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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
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 \(|z_{1,N-1}z_{1,N}z_{N-1,N}|^2\).
References
Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Goncharov, A.B., Postnikov, A., Trnka, J.: Scattering amplitudes and the positive grassmannian. arXiv:1212.5605 [hep-th]
Arkani-Hamed, N., Trnka, J.: The Amplituhedron. JHEP 1410, 30 (2014). arXiv:1312.2007 [hep-th]
Bern, Z., Dixon, L.J., Perelstein, M., Rozowsky, J.S.: Multileg one loop gravity amplitudes from gauge theory. Nucl. Phys. B 546, 423 (1999). [hep-th/9811140]
Bern, Z., Carrasco, J.J.M., Johansson, H.: New relations for gauge-theory amplitudes. Phys. Rev. D 78, 085011 (2008). arXiv:0805.3993 [hep-ph]
Beukers, F.: Algebraic A-hypergeometric functions. Invent. Math. 180, 589–610 (2010)
Bjerrum-Bohr, N.E.J., Damgaard, P.H., Vanhove, P.: Minimal basis for gauge theory amplitudes. Phys. Rev. Lett. 103, 161602 (2009). arXiv:0907.1425 [hep-th]
Bjerrum-Bohr, N.E.J., Damgaard, P.H., Sondergaard, T., Vanhove, P.: The momentum kernel of gauge and gravity theories. JHEP 1101, 001 (2011). arXiv:1010.3933 [hep-th]
Blümlein, J., Broadhurst, D.J., Vermaseren, J.A.M.: The multiple zeta value data mine. Comput. Phys. Commun. 181, 582 (2010). arXiv:0907.2557 [math-ph]
Boels, R.H.: On the field theory expansion of superstring five point amplitudes. Nucl. Phys. B 876, 215 (2013). arXiv:1304.7918 [hep-th]
Bogner, C., Weinzierl, S.: Periods and Feynman integrals. J. Math. Phys. 50, 042302 (2009). arXiv:0711.4863 [hep-th]
Broadhurst, D.J., Kreimer, D.: Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops. Phys. Lett. B 393, 403 (1997). [hep-th/9609128]
Broedel, J., Schlotterer, O., Stieberger, S.: Polylogarithms, multiple zeta values and superstring amplitudes. Fortsch. Phys. 61, 812 (2013). arXiv:1304.7267 [hep-th]
Broedel, J., Schlotterer, O., Stieberger, S., Terasoma, T.: Notes on Lie Algebra structure of Superstring Amplitudes. unpublished
Brown, F.: Single-valued multiple polylogarithms in one variable. C.R. Acad. Sci. Paris, Ser. I 338, 527–532 (2004)
Brown, F.: Multiple zeta values and periods of moduli spaces \(M_{0 ,n} ( {{\bf R}} )\). Ann. Sci. Ec. Norm. Sup. 42, 371 (2009). arXiv:math/0606419 [math.AG]
Brown, F.C.S., Carr, S., Schneps, L.: Algebra of cell-zeta values. Compositio Math. 146, 731–771 (2010)
Brown, F.: On the decomposition of motivic multiple zeta values. Galois-Teichmüller Theory Arithmetic Geometry Adv. Stud. Pure Math. 63, 31–58 (2012). arXiv:1102.1310 [math.NT]
Brown, F.C.S., Levin, A.: Multiple Elliptic Polylogarithms. arXiv:1110.6917 [math.NT]
Brown, F.: Mixed tate motives over \({ Z}\). Ann. Math. 175, 949–976 (2012)
Brown, F.: Single-valued Motivic periods and multiple zeta values. SIGMA 2, e25 (2014) arXiv:1309.5309 [math.NT]
Brown, F.: Periods and Feynman amplitudes. arXiv:1512.09265 [math-ph]
Brown, F.: A class of non-holomorphic modular forms I. arXiv:1707.01230 [math.NT]
Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois groups over \({\bf Q}\), Springer, MSRI publications 16 (1989), 72-297; “Periods for the fundamental group,” Arizona Winter School (2002)
Drinfeld, V.G.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \(Gal(\overline{Q},{Q}) \). Alg. Anal. 2, 149: English translation: leningrad Math. J. 2(1991), 829–860 (1990)
Drummond, J.M., Ragoucy, E.: Superstring amplitudes and the associator. JHEP 1308, 135 (2013). arXiv:1301.0794 [hep-th]
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Generalized Euler integrals and \(A\)-hypergeometric functions. Adv. Math. 84, 255–271 (1990)
Golden, J., Goncharov, A.B., Spradlin, M., Vergu, C., Volovich, A.: Motivic amplitudes and cluster coordinates. JHEP 1401, 091 (2014). arXiv:1305.1617 [hep-th]
Goncharov, A.B.: Multiple zeta-values, Galois groups, and geometry of modular varieties. arXiv:math/0005069v2 [math.AG]
Goncharov, A.B.: Multiple polylogarithms and mixed Tate motives. arXiv:math/ 0103059v4 [math.AG]
Goncharov, A., Manin, Y.: Multiple \(\zeta \)-motives and moduli spaces \(\overline{{\cal{M}}}_{0, n}\). Compos. Math. 140, 1–14 (2004). arXiv:math/0204102
Goncharov, A.B.: Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Math. J. 128, 209–284 (2005). arXiv:math/0208144v4 [math.AG]
Goncharov, A.B.: Private communication
Ihara, Y.: Some arithmetic aspects of Galois actions in the pro-p fundamental group of \({ P}^1-\{0, 1, \infty \}\). In: Arithmetic Fundamental Groups and Noncommutative Algebra. Proceedings of Symposia in Pure Mathematics 70 (2002)
Kawai, H., Lewellen, D.C., Tye, S.H.H.: A relation between tree amplitudes of closed and open strings. Nucl. Phys. B 269, 1 (1986)
Kontsevich, M., Zagier, D.: Periods. In: Engquist, B., Schmid, W. (eds.) Mathematics unlimited—2001 and beyond, Berlin, pp. 771–808. Springer, New York (2001)
Le, T.Q.T., Murakami, J.: Kontsevich’s integral for the Kauffman polynomial. Nagoya Math. J. 142, 39–65 (1996)
Mafra, C.R., Schlotterer, O., Stieberger, S.: Complete N-point superstring disk amplitude I. Pure spinor computation. Nucl. Phys. B 873, 419 (2013). arXiv:1106.2645 [hep-th]
Mafra, C.R., Schlotterer, O., Stieberger, S.: Complete N-point superstring disk amplitude II. Amplitude and hypergeometric function structure. Nucl. Phys. B 873, 461 (2013). arXiv:1106.2646 [hep-th]
Oprisa, D., Stieberger, S.: Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums. [hep-th/0509042]
Puhlfürst, G., Stieberger, S.: Differential equations, associators, and recurrences for amplitudes. Nucl. Phys. B 902, 186 (2016) arXiv:1507.01582 [hep-th]. A feynman integral and its recurrences and associators. Nucl. Phys. B 906, 168 (2016). arXiv:1511.03630 [hep-th]
Schlotterer, O., Stieberger, S.: Motivic multiple zeta values and superstring amplitudes. J. Phys. A 46, 475401 (2013). arXiv:1205.1516 [hep-th]
Schnetz, O.: Graphical functions and single-valued multiple polylogarithms. Commun. Num. Theor. Phys. 08, 589 (2014). arXiv:1302.6445 [math.NT]
Stieberger, S., Taylor, T.R.: Multi-gluon scattering in open superstring theory. Phys. Rev. D 74, 126007 (2006). [hep-th/0609175]
Stieberger, S.: Open & Closed vs. Pure Open String Disk Amplitudes. arXiv:0907.2211 [hep-th]
Stieberger, S.: Constraints on tree-level higher order gravitational couplings in superstring theory. Phys. Rev. Lett. 106, 111601 (2011). arXiv:0910.0180 [hep-th]
Stieberger, S.: Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator. J. Phys. A 47, 155401 (2014). arXiv:1310.3259 [hep-th]
Stieberger, S., Taylor, T.R.: Closed string amplitudes as single-valued open string amplitudes. Nucl. Phys. B 881, 269 (2014). arXiv:1401.1218 [hep-th]
Terasoma, T.: Selberg integrals and multiple zeta values. Compos. Math. 133, 1–24 (2002)
Tsunogai, H.: On ranks of the stable derivation algebra and Deligne’s problem. Proc. Japan Acad. Ser. A 73, 29–31 (1997)
Zagier, D.: Values of zeta functions and their applications. In: Joseph, A., et al. (eds.) First European Congress of Mathematics (Paris, 1992), vol. II, pp. 497–512. Basel, Birkh��user (1994)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-030-37031-2_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-37030-5
Online ISBN: 978-3-030-37031-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)