Polylogarithm identities in a conformal field theory in three dimensions

Remarkably, it turns out that 2 − τ is one of only **three** real, positive, z for which both

Li 2 (z) andLi 3 (z) can be expressed **in** terms of elementary functions [8] (the other po**in**ts are

z =1andz =1/2). As shown **in** the book by Lew**in** [8], the value of Li 2 (2 − τ) follows from

a comb**in**ed analysis of the follow**in**g **identities**

Li 2 (z)+Li 2 (1 − z) = π2

− log z log(1 − z)

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( ) −z

Li 2 (z)+Li 2 = − 1 1 − z 2 log2 (1 − z)

( )

1

−z

2 Li 2(z 2 )+Li 2 − Li 2 (−z) = − 1 1 − z

2 log2 (1 − z) (12)

To see the special role of the golden mean **in** these **identities**, note that two of the arguments

z 2 and −z/(1 − z) co**in**cide when z 2 − z − 1 = 0. The solutions of this are z = τ,1 − τ. Itis

not difficult to show that the above **identities** evaluated at z =2− τ, τ − 1, and 1 − τ, can

be comb**in**ed to uniquely determ**in**e Li 2 (2 − τ) [8]:

Similarly, the **identities** [8]

Li 2 (2 − τ) = π2

15 − 1 4 log2 (2 − τ) (13)

1

4 Li 3(z 2 ) = Li 3 (z)+Li 3 (−z)

( ) −z

Li 3 (z)+Li 3 +Li 3 (1 − z) = Li 3 (1) + π2

log(1 − z)

1 − z

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− 1 2 log z log2 (1 − z)+ 1 6 log3 (1 − z) (14)

evaluated at z =2− τ and τ − 1 yield [8]

Li 3 (2 − τ) = 4 5 Li 3(1) + π2

15 log(2 − τ) − 1 12 log3 (2 − τ) (15)

Insert**in**g (13) and (15) **in**to (10), we get one of our ma**in** results

˜c

N = 4 5

(16)

5