Functional equations for multiple polylogarithms · Functional equations for multiple polylogarithms Herbert Gangl Durham University Amplitudes 2013, 30.4.13. Outline functional equations

Functional equations for multiple polylogarithms

Herbert Gangl

Durham University

Amplitudes 2013, 30.4.13

Outline

• functional equations for classical polylogs

• relate multiple polylogs to classical ones, in low weight

• new phenomena in weight 4

• relating MPL’s of different depths

• [if time] glimpse of a mathematical application

Outline






Outline






Outline






Outline






Outline






The classical case

For classical polylogarithms, (non-trivial) functional equationsare known for small “weight”(=“transcendentality” for physicists)

• Weight 1: one has the 1-logarithm function∑k≥1

zk

k= − log(1− z) = Li1(z) .

The standard functional equation for log

log(ab) = log(a) + log(b) (1)

translates into an equation for Li1(z):

Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .

Typical here: arguments are rational functions in several variables,coefficients are integers.

Hence for weight 1 can “combine” terms, by virtue of (1), thussuccessively reducing number of terms in a given expression.

The classical case



zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case


• Weight 1: one has the 1-logarithm function

∑k≥1

zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k

= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k= − log(1− z)

= Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



The classical case



zk

k= − log(1− z) = Li1(z) .




Li1(x) + Li1(y)− Li1(x + y − x y) = 0 .



• Weight 2: the dilogarithm Li2(z),

or rather its single-valuedvariant L2(z) = =

(Li2(z)− log |z | Li1(z)

), the Bloch-Wigner dilog,

satisfies the famous five term relation L2

(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)

better memorised by writing the arguments as suitable cross-ratios(take five points on projective line, drop one each and form thecross-ratio of the remaining four).

No obvious way of “combining” terms to a simpler expression asfor weight 1 case.

There are also many other (less well-known) equations: e.g.,

given integers a, b > 0, take solutions {pa,bj (t)}j (algebraic fcts in

t) ofxa(1− x)b = t .

Then ∑j

L2

(pa,bj (t)

)= 0 .

• Weight 2: the dilogarithm Li2(z), or rather its single-valuedvariant L2(z) = =




(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)






Then ∑j

L2

(pa,bj (t)

)= 0 .





(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)






Then ∑j

L2

(pa,bj (t)

)= 0 .





(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)

better memorised by writing the arguments as suitable cross-ratios

(take five points on projective line, drop one each and form thecross-ratio of the remaining four).





Then ∑j

L2

(pa,bj (t)

)= 0 .





(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)






Then ∑j

L2

(pa,bj (t)

)= 0 .





(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)






Then ∑j

L2

(pa,bj (t)

)= 0 .





(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)



There are also many other (less well-known) equations:

e.g.,



Then ∑j

L2

(pa,bj (t)

)= 0 .





(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)






Then ∑j

L2

(pa,bj (t)

)= 0 .





(V (x , y)

)= 0, where

V (x , y) = [x ] + [y ] +[ 1− x

1− xy

]+ [1− xy ] +

[ 1− y

1− xy

], (2)






Then ∑j

L2

(pa,bj (t)

)= 0 .

In fact, these sets {pj = pa,bj (t)} also suffice to produce (as a, b

vary, infinitely many different) equations for weight 3, like∑j

(aL3

(pj

)− bL3

(1− pj

))= 0 ,

(Ln a single–valued version of Lin) and also for weight 4∑j

(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .

Our knowledge on fctl eqs is still quite limited, though: firstnon-trivial equations (mostly in two variables) were found for• weight 3: Spence (1809), Kummer (1840);• weight 4, 5: Kummer (1840)• weight 6, 7: G. (1990, 1992).• weight > 7: still unknown.

Significance: explicit presentation of algebraic K-groups.

For n > 3, the currently known equations are consideredinsufficient for such a presentation. Motivates to look at. . .



(aL3

(pj

)− bL3

(1− pj

))= 0 ,

(Ln a single–valued version of Lin)

and also for weight 4∑j

(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .






(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .






(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .

Our knowledge on fctl eqs is still quite limited, though: firstnon-trivial equations (mostly in two variables) were found for

• weight 3: Spence (1809), Kummer (1840);• weight 4, 5: Kummer (1840)• weight 6, 7: G. (1990, 1992).• weight > 7: still unknown.





(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .

Our knowledge on fctl eqs is still quite limited, though: firstnon-trivial equations (mostly in two variables) were found for• weight 3: Spence (1809), Kummer (1840);

• weight 4, 5: Kummer (1840)• weight 6, 7: G. (1990, 1992).• weight > 7: still unknown.





(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .

Our knowledge on fctl eqs is still quite limited, though: firstnon-trivial equations (mostly in two variables) were found for• weight 3: Spence (1809), Kummer (1840);• weight 4, 5: Kummer (1840)

• weight 6, 7: G. (1990, 1992).• weight > 7: still unknown.





(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .

Our knowledge on fctl eqs is still quite limited, though: firstnon-trivial equations (mostly in two variables) were found for• weight 3: Spence (1809), Kummer (1840);• weight 4, 5: Kummer (1840)• weight 6, 7: G. (1990, 1992).

• weight > 7: still unknown.





(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .






(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .






(aL3

(pj

)− bL3

(1− pj

))= 0 ,


(a(a + b)L4

(pj

)− b(a + b)L4

(1− pj

)− abL4

(1− 1

pj

))= 0 .




The multiple caseMultiple polylogarithms (MPL’s, Goncharov polylogs)(Lappo-Danilevsky, Chen, Goncharov).

Classical polylogs are somehow too special—it is important toconsider multivariable variants. Altogether they should be part ofan overarching coLie algebra structure (Beilinson, Goncharov).

Examples: Non-classical polylogs in depth 2 (for p, q > 0) are

Lip,q(x , y) =∑

0<m<n

xm

mp

yn

nq,

Combined with the classical ones they also satisfy fctl eqns, e.g.since

∑0<m<n +

∑0<n<m +

∑n=m =

∑n,m we get

Lip,q(x , y) + Liq,p(y , x) + Lip+q(xy) = Lip(x)Liq(y) .

• Nothing new for weights 2 and 3: MPL’s of weight ≤ 3 arewell-known to be expressible in terms of the classical ones, e.g.

Li1,1(x , y) = Li2( 1− x

1− y−1

)− Li2

( 1

1− y−1

)− Li2(xy) ,


Classical polylogs are somehow too special—it is important toconsider multivariable variants.

Altogether they should be part ofan overarching coLie algebra structure (Beilinson, Goncharov).


Lip,q(x , y) =∑

0<m<n

xm

mp

yn

nq,


∑0<m<n +

∑0<n<m +

∑n=m =

∑n,m we get



Li1,1(x , y) = Li2( 1− x

1− y−1

)− Li2

( 1

1− y−1

)− Li2(xy) ,




Lip,q(x , y) =∑

0<m<n

xm

mp

yn

nq,


∑0<m<n +

∑0<n<m +

∑n=m =

∑n,m we get



Li1,1(x , y) = Li2( 1− x

1− y−1

)− Li2

( 1

1− y−1

)− Li2(xy) ,




Lip,q(x , y) =∑

0<m<n

xm

mp

yn

nq,


∑0<m<n +

∑0<n<m +

∑n=m =

∑n,m we get



Li1,1(x , y) = Li2( 1− x

1− y−1

)− Li2

( 1

1− y−1

)− Li2(xy) ,




Lip,q(x , y) =∑

0<m<n

xm

mp

yn

nq,


∑0<m<n +

∑0<n<m +

∑n=m =

∑n,m we get



Li1,1(x , y) = Li2( 1− x

1− y−1

)− Li2

( 1

1− y−1

)− Li2(xy) ,




Lip,q(x , y) =∑

0<m<n

xm

mp

yn

nq,


∑0<m<n +

∑0<n<m +

∑n=m =

∑n,m we get



Li1,1(x , y) = Li2( 1− x

1− y−1

)− Li2

( 1

1− y−1

)− Li2(xy) ,




Lip,q(x , y) =∑

0<m<n

xm

mp

yn

nq,


∑0<m<n +

∑0<n<m +

∑n=m =

∑n,m we get



Li1,1(x , y) = Li2( 1− x

1− y−1

)− Li2

( 1

1− y−1

)− Li2(xy) ,

and there are similar, more complicated expressions of Li2,1(x , y)and Li1,1,1(x , y , z) in terms of Li3(z) (and products of lower weightpolylogarithms).

From weight 4 on, things are different: Li4 not sufficient. . .we need a function of at least two variables if we want to expressall weight 4 MPL’s:

e.g. Li2,2(x , y), Li3,1(x , y) or even Li1,1,1,1(x , y , z ,w)(Li4 being a kind of limiting case of these, modulo products).

It is often convenient to work with a notation corresponding toiterated integrals (similar to physicists’ G -fct)

Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)

where x1 = (z1 . . . zk)−1 , x2 = (z2 . . . zk)−1 , . . . , xk = z−1k ,

= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .


From weight 4 on, things are different: Li4 not sufficient. . .

we need a function of at least two variables if we want to expressall weight 4 MPL’s:



Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)


= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .





Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)


= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .



e.g. Li2,2(x , y), Li3,1(x , y) or even Li1,1,1,1(x , y , z ,w)

(Li4 being a kind of limiting case of these, modulo products).


Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)


= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .





Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)


= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .





Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)


= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .





Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)


= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .





Im1,...,mk(x1, . . . , xk) = (−1)kLim1,...,mk

(z1, . . . , zk)


= G (0, . . . , 0︸︷︷︸mk−1

, xk , . . . , 0, . . . , 0︸︷︷︸m1−1

, x1; 1) .

Functional equations in weight 4; case of depth 2 (write =for “mod products”):

I3,1(x , y) + I3,1(y , x) = 0 ,

furthermore, if we simultaneously transform both variables underthe usual S3-action, we get that modulo Li4-terms (denoted by ≡)

I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .

One can be more precise and give the Li4-expressions explicitly:for the first congruence, there is actually no such term needed, butfor the second one we find

I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),

and similar for the remaining ones.Rewrite equations. It is convenient to introduce cross-ratios

(abcd) = cr(a, b, c, d) =a− c

a− d· b − d

b − c

in the arguments for the I···—to make symmetries more apparent.


I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .


I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),



a− d· b − d

b − c



I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .


I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),



a− d· b − d

b − c



I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .


I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),



a− d· b − d

b − c



I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .

One can be more precise and give the Li4-expressions explicitly:

for the first congruence, there is actually no such term needed, butfor the second one we find

I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),



a− d· b − d

b − c



I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .

One can be more precise and give the Li4-expressions explicitly:for the first congruence, there is actually no such term needed,

butfor the second one we find

I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),



a− d· b − d

b − c



I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .


I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),

and similar for the remaining ones.

Rewrite equations. It is convenient to introduce cross-ratios


a− d· b − d

b − c



I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .


I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),



a− d· b − d

b − c



I3,1(x , y) + I3,1(y , x) = 0 ,


I3,1(x , y) ≡ I3,1(1− x , 1− y) ≡ I3,1(1

x,

1

y

)≡ . . .


I3,1(x , y) − I3,1(1

x,

1

y

)= Li4

([x ] − [y ] + 3

[x

y

]),



a− d· b − d

b − c


• Abbreviate (abcde)31 = I31

((abcd), (abce)

), then above reads

(abcde)31 ≡ (acbde)31 ≡ (bacde)31 ≡ −(abced)31 ,

(symmetric in first three slots, antisymmetric in last two).

• Similarly, for (abcde)22 = I22

(abcd , abce

)find, mod Li4-terms,

(abcde)22 ≡ (bacde)22 ≡ −(abdce)22 ≡ −(badce)22 ,

(symmetric in first two slots, antisymmetric in slots 3 and 4).

• Particularly nice here (swapping both slots simultaneously):

(abcde)22 + (badce)22 = (abcd)4 .

• Moreover, get 4-term equations (e.g. cyclic symmetry in lastfour) (

e(abcd)cyc)22≡ 0 ,

((abcd)cyce

)31≡ 0 .

• Among the 120 terms (aσ(1) . . . aσ(5))31, σ ∈ S5, there are only 6linearly independent ones mod Li4. Similarly for I22 instead of I31.


((abcd), (abce)

), then above reads




(abcd , abce








((abcd)cyce

)31≡ 0 .



((abcd), (abce)

), then above reads




(abcd , abce








((abcd)cyce

)31≡ 0 .



((abcd), (abce)

), then above reads




(abcd , abce








((abcd)cyce

)31≡ 0 .



((abcd), (abce)

), then above reads




(abcd , abce








((abcd)cyce

)31≡ 0 .



((abcd), (abce)

), then above reads




(abcd , abce






• Moreover, get 4-term equations

(e.g. cyclic symmetry in lastfour) (


((abcd)cyce

)31≡ 0 .



((abcd), (abce)

), then above reads




(abcd , abce








((abcd)cyce

)31≡ 0 .



((abcd), (abce)

), then above reads




(abcd , abce








((abcd)cyce

)31≡ 0 .

• Among the 120 terms (aσ(1) . . . aσ(5))31, σ ∈ S5, there are only 6linearly independent ones mod Li4.

Similarly for I22 instead of I31.


((abcd), (abce)

), then above reads




(abcd , abce








((abcd)cyce

)31≡ 0 .


Functional equations in depth 3:

Similar convention:(abcdef )211 = I211(cr(a, b, c, d), cr(a, b, c , e), cr(a, b, c, f )) etc.

• For I211, get that the alternating sum of six terms∑σ∈S3

(−1)|σ|((abc)σdef

)211

is symmetric, mod Li4, in last three slots, its difference to “fde”,say, being expressed as a sum of six Li4–terms (coefficients ±4).

• Similar expressions for I121 (but not for I112).

• Can represent I211 in terms of I31:

2 (a1a2a3a4a5a6)211 ≡ sum of 36 terms of form± (ai1 . . . ai5)31

with ik ∈ {1, . . . , 6}. (Note coefficients all ±1.)

• Can also represent I211 via 36 I22-terms, coefficients ±16 , ±1

2 .

Functional equations in depth 3: Similar convention:(abcdef )211 = I211(cr(a, b, c, d), cr(a, b, c , e), cr(a, b, c, f )) etc.



)211







2 .




)211

is symmetric, mod Li4, in last three slots,

its difference to “fde”,say, being expressed as a sum of six Li4–terms (coefficients ±4).






2 .




)211







2 .




)211







2 .




)211







2 .




)211







2 .

Functional equations in depth 4: (convention now obvious)

For I1111, the quadruple logarithm, find• simple 2-fold symmetries

(abcdefg)1111 = −(abcgfed)1111

and(abcdefg)1111 = −(acbdefg)1111 ,

• functional equations with four terms (qq=shuffle)(abc((def )qq g

))1111

= 0 =(

abc(defg)cyc

))1111

,

• and several ones with 6 terms, e.g.(a(bcd)cycefg

)1111

is symmetric in e and g .

But all the above are ‘too simple’ to combine two different depths:they do not involve I31–terms or Li4–terms.

Functional equations in depth 4: (convention now obvious)For I1111, the quadruple logarithm, find• simple 2-fold symmetries




))1111

= 0 =(

abc(defg)cyc

))1111

,


)1111







))1111

= 0 =(

abc(defg)cyc

))1111

,


)1111






• functional equations with four terms (qq=shuffle)

(abc((def )qq g

))1111

= 0 =(

abc(defg)cyc

))1111

,


)1111







))1111

= 0 =(

abc(defg)cyc

))1111

,


)1111







))1111

= 0 =(

abc(defg)cyc

))1111

,

• and several ones with 6 terms,

e.g.(a(bcd)cycefg

)1111







))1111

= 0 =(

abc(defg)cyc

))1111

,


)1111







))1111

= 0 =(

abc(defg)cyc

))1111

,


)1111



More interesting: eqs with 18 I1111–terms, adding up to acombination of Li4–terms; arguments of 3 types, indices mod 3:

(aibjbj+1bj+2ai+1cai+2) , (bjbj+1bj+2a1ca2a3) , (cbjbj+1a3bj+2a1a2).

Most interesting perhaps is the following one:

Theorem. The alternating sum over 1,2,3,4 of (a1a2a3a4bcd)1111

is antisymmetric, mod Li4-terms, under exchanging the first entrywith the sixth.

Moreover, its antisymmetrisation equals the same alternating sum(over 1,2,3,4) of (a1a2a3b)4 + (a1a2a3d)4.

We believe that all these functional equations are new.Earlier results by N.Dan (explicitly relating I1111, I31 and Li4),by F.Brown (developed rep theory of polylogs, implicit equations)and in Rhodes’s thesis (weight ≤ 5, not neglecting products).

More interesting: eqs with 18 I1111–terms, adding up to acombination of Li4–terms; arguments of 3 types, indices mod 3:(aibjbj+1bj+2ai+1cai+2) , (bjbj+1bj+2a1ca2a3) , (cbjbj+1a3bj+2a1a2).





























We believe that all these functional equations are new.

Earlier results by N.Dan (explicitly relating I1111, I31 and Li4),by F.Brown (developed rep theory of polylogs, implicit equations)and in Rhodes’s thesis (weight ≤ 5, not neglecting products).






We believe that all these functional equations are new.Earlier results by N.Dan (explicitly relating I1111, I31 and Li4),

by F.Brown (developed rep theory of polylogs, implicit equations)and in Rhodes’s thesis (weight ≤ 5, not neglecting products).






We believe that all these functional equations are new.Earlier results by N.Dan (explicitly relating I1111, I31 and Li4),by F.Brown (developed rep theory of polylogs, implicit equations)

and in Rhodes’s thesis (weight ≤ 5, not neglecting products).







Check time!

How can one find these equations?

In one word: compute the symbols of potential candidate termsand use (multi-)linear algebra;

convenient Mathematica implementation by C.Duhr, establishedfor our previous joint work (JHEP 2012, Duhr, G.,Rhodes).

How can one find these equations?

In one word: compute the symbols of potential candidate termsand use (multi-)linear algebra;convenient Mathematica implementation by C.Duhr, establishedfor our previous joint work (JHEP 2012, Duhr, G.,Rhodes).

Mathematical ImpactA Conjecture of Goncharov

Method successfully used also to tackle an old (∼ 1992) questionof Goncharov, who reduced Zagier’s Conjecture on polylogarithmsfor weight 4, say, to two highly delicate combinatorial problems.

The first of these two problems boils down to the conjecture that acertain expression can be expressed in terms of Li4. By a remark ofN.Dan, one can take this expression to be I31(V (x , y), z), i.e. asum of five I31-terms, where V (x , y) denotes any variant of the5-term equation for the dilogarithm.

Theorem. There is an Li4–expression in three variables x, y , z,whose “symbol” agrees with the one for I31(V (x , y), z).Moreover, there is an explicit solution consisting of 122 terms.

Goncharov outlined how one should get from such an expression toa functional equation for Li4. Indeed:

Theorem. There is a resulting functional equation for Li4 in fourvariables consisting of . . . 931 terms.









The first of these two problems boils down to the conjecture that acertain expression can be expressed in terms of Li4.

By a remark ofN.Dan, one can take this expression to be I31(V (x , y), z), i.e. asum of five I31-terms, where V (x , y) denotes any variant of the5-term equation for the dilogarithm.






The first of these two problems boils down to the conjecture that acertain expression can be expressed in terms of Li4. By a remark ofN.Dan, one can take this expression to be I31(V (x , y), z), i.e. asum of five I31-terms,

where V (x , y) denotes any variant of the5-term equation for the dilogarithm.













Theorem. There is an Li4–expression in three variables x, y , z,whose “symbol” agrees with the one for I31(V (x , y), z).

Moreover, there is an explicit solution consisting of 122 terms.




















Theorem. There is a resulting functional equation for Li4 in fourvariables consisting of . . .

931 terms.







A typical more complicated term has the form

(−1 + w)(−1 + xy)(1− y − z + xyz)

w(−1 + x)(−1 + y)y(−1 + wz)

After symmetrisation can reduce to 9 terms under a big symmetrygroup (S5 × S5 × S2):

Theorem. Let {cj}j = (−1, 2, 2, 4, 4, 8, 2, 3,−6); let Ai , Bj ∈ C(i , j = 1, . . . , 5). Then

9∑j=1

∑σ,τ∈S5

±L4

(fj(Aσ(1), . . . ,Aσ(5), cr(Bτ(1), . . . ,Bτ(4))

))is antisymmetric under A↔ B, where {fj}9j=1 are given by

f1(a, b, c , d , e, g) = −g(eabc − g)ecbd · eabd

(eabd − g)2,

f2(a, b, c , d , e, g) =g2

1− g

cade · cabe

eabd − g, . . .


(−1 + w)(−1 + xy)(1− y − z + xyz)

w(−1 + x)(−1 + y)y(−1 + wz)



9∑j=1

∑σ,τ∈S5

±L4

(fj(Aσ(1), . . . ,Aσ(5), cr(Bτ(1), . . . ,Bτ(4))



(eabd − g)2,

f2(a, b, c , d , e, g) =g2

1− g

cade · cabe

eabd − g, . . .


(−1 + w)(−1 + xy)(1− y − z + xyz)

w(−1 + x)(−1 + y)y(−1 + wz)



9∑j=1

∑σ,τ∈S5

±L4

(fj(Aσ(1), . . . ,Aσ(5), cr(Bτ(1), . . . ,Bτ(4))



(eabd − g)2,

f2(a, b, c , d , e, g) =g2

1− g

cade · cabe

eabd − g, . . .


(−1 + w)(−1 + xy)(1− y − z + xyz)

w(−1 + x)(−1 + y)y(−1 + wz)



9∑j=1

∑σ,τ∈S5

±L4

(fj(Aσ(1), . . . ,Aσ(5), cr(Bτ(1), . . . ,Bτ(4))



(eabd − g)2,

f2(a, b, c , d , e, g) =g2

1− g

cade · cabe

eabd − g, . . .


(−1 + w)(−1 + xy)(1− y − z + xyz)

w(−1 + x)(−1 + y)y(−1 + wz)



9∑j=1

∑σ,τ∈S5

±L4

(fj(Aσ(1), . . . ,Aσ(5), cr(Bτ(1), . . . ,Bτ(4))



(eabd − g)2,

f2(a, b, c , d , e, g) =g2

1− g

cade · cabe

eabd − g, . . .

A higher Bloch group candidate in weight 4.

Zagier conjectures that, for F a number field and n ≥ 2, there is apresentation of some mysterious mathematical object. . . K2n−1(F )in terms of generators and relations as follows:

generators (roughly): elements in kernel of symbol map for Lin;relations: “universal” such elements, i.e. functional eqns for Lin.

Such a presentation, denoted the n-th higher Bloch group of F , isknown for n = 2 by combining work of Bloch, Borel and Suslin,and for n = 3 an explicit candidate was given by Goncharov in hisproof of an important corollary about the Dedekind zeta valueζF (3) (often also called Zagier’s Conjecture).

For higher n, no candidate has been given.

Punchline: We can now give an explicit candidate for K7(F ),using the functional equation in 931 terms mentioned above.


Zagier conjectures that, for F a number field and n ≥ 2, there is apresentation of some mysterious mathematical object. . .

K2n−1(F )in terms of generators and relations as follows:













generators (roughly): elements in kernel of symbol map for Lin;

relations: “universal” such elements, i.e. functional eqns for Lin.













Such a presentation, denoted the n-th higher Bloch group of F , isknown for n = 2 by combining work of Bloch, Borel and Suslin,

and for n = 3 an explicit candidate was given by Goncharov in hisproof of an important corollary about the Dedekind zeta valueζF (3) (often also called Zagier’s Conjecture).





















Summary

• non-trivial fctl eqs not easy to come by, even for classical polylogs

• in weights ≤ 3, classical polylogs cover all MPL’s

• for weight ≥ 4, need a grip on new functions, e.g. I31, I1111

• found new fctl eqs for MPL’s in weight 4 (different depths)

• solved an old question of Goncharov, amounting to expressI31(V (x , y), z) via Li4.

Summary






Summary






Summary






Summary






Summary






Documents

Functional equations for multiple polylogarithms · Functional equations for multiple polylogarithms Herbert Gangl Durham University Amplitudes 2013, 30.4.13. Outline functional equations