Abstract
Scattering amplitudes which describe the interaction of physical states play an important role in determining physical observables. In string theory the physical states are given by vibrations of open and closed strings and their interactions are described (at the leading order in perturbation theory) by a world–heet given by the topology of a disk or sphere, respectively. Formally, for scattering of N strings this leads to \(N\!-\!3\)–dimensional iterated real integrals along the compactified real axis or \(N\!-\!3\)–dimensional complex sphere integrals, respectively. As a consequence the physical observables are described by periods on \({\mathscr {M}}_{0,N}\)–the moduli space of Riemann spheres of N ordered marked points. The mathematical structure of these string amplitudes share many recent advances in arithmetic algebraic geometry and number theory like multiple zeta values, single–valued multiple zeta values, Drinfeld, Deligne associators, Hopf algebra and Lie algebra structures related to Grothendiecks Galois theory. We review these results, with emphasis on a beautiful link between generalized hypergeometric functions describing the real iterated integrals on \({\mathscr {M}}_{0,N}(\mathbf{R})\) and the decomposition of motivic multiple zeta values. Furthermore, a relation expressing complex integrals on \({\mathscr {M}}_{0,N}(\mathbf{C})\) as single–valued projection of iterated real integrals on \({\mathscr {M}}_{0,N}(\mathbf{R})\) is exhibited.
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Notes
- 1.
- 2.
The initial data for a GKZ–system is an integer matrix \(A\in \mathbf{Z}^{r\times n}\) together with a parameter vector \(\gamma \in \mathbf{C}^r\). For a given matrix A the structure of the GKZ–system depends on the properties of the vector \(\gamma \) defining non–resonant and resonant systems. Example a non–resonant system of A–hypergeometric equations is irreducible [5].
- 3.
More precisely, at an algebraic value of their argument their value is \(\frac{1}{\pi }\wp \), with \(\wp \) being the set of periods.
- 4.
The matrix S with entries \(S_{\rho ,\sigma }=S[\rho |\sigma ]\) is defined as a \((N-3)!\times (N-3)!\) matrix with its rows and columns corresponding to the orderings \(\rho \equiv \{\rho (2),\ldots ,\rho (N-2)\}\) and \(\sigma \equiv \{\sigma (2),\ldots ,\sigma (N-2)\}\), respectively. The matrix S is symmetric, i.e. \(S^t=S\).
- 5.
The ordering colons \(:\ldots :\) are defined such that matrices with larger subscript multiply matrices with smaller subscript from the left, i.e. \(: \, M_{i} \ M_{j} \, : = \left\rbrace \begin{array}{lcl}M_{i} \ M_j\ , &{} i \ge j\ ,\\ M_{j} \ M_i\ , &{} i<j\ .\end{array}\right.\) The generalization to iterated matrix products \(: M_{i_1} M_{i_2} \ldots M_{i_p}:\) is straightforward.
- 6.
- 7.
A Hopf algebra is an algebra \({\mathscr {A}}\) with multiplication \(\mu : {\mathscr {A}}\otimes {\mathscr {A}}\rightarrow {\mathscr {A}}\), i.e. \(\mu (x_1\otimes x_2)=x_1\cdot x_2\) and associativity. At the same time it is also a coalgebra with coproduct \(\varDelta : {\mathscr {A}}\rightarrow {\mathscr {A}}\otimes {\mathscr {A}}\) and coassociativity such that the product and coproduct are compatible: \(\varDelta (x_1\cdot x_2)=\varDelta (x_1)\cdot \varDelta (x_2)\), with \(x_1,x_2\in {\mathscr {A}}\).
- 8.
Note, that there is no canonical choice of \(\phi \) and the latter depends on the choice of motivic generators of \({\mathscr {H}}\).
- 9.
The choice of \(\phi \) describes for each weight \(2r+1\) the motivic derivation operators \(\partial _{2r+1}^\phi \) acting on the space of motivic MZVs \(\partial _{2r+1}^\phi :{\mathscr {H}}\rightarrow {\mathscr {H}}\) [17] as:
$$\begin{aligned} \partial _{2r+1}^\phi =(c_{2r+1}^\phi \otimes id) \circ D_{2r+1}\ , \end{aligned}$$(55)with the coefficient function \(c_{2r+1}^\phi \).
- 10.
Note the useful relation \(\phi ^B(Q_8^{\mathfrak {m}})=f_5f_3\ [M_3,M_5]\) for \(Q_8^{\mathfrak {m}}={1\over 5}\zeta _{3,5}^{\mathfrak {m}} \ [M_5,M_3]\).
- 11.
For instance instead of taking a basis containing the depth one elements \(\zeta ^{\mathfrak {m}}_{2n+1}\) one also could choose the set of Lyndon words in the Hoffman elements \(\zeta ^{\mathfrak {m}}_{n_1,\ldots ,n_r}\), with \(n_i=2,3\) and define the corresponding matrices (33).
- 12.
The relation (96) implies, that any commutator \({\mathscr {Q}}_{(2)}\) is similar to an anti–symmetric matrix, and hence (97) implies \([\; [M_a,M_b],[M_c,M_d]\; ]=0\), which in turn as a result of the Jacobi relation yields the following identity: \([M_a,[M_b,[M_c,M_d]]]-[M_b,[M_a,[M_c,M_d]]] = [\; [M_a,M_b],[M_c,M_d]\; ]=0\). Furthermore, (96) implies that the commutator \({\mathscr {Q}}_{(3)}\) is similar to a symmetric and traceless matrix. As a consequence from (98), we obtain the following anti–commutation relation: \(\{\; [M_a,M_b],[M_c,[M_d,M_e]\; \}=0\). Relations for \(N=5\) between different matrices \(M_{2i+1}\) have also been discussed in [9].
- 13.
The factor \(V_\mathrm{CKG}\) accounts for the volume of the conformal Killing group of the sphere after choosing the conformal gauge. It will be canceled by fixing three vertex positions according to (1) and introducing the respective c–ghost factor \(|z_{1,N-1}z_{1,N}z_{N-1,N}|^2\).
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Acknowledgements
We wish to thank the organizers (especially José Burgos, Kurush Ebrahimi-Fard, and Herbert Gangl) of the workshop Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory and the conference Multiple Zeta Values, Modular Forms and Elliptic Motives II at ICMAT, Madrid for inviting me to present the work exhibited in this publication and creating a stimulating atmosphere.
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Stieberger, S. (2020). Periods and Superstring Amplitudes. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_3
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