to this set. The algebraic relations between the finite harmonic
sums can be used to reduce the high complexity of the expressions for the
Mellin moments of the Wilson coefficients and splitting functions significantly
for massless field theories as QED and QCD up to three loop and higher orders
in the coupling constant and are also of importance for processes depending on
more scales. The ratio of the number of independent sums thus obtained to the
number of all sums for a given index set is found to be with the
depth of the sum independently of the weight. The corresponding counting
relations are given in analytic form for all classes of harmonic sums to
arbitrary depth and are tabulated up to depth .Comment: 39 pages LATEX, 1 style fil
BPS counting for knots and combinatorics on words
We discuss relations between quantum BPS invariants defined in terms of a
product decomposition of certain series, and difference equations (quantum
A-polynomials) that annihilate such series. We construct combinatorial models
whose structure is encoded in the form of such difference equations, and whose
generating functions (Hilbert-Poincar\'e series) are solutions to those
equations and reproduce generating series that encode BPS invariants.
Furthermore, BPS invariants in question are expressed in terms of Lyndon words
in an appropriate language, thereby relating counting of BPS states to the
branch of mathematics referred to as combinatorics on words. We illustrate
these results in the framework of colored extremal knot polynomials: among
others we determine dual quantum extremal A-polynomials for various knots,
present associated combinatorial models, find corresponding BPS invariants
(extremal Labastida-Mari\~no-Ooguri-Vafa invariants) and discuss their
integrality.Comment: 41 pages, 1 figure, a supplementary Mathematica file attache
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