(Elliptic) multiple zeta values in open superstring amplitudes
Transcript of (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 . . .
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 2/16
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.
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 3/16
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?
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 4/16
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)
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 5/16
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?
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 6/16
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)
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 7/16
Collect tree-level resultsTree-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?
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 8/16
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
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 9/16
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)
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 10/16
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
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 11/16
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)
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 12/16
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)
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 13/16
ComparisonTree-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 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!
Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 16/16