(Elliptic) multiple zeta values in open superstring amplitudes

Transcription

1 (Elliptic) multiple zeta values in open superstring amplitudes Johannes Broedel Humboldt University Berlin based on joint work with Carlos Mafra, Nils Matthes and Oliver Schlotterer arxiv: , arxiv: Selected Topics in Theoretical High Energy Physics tbilisi, sakartvelo, September 2 st, 25

2 Introduction Goal: 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 )-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 /6

3 State of the art: open string theory Tree-level: Calculation of all amplitudes based on multiple polylogarithms and multiple zeta values and the algebraic structure of string corrections at tree-level. Broedel, Schlotterer [ Stieberger ] [Goncharov][Brown][Zagier] [ Schlotterer Stieberger ] Broedel, Schlotterer Drinfeld associator avoids necessity of solving integrals at all.[ Stieberger, Terasoma] Complete calculation boiled down to recursive application of linear algebra. Loop-level: calculation based on elliptic iterated integrals and elliptic multiple zeta Broedel, Mafra values [ Broedel Matthes, Schlotterer][ Matthes, Schlotterer] no analogue of the Drinfeld method so far: integrals can not be replaced completely yet... Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 2/6

4 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 Adams, Bogner diagrams. [ Weinzierl ][ 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 theory with an amazing degree of symmetry should produce simple answers. Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 3/6

5 Outline Tree-level One-loop z N = t z z 2 z N 2 N 2 i=2 i+ dz i multiple polylogarithms G(a,a 2,..., a n ; z) = dt z N t a G(a 2,..., a n ; t) partial fraction multiple zeta values ζ Drinfeld method - no integrals z z 2 z N z N N i+ dz N dz i δ(z ) i= elliptic iterated integrals Γ ( n n 2... n r a a 2... a r ; z) = dt f (n ) (t a ) Γ ( n 2... n r a 2... a r ; t) Fay-identities elliptic multiple zeta values ω elliptic (KZB) associator? Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 4/6

6 Tree-level, basics N-point tree-level open-string amplitude: Mafra, Schlotterer [Veneziano]... [ Stieberger ] A open string = F.A YM dependence on external states in A YM F = F (s ij ), s ij = α (k i + k j ) 2 coefficients are multiple zeta values (MZVs) F,2,...,N = N 2 i=2 N 2 i=2 i+ i+ dz i Multiple polylogarithms G(a, a 2,..., a n ; z) = dz i N i<j z i a i z ij sij { s2 z 2 N i<j z ij sij }{{} expand... ( s3 + s ) z 3 z 23 N i<j n ij= z N = z z 2 z N 2 z N ( s,n s N 3,N 2 z,n 2 z N 3,N 2 (ln z (s ij ) nij ij ) nij n ij! }{{} multiple polylogs dt t a G(a 2,..., a n ; t), G(; z) =, G( a; ) = G(; ) = G(,,..., ; z) = }{{} w! (ln z)w G(,..., ; z) = }{{} w! lnw ( z) w w Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 5/6 ) }

7 N 2 i+ N dz i (s ij ) n ij G({,, z l }, z k ) i=2 z i a i i<j n ij = }{{} integrate step by step... N (s ij ) n ij G({, }, ) i<j n ij = }{{} rewrite polylogs as multiple ζ s ζ n,...,n r = k n <k < <k r r knr = ( ) r G(,,...,,,...,,,...,, ; ) = ζ }{{}}{{} (w) n r n 5-point-example: F (23) = ζ 2 (s 2 s 23 + s 2 s 24 + s 2 s 34 + s 3 s 34 + s 23 s 34 ) + ζ 3 (s 2 2s 23 + s 2 s s 2 2s s 2 s 23 s 24 + s 2 s ) + + ζ 3,5 (...) + Pretty cumbersome - isn t there an easier way to obtain the result? Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 6/6

8 Drinfeld-method Knizhnik-Zamolodchikov equation [ Knizhnik Zamolodchikov] ( dˆf(z ) e = e ) ˆF(z ). dz z z Broedel, Schlotterer [ Stieberger, Terasoma] z N = z C\{, }, Lie-algebra generators e, e Regularized boundary values C lim z z e ˆF(z ) z z 2 z N 2 z C lim z ( z ) e ˆF(z ) z N z N = z N = (N )-point z z z2 z z N N-point z z 2 z N 2 z z z N are related by the Drinfeld associator Φ: [Drinfeld][ Le Murakami][Furusho][ Drummond Ragoucy ] C = Φ(e, e ) C, Φ(e, e ) = w[e, e ]ζ (w) w {,} Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 7/6

9 Collect tree-level results Tree-level One-loop z N = t z z 2 z N 2 N 2 i=2 i+ dz i multiple polylogarithms G(a,a 2,..., a n ; z) = dt z N t a G(a 2,..., a n ; t) partial fraction multiple zeta values ζ Drinfeld method - no integrals z z 2 z N z N N i+ dz N dz i δ(z ) i= elliptic iterated integrals Γ ( n n 2... n r a a 2... a r ; z) = dt f (n ) (t a ) Γ ( n 2... n r a 2... a r ; t) Fay-identities elliptic multiple zeta values ω elliptic (KZB) associator? Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 8/6

10 One-loop open string topologies: all genus-one worldsheets with boundaries. here: cylinder with insertions on one boundary only: Im(z) = one imaginary parameter: τ General form of the integral: t z z 2 z N z N A -loop string (, 2, 3, 4) = s 2s 23 A tree YM(, 2, 3, 4) I 4pt (, 2, 3, 4)(τ) dτ I 4pt (, 2, 3, 4)(τ) dz 4 dz 3 dz 2 dz δ(z ) 4 [ χjk (τ) ] s jk j<k }{{} Koba-Nielsen Green s function of the free boson on a genus-one surface with modulus τ: ln χ ij (τ) zij =τx ij Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 9/6

11 Compare tree-level N 2 i=2 i+ dz i z i a i N i<j n ij = n ij! (s ij) n ij (ln z ij ) n ij }{{} multiple polylogs with one-loop situation: N dz N i= i+ Suitable (iterated) object: N dz i δ(z ) n i<j n ij = ij! (s ij) n ij (ln χ ij (τ)) n ij }{{}??? ln χ ij (τ) = i z j dw f () (w z j, τ) Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes /6

12 Natural weights for differentials on an elliptic curve: [Enriquez][ Brown Levin ] f (n) (z, τ) = f (n) (z +, τ) and f (n) (z, τ) = f (n) (z + τ, τ). Explicitly: (simplification in our situation because Im(z) = ) f () (z, τ) f (2) (z, τ) 2 Parity: f (n) ( z, τ) = ( ) n f (n) (z, τ) f () (z, τ) ln θ (z, τ) + 2πi Imz Imτ [( ln θ (z, τ) + 2πi Imz ) ln θ (z, τ) θ (, τ) ] Imτ 3 θ (, τ) Relation to Eisenstein Kronecker-series: [Kronecker][ Brown Levin ] ( αω(z, α, τ) α exp F (z, α, τ) θ (, τ)θ (z + α, τ) θ (z, τ)θ (α, τ) 2πiα Im(z) Im(τ) ) F (z, α, τ) =, f (n) (z, τ)α n n= Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes /6

13 Elliptic iterated integrals (suppress τ-dependence from here... ) Γ ( n n 2... n r a a 2... a r ; z) dw f (n) (w a ) Γ ( n 2... n r a 2... a r ; w) can rewrite any integral into an elliptic iterated integral. Products of differential weights (tree-level) partial fraction: dw w a w a 2 (w a )(w a 2 ) = (w a )(a a 2 ) + (w a 2 )(a 2 a ) Products of differential weights (one-loop) Fay identities dw f (n ) (w x)f (n 2) (w) f () (w x)f () (w) =f () (w x)f () (x) f () (w)f ( + f (2) (w) + f (2) (x) + f (2) (w The Fay identity is a form of the trisecant equation for Eisenstein Kronecker series: F (z, α )F (z 2, α 2 ) = F (z, α + α 2 )F (z 2 z, α 2 ) + F (z 2, α + α 2 )F (z z 2, α ) Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 2/6

14 Elliptic multiple zeta values (emzv s) ω(n, n 2,..., n r ) f (n) (z )dz f (n2) (z 2 )dz 2... f (nr) (z r )dz r Four-point result with z i z i+ = Γ(n r,..., n 2, n ; ) = Γ ( nr n r... n... ; ) I 4pt (, 2, 3, 4)(τ) = ω(,, ) 2 ω(,,, ) (s 2 + s 23 ) β 5 = ω(,,,, ) ( s s 2 23) 2 ω(,,,, ) s2 s 23 + β 5 (s s 2 2s s 2 s s 3 23) + β 2,3 s 2 s 23 (s 2 + s 23 ) + O(α 4 ) [ ω(,,,,, 2) + ω(,,,,, ) ω(2,,,,, ) ζ2 ω(,,, ) ] β 2,3 = 3 ω(,,,, 2, ) 3 2 ω(,,,,, 2) 2 ω(,,,,, ) 2 ω(2,,,,, ) 4 3 ω(,,,,, 2) 3 ζ 2 ω(,,, ) Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 3/6

15 Comparison Tree-level One-loop z N = t z z 2 z N 2 N 2 i=2 i+ dz i multiple polylogarithms G(a,a 2,..., a n ; z) = dt z N t a G(a 2,..., a n ; t) partial fraction multiple zeta values ζ Drinfeld method - no integrals z z 2 z N z N N i+ dz N dz i δ(z ) i= elliptic iterated integrals Γ ( n n 2... n r a a 2... a r ; z) = dt f (n ) (t a ) Γ ( n 2... n r a 2... a r ; t) Fay-identities elliptic multiple zeta values ω elliptic associator? [ Knizhnik, Bernard Zamolodchikov ] Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 4/6

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 x What 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: x Goal closed/recursive form of the integrand for the one-loop open-string amplitude in terms of iterated Eisenstein integrals (analogue of Drinfeld-method) Adams, Bogner relation to functions ELi occurring in [ Weinzierl ] Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 5/6

17 Thanks! Johannes Broedel: (Elliptic) multiple zeta values in open superstring amplitudes 6/6