(Elliptic) multiple zeta values in open superstring amplitudes

(Elliptic) multiple zeta valuesin open superstring amplitudes

Johannes BroedelHumboldt University Berlin

based on joint work with Carlos Mafra, Nils Matthes and Oliver SchlottererarXiv:1412.5535, arXiv:1507.02254

Selected Topics in Theoretical High Energy Physicstbilisi, sakartvelo, September 21st, 2015

IntroductionGoal: scattering amplitudes/cross sections in a field or string theory• standard method: Feynman/worldsheet graphs, useful and cumbersome• alternative idea: add symmetry→ obtain a more symmetric/constrained theory

→ learn about structure→ remove symmetry→ what remains• typical results: new language: special functions for particular theory

(e.g. spinor-helicity for massless theories)recursion relations: relate N -point to (N − 1)-point

• best scenario: avoid Feynman calculations completely/S-matrix approach

This talk:Open string theory as a simple (and very symmetric) testing ground• tree-level: polylogarithms (language) and Drinfeld associator (recursion)• one-loop elliptic iterated integrals (language) and elliptic multiple zeta values• Outlook: link to number theory/algebraic geometry

Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 1/16

State of the art: open string theoryTree-level:• Calculation of all amplitudes based on [Broedel, Schlotterer

Stieberger ]multiple polylogarithms and multiple zeta values [Goncharov][Brown][Zagier]and the algebraic structure of string corrections at tree-level. [Schlotterer

Stieberger ]• Drinfeld associator avoids necessity of solving integrals at all.[ Broedel, Schlotterer

Stieberger, Terasoma]Complete calculation boiled down to recursive application of linear algebra.

Loop-level:• calculation based on elliptic iterated integrals and elliptic multiple zeta

values [ Broedel, MafraMatthes, Schlotterer][ Broedel

Matthes, Schlotterer]• no analogue of the Drinfeld method so far: integrals can not be replaced com-

pletely yet . . .


Why open string theory?Iterated integrals are essential in calculations in field and string theory:• same building blocks

field theory: multiple polylogarithms at loop level. Divergences appear.string theory: multiple polylogarithms at tree level. No divergences.• field theory: elliptic iterated integrals make an appearance in particular Feynman

diagrams. [Adams, BognerWeinzierl ][Caron-Huot

Larsen ]string theory: one-loop amplitudes are natural for elliptic iterated integrals.• field theory results can be obtained from open string theory in the low-energy

limit.

After all, string theory is a heavily constrained theorywith an amazing degree of symmetry⇒ should produce simple answers.


OutlineTree-level

0

z1 z2 zN−2

1

zN−1

zN =∞

· · ·N−2∏i=2

∫ zi+1

0dzi

multiple polylogarithmsG(a1,a2, . . . , an; z)

=∫ z

0dt 1t− a1

G(a2, . . . , an; t)

partial fractionmultiple zeta values ζDrinfeld method - no integrals

One-loop

z1 z2 zN−1 zN

t

1· · ·∫ 1

0dzN

N−1∏i=1

∫ zi+1

0dzi δ(z1)

elliptic iterated integralsΓ ( n1 n2 ... nr

a1 a2 ... ar ; z)

=∫ z

0dt f (n1)(t− a1) Γ ( n2 ... nr

a2 ... ar ; t)

Fay-identitieselliptic multiple zeta values ωelliptic (KZB) associator?


Tree-level, basicsN -point tree-level open-string amplitude: [Veneziano]. . . [Mafra, Schlotterer

Stieberger ]

Aopenstring = F .AYM

• dependence on external states in AYM• F = F (sij), sij = α′(ki + kj)2

• coefficients are multiple zeta values (MZVs) 0

z1 z2 zN−2

1

zN−1

zN =∞

· · ·

F 1,2,...,N =N−2∏i=2

∫ zi+1

0dzi

N−1∏i<j

|zij |sij

s12

z12

(s13

z13+ s23

z23

). . .

(s1,N−2

z1,N−2+ . . .+ sN−3,N−2

zN−3,N−2

)

N−2∏i=2

∫ zi+1

0

dzi

zi − ai

N−1∏i<j

|zij |sij︸︷︷︸expand. . .

⇒N−1∏i<j

∞∑nij=0

(sij)nij(ln |zij |)nij

nij !︸︷︷︸multiple polylogsMultiple polylogarithms

G(a1, a2, . . . , an; z) =∫ z

0

dtt− a1

G(a2, . . . , an; t), G(; z) = 1, G(~a; 0) = G(; 0) = 0

G(0, 0, . . . , 0︸︷︷︸w

; z) = 1w! (ln z)w G(1, 1 . . . , 1︸︷︷︸

w

; z) = 1w! lnw(1− z)


N−2∏i=2

∫ zi+1

0

dzizi − ai

N−1∏i<j

∞∑nij=0

(sij)nijG(0, 1, zl, zk)︸︷︷︸integrate step by step. . .

N−1∏i<j

∞∑nij=0

(sij)nijG(0, 1, 1)

︸︷︷︸rewrite polylogs as multiple ζ’s

ζn1,...,nr =∑

0<k1<···<kr

1kn1

1 · · · knrr

= (−1)rG(0, 0, . . . , 0, 1︸︷︷︸nr

, . . . , 0, 0, . . . , 0, 1︸︷︷︸n1

; 1) = ζ(w)

5-point-example:

F (23) = 1− ζ2(s12s23 + s12s24 + s12s34 + s13s34 + s23s34)+ ζ3(s2

12s23 + s12s223 + s2

12s24 + 2s12s23s24 + s12s224 + · · · ) + · · ·

+ ζ3,5(. . .) + · · ·

Pretty cumbersome - isn’t there an easier way to obtain the result?


Drinfeld-method [ Broedel, SchlottererStieberger, Terasoma]

Knizhnik-Zamolodchikov equation [ KnizhnikZamolodchikov]

dF(z0)dz0

=(e0z0− e1z0 − 1

)F(z0) .

• z0 ∈ C\0, 1, Lie-algebra generators e0, e1 0

z1 z2 zN−2 z0

1

zN−1

zN =∞

· · ·Regularized boundary values

C0 ≡ limz0→0

z−e00 F(z0)

(N− 1)-point

0

z1z2

z0 z0

1

zN−1

zN =∞

C1 ≡ limz0→1

(1− z0)e1F(z0)

N-point

0

z1 z2 zN−2 z0 z0

1

zN−1

zN =∞

· · ·

are related by the Drinfeld associator Φ: [Drinfeld][ LeMurakami][Furusho][Drummond

Ragoucy ]

C1 = Φ(e0, e1)C0, Φ(e0, e1) =∑

w∈0,1w[e0, e1]ζ(w)


Collect tree-level resultsTree-level

0

z1 z2 zN−2

1

zN−1

zN =∞

· · ·N−2∏i=2

∫ zi+1

0dzi


=∫ z

0dt 1t− a1

G(a2, . . . , an; t)


One-loop

z1 z2 zN−1 zN

t

1· · ·∫ 1

0dzN

N−1∏i=1

∫ zi+1

0dzi δ(z1)


a1 a2 ... ar ; z)

=∫ z

0dt f (n1)(t− a1) Γ ( n2 ... nr

a2 ... ar ; t)

Fay-identitieselliptic multiple zeta values ωelliptic (KZB) associator?


z1 z2 zN−1 zN

t

1· · ·

One-loop open string• topologies: all genus-one worldsheets

with boundaries.• here: cylinder with insertions on

one boundary only: Im(z) = 0• one imaginary parameter: τ

General form of the integral:

A1-loopstring (1, 2, 3, 4) = s12s23A

treeYM(1, 2, 3, 4)

∫ ∞0

dτ I4pt(1, 2, 3, 4)(τ)

I4pt(1, 2, 3, 4)(τ) ≡∫ 1

0dz4

∫ z4

0dz3

∫ z3

0dz2

∫ z2

0dz1 δ(z1)

4∏j<k

[χjk(τ)

]sjk

︸︷︷︸Koba-Nielsen

Green’s function of the free boson on a genus-one surface with modulus τ :

lnχij(τ)∣∣∣zij=τxij


Compare tree-level

N−2∏i=2

∫ zi+1

0

dzizi − ai

N−1∏i<j

∞∑nij=0

1nij !

(sij)nij (ln |zij |)nij︸︷︷︸multiple polylogs

with one-loop situation:

∫ 1

0dzN

N−1∏i=1

∫ zi+1

0dzi δ(z1)

N∏i<j

∞∑nij=0

1nij !

(sij)nij (lnχij(τ))nij︸︷︷︸???

Suitable (iterated) object:

lnχij(τ) =∫ zi

zj

dw f (1)(w − zj , τ)


Natural weights for differentials on an elliptic curve: [Enriquez][BrownLevin ]

f (n)(z, τ) = f (n)(z + 1, τ) and f (n)(z, τ) = f (n)(z + τ, τ) .

Explicitly: (simplification in our situation because Im(z) = 0)

f (0)(z, τ) ≡ 1 f (1)(z, τ) ≡ ∂ ln θ1(z, τ) + 2πi ImzImτ

f (2)(z, τ) ≡ 12[(∂ ln θ1(z, τ) + 2πi ImzImτ

)2+ ∂2 ln θ1(z, τ)− 1

3θ′′′1 (0, τ)θ′1(0, τ)

]Parity: f (n)(−z, τ) = (−1)nf (n)(z, τ)

Relation to Eisenstein–Kronecker-series: [Kronecker][BrownLevin ]

F (z, α, τ) ≡ θ′1(0, τ)θ1(z + α, τ)θ1(z, τ)θ1(α, τ) ,

αΩ(z, α, τ) ≡ α exp(

2πiα Im(z)Im(τ)

)F (z, α, τ) =

∞∑n=0

f (n)(z, τ)αn


Elliptic iterated integrals (suppress τ -dependence from here. . . )

Γ ( n1 n2 ... nra1 a2 ... ar ; z) ≡

∫ z

0dw f (n1)(w − a1) Γ ( n2 ... nr

a2 ... ar ;w)

⇒ can rewrite any integral∫

1234 . . . into an elliptic iterated integral .

Products of differential weights (tree-level) ⇒ partial fraction:∫ z

0dw 1

w − a1

1w − a2

· · · ⇒ 1(w−a1)(w − a2) = 1

(w−a1)(a1−a2)+ 1(w−a2)(a2−a1)

Products of differential weights (one-loop) ⇒ Fay identities∫ z

0dw f (n1)(w − x)f (n2)(w) · · · ⇒ f (1)(w−x)f (1)(w) =f (1)(w−x)f (1)(x)−f (1)(w)f (1)(x)

+ f (2)(w) + f (2)(x) + f (2)(w − x)

The Fay identity is a form of the trisecant equation for Eisenstein–Kronecker series:

F (z1, α1)F (z2, α2) = F (z1, α1 + α2)F (z2 − z1, α2)+ F (z2, α1 + α2)F (z1 − z2, α1)


Elliptic multiple zeta values (eMZV’s)

ω(n1, n2, . . . , nr) ≡∫

0≤zi≤zi+1≤1

f (n1)(z1)dz1 f(n2)(z2)dz2 . . . f

(nr)(zr)dzr

= Γ(nr, . . . , n2, n1; 1) = Γ ( nr nr−1 ... n10 0 ... 0 ; 1)

Four-point result

I4pt(1, 2, 3, 4)(τ) =ω(0, 0, 0) − 2ω(0, 1, 0, 0) (s12 + s23)+ 2ω(0, 1, 1, 0, 0)

(s2

12 + s223)− 2ω(0, 1, 0, 1, 0) s12s23

+ β5 (s312 + 2s2

12s23 + 2s12s223 + s3

23)+ β2,3 s12s23(s12 + s23) + O(α′4)

with

β5 = 43[ω(0, 0, 1, 0, 0, 2) + ω(0, 1, 1, 0, 1, 0)− ω(2, 0, 1, 0, 0, 0)− ζ2 ω(0, 1, 0, 0)

]β2,3 = 1

3 ω(0, 0, 1, 0, 2, 0)− 32 ω(0, 1, 0, 0, 0, 2)− 1

2 ω(0, 1, 1, 1, 0, 0)

− 2 ω(2, 0, 1, 0, 0, 0)− 43 ω(0, 0, 1, 0, 0, 2)− 10

3 ζ2 ω(0, 1, 0, 0)


ComparisonTree-level

0

z1 z2 zN−2

1

zN−1

zN =∞

· · ·N−2∏i=2

∫ zi+1

0dzi


=∫ z

0dt 1t− a1

G(a2, . . . , an; t)


One-loop

z1 z2 zN−1 zN

t

1· · ·∫ 1

0dzN

N−1∏i=1

∫ zi+1

0dzi δ(z1)


a1 a2 ... ar ; z)

=∫ z

0dt f (n1)(t− a1) Γ ( n2 ... nr

a2 ... ar ; t)

Fay-identitieselliptic multiple zeta values ωelliptic associator? [Knizhnik, Bernard

Zamolodchikov ]Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 14/16

Summary• eMZVs are natural language for one-loop amplitudes in open string theory• full amplitude only after τ -integration and consideration of other topologies

eMZVs might not be the only ingredient: Euler sums?• open-string result does not contain divergent eMZVs

xWhat else?• eMZVs: can be represented as iterated Eisenstein integrals• iterated Eisenstein integrals nicely related to special derivation algebra, [Pollack]

available cusp forms on the elliptic curve ⇔ number of ”basis” eMZVs [Brown]• number of ”basis” eMZVs + canonical choice known [Hain][Broedel, Matthes

Schlotterer ]• using our formalism, one can derive new relations in the derivation algebra u,

which match the known pattern of cusp forms• numerous relations for eMZVs: https://tools.aei.mpg.de/emzv

xGoal• closed/recursive form of the integrand for the one-loop open-string amplitude

in terms of iterated Eisenstein integrals (analogue of Drinfeld-method)• relation to functions ELi occurring in [Adams, Bogner

Weinzierl ]Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 15/16

Thanks!


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(Elliptic) multiple zeta values in open superstring amplitudes