Monodromy of Polylogarithms

Minh Hoang Ngoc

LIFL, Université de Lille I

Algorithms Seminar

July 6, 1998

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Abstract
Generalized polylogarithms are complex, multivalued functions with singularities at z=0 and z=1. We calculate the monodromy at the two singularities. As opposed to the classical polylogs [11, 12], the monodromy of generalized polylogs involves the so-called ``multiple zeta values,''  which play an important role in number theory, knot theory [4, 6, 5, 10], and physics [7, 9]. Via monodromy of polylogs, Radford  showed that the C-algebra of polylogs is isomorphic to the C-algebra of non-commutative polynomials in two variables---a ``shuffle algebra'' freely generated by the so-called Lyndon words. Here, monodromy is used to give an induction proof of the linear independence of the polylogarithms. We also obtain a Gröbner basis of the polynomial relations between ``multiple zeta values'' using the techniques of non-commutative algebra. By expressing multiple zeta values in terms of the Gröbner basis, one obtains symbolic algebraic proofs of relations between multiple zeta values.

## 1   Polylogarithms and Combinatorics on Words

Let X={x0,x1}. To any word w=x0s1-1x1x0s2-1x1 ··· x0sk-1x1 we associate the multi-index s=(s1,s2,...,sk) and define the generalized polylogarithm
Liw(z)=Li s(z)=
 å n1>n2>...>nk>0
z
 n1
n
 s1 1
n
 s2 2
··· n
 sk k
.
The associated multiple zeta value is zw=z(s)=Liw(1)=Lis(1). The shuffle product is defined on words by the recursion
xu W yv = x(u W yv) + y(xu W v),
where x,yÎ X and u and v are words on X. We can extend the shuffle product linearly to the non-commutative polynomials Qá Xñ. The resulting polynomial algebra, denoted ShQ(X) is commutative and associative.

The Lyndon words L are those non-empty words on X that are inferior to each of their right factors in the lexicographical order. They are algebraically independent and generate ShQ(X), thus forming a transcendence basis. More precisely, a theorem of Radford  states that the algebra ShQ(X) is isomorphic to the polynomial algebra generated by the Lyndon words, i.e. Q[L].

## 2   Relations between Multiple Zeta Values

There are countless relations between multiple zeta values [1, 3, 2]. We content ourselves here with providing only two examples:
z(2,1)=z(3)   and   z(2,2,1)=-
11
2
z(5) +3z(2)z(3).
It turns out that a large class of relations can be explained by the collision of two distinct shuffles obeyed by the multiple zeta values. We've already seen one type of shuffle. It provides relations of the form zuW v=zu zv. A second type of shuffle provides relations of the form zu*v=zuzv and is defined by the recursion
(s1,s) * (t1,t) = (s1,s*(t1,t)) + (t1,(s1,s)*t)+(s1+t1,s*t),
where we have used the multi-index notation s=(s2,s3,...,sk), t=(t2,t3,...,tr) of Section 1. With a slight abuse of notation, we define a map z: w® zw, extended linearly in the natural way to Qá Xñ. Then z is a Q-algebra homomorphism which respects both shuffle products. Thus, if I is the ideal generated by the words u W v - u*v, then I Í kerz. We can compute a Gröbner basis for the ideal I up to any given order using only symbolic computation. The first relation above is the unique basis element of order 3. The second relation above is one of five basis elements of order 5.

## 3   Monodromy of Polylogarithms

To compute the monodromy, we use the standard keyhole contours about the two singularities z=0 and z=1. The monodromy is given by
M0Li
 wx0
= Li
 wx0
+ 2p iLiw + ···
M1Li
 wx1
= Li
 wx1
- 2p iLiw +···,
where the remaining terms are linear combinations of polylogarithms coded by words of lengths less than the length of w. For example, using the computational package Axiom, we find that
M1Li
 x0
=Li
 x0
,   M1Li
 x1
=Li
 x1
-2p i,   M1Li
 x0x1
=Li
 x0x1
-2p iLi
 x0
,
and so on. The generating series of the generalized polylogarithms is
L(z) =
 å wÎ X*
wLiw(z),
with the convention that Lix0n(z) = (log z)n/n!. Drinfel'd's differential equation [8, 9]
d
dz
L(z) = æ
ç
ç
è
x0
z
+
x1
1-z
ö
÷
÷
ø
L(z),
is satisfied, with boundary condition L(e)=exp(x0log e)+O((e)1/2) as e®0+. It turns out that L is a Lie exponential, and this fact can be used to obtain asymptotic expansions of the generalized polylogarithms at z=1.

## 4   Independence of Polylogarithms

Theorem 1   The functions Liw with wÎ X* are C-linearly independent.
Corollary 1   The C-algebra generated by the Liw is isomorphic to ShC(X). By Radford's theorem, the generalized polylogarithms coded by Lyndon words form an infinite transcendence basis.
Corollary 2   Each generalized polylogarithm Liw has a unique representation as a Q-polynomial in polylogarithms coded by Lyndon words. The classical [11, 12] polylogarithms Lik, which are coded by the Lyndon words x0k-1x1, are algebraically independent.

Proof.[Proof of Theorem 1] Given n³ 0, assume that
 å |w|£ n
lw Liw = 0,     lwÎC,     (1)
where |w| denotes the length of the word w. We prove by induction on n that lw=0 for all w, the case n=0 being trivial. Rewrite (1) as
l1 +
 å |u|
l
 ux0
Li
 ux0
+
 å |u|
l
 ux1
Li
 ux1
=0.
Applying the operators (M0-Id) and (Id-M1) on this latter expression, yields two new linear relations
ì
ï
ï
í
ï
ï
î
2p i
 å |u|=n-1
l
 ux0
Li u +
 å |u|
µuLiu
=0,
2p i
 å |u|=n-1
l
 ux1
Li u +
 å |u|
nuLiu
=0,
for certain coefficients µu and nu. By the induction hypothesis, the coefficients lux0 and lux1 with |u|=n-1 all vanish (as well as the coefficients µu and nu). Consequently,
 å |w|£ n-1
lw Liw=0,
whence lw=0 for all w, again by the induction hypothesis.

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