11 research outputs found
Demi-shuffle duals of Magnus polynomials in free associative algebra
We study two linear bases of the free associative algebra : one is formed by the Magnus polynomials of type
and the other is its
dual basis (formed by what we call the `demi-shuffle' polynomials) with respect
to the standard pairing on the monomials of . As
an application, we show a formula of Le-Murakami, Furusho type that expresses
arbitrary coefficients of a group-like series by the `regular' coefficients of .Comment: Note 4.3 has been adde
Elliptic multizetas and the elliptic double shuffle relations
We define an elliptic generating series whose coefficients, the elliptic
multizetas, are related to the elliptic analogues of multiple zeta values
introduced by Enriquez as the coefficients of his elliptic associator; both
sets of coefficients lie in , the ring of functions
on the Poincar\'e upper half-plane . The elliptic multizetas
generate a -algebra which is an elliptic analogue of
the algebra of multiple zeta values. Working modulo , we show that the
algebra decomposes into a geometric and an arithmetic part and
study the precise relationship between the elliptic generating series and the
elliptic associator defined by Enriquez. We show that the elliptic multizetas
satisfy a double shuffle type family of algebraic relations similar to the
double shuffle relations satisfied by multiple zeta values. We prove that these
elliptic double shuffle relations give all algebraic relations among elliptic
multizetas if (a) the classical double shuffle relations give all algebraic
relations among multiple zeta values and (b) the elliptic double shuffle Lie
algebra has a certain natural semi-direct product structure analogous to that
established by Enriquez for the elliptic Grothendieck-Teichm\"uller Lie
algebra.Comment: major revision, to appear in: Int. Math. Res. No
Generalised ladders and single-valued polylogarithms
We introduce and solve an infinite class of loop integrals which generalises
the well-known ladder series. The integrals are described in terms of
single-valued polylogarithmic functions which satisfy certain differential
equations. The combination of the differential equations and single-valued
behaviour allow us to explicitly construct the polylogarithms recursively. For
this class of integrals the symbol may be read off from the integrand in a
particularly simple way. We give an explicit formula for the simplest
generalisation of the ladder series. We also relate the generalised ladder
integrals to a class of vacuum diagrams which includes both the wheels and the
zigzags.Comment: 27 pages, 7 figure
Elliptic multiple polylogarithms in open string theory
In dieser Dissertation wird eine Methode zur Berechnung der genus-eins Korrekturen von offenen Strings zu Feldtheorie-Amplituden konstruiert. Hierzu werden Vektoren von Integralen definiert, die ein elliptisches Knizhnik-Zamolodchikov-Bernard (KZB) System auf dem punktierten Torus erfĂŒllen, und die entsprechenden Matrizen aus dem KZB System berechnet. Der elliptische KZB Assoziator erzeugt eine Relation zwischen zwei regulierten Randwerten dieser Vektoren. Die Randwerte enthalten die genus-null und genus-eins Korrekturen. Das fĂŒhrt zu einer Rekursion im Genus und der Anzahl externer ZustĂ€nde, die einzig algebraische Operationen der bekannten Matrizen aus dem KZB System umfasst. Geometrisch werden zwei externe ZustĂ€nde der genus-null WeltflĂ€che der offenen Strings zu einer genus-eins WeltflĂ€che zusammengeklebt.
Die Herleitung dieser genus-eins Rekursion und die Berechnung der relevanten Matrizen wird durch eine graphische Methode erleichtert, mit der die Kombinatorik strukturiert werden kann. Sie wurde durch eine erneute Untersuchung der auf Genus null bekannten Rekursion entwickelt, bei welcher der Drinfeld Assoziator Korrekturen offener Strings auf Genus null auf solche mit einem zusĂ€tzlichen externen Zustand abbildet. Diese genus-null Rekursion umfasst ebenfalls ausschliesslich Matrixoperationen und basiert auf einem Vektor von Integralen, der eine Knizhnik-Zamolodchikov (KZ) Gleichung erfĂŒllt. Die in der Rekursion gebrauchten Matrizen aus der KZ Gleichung werden als Darstellungen einer Zopfgruppe identifiziert und rekursiv berechnet.
Der elliptische KZB Assoziator ist die Erzeugendenreihe der elliptischen Multiplen Zeta-Werte. Die Konstruktion der genus-eins Rekursion benötigt verschiedene Eigenschaften dieser Werte und ihren definierenden Funktionen, den elliptischen Multiplen Polylogarithmen. So werden Relationen verschiedener Klassen von elliptischen Polylogarithmen und Funktionalrelationen erzeugt durch elliptische Funktionen hergeleitet.In this thesis, a method to calculate the genus-one, open-string corrections to the field-theory amplitudes is constructed. For this purpose, vectors of integrals satisfying an elliptic Knizhnik-Zamolodchikov-Bernard (KZB) system on the punctured torus are defined and the matrices from the KZB system are calculated. The elliptic KZB associator is used to relate two regularised boundary values of these vectors. The boundary values are shown to contain the open-string corrections at genus zero and genus one. This yields a recursion in the genus and the number of external states, solely involving algebraic operations on the known matrices from the KZB system. Geometrically, two external states of the genus-zero, open-string worldsheet are glued together to form a genus-one, open-string worldsheet.
The derivation of this genus-one recursion and the calculation of the relevant matrices is facilitated by a graphical method to structure the combinatorics involved. It is motivated by the reinvestigation of the recursion in the number of external states known at genus zero, where the Drinfeld associator maps the genus-zero, open-string corrections to the corrections with one more external state. This genus-zero recursion also involves matrix operations only and is based on a vector of integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. The matrices in the KZ equation and used in the recursion are shown to be braid matrices and a recursive method for their calculation is provided.
The elliptic KZB associator is the generating series of elliptic multiple zeta values. The construction of the genus-one recursion requires various properties of these values and their defining functions, the elliptic multiple polylogarithms. Thus, the third part of this thesis consists of an analysis of elliptic multiple polylogarithms, which in particular leads to relations among different classes of elliptic polylogarithms and functional relations generated by elliptic functions