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A Tree Expansion Formula of a Homology Intersection Number on the Configuration Space \(\mathcal {M}_{0,n}\)

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Abstract

In Mizera (J High Energy Phys 8:097, 2017) , Sebastian Mizera discovered a tree expansion formula of a homology intersection number on the configuration space \(\mathcal {M}_{0,n}\). The formula originates in a study of Kawai–Lewellen–Tye relation in string theory. In this paper, we give an elementary proof of the formula. The basic ingredients are the combinatorics of the real moduli space \(\overline{\mathcal {M}}_{0,n}(\mathbb {R})\) and a combinatorial identity related to the face number of the associahedron.

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Notes

  1. In this paper, we assume that \(s_{ij}\) are generic parameters in a sense that will be clarified later (see Sect. 4). Under this condition, the twisted homology group is canonically isomorphic to its Borel-Moore counterpart.

  2. In [28], the Deligne–Knudsen–Mumford compactification \(\overline{\mathcal {M}}_{0,n}\) is denoted by \(\widetilde{\mathcal {M}}_{0,n}\)

  3. More precisely, \(m(\alpha |\beta )\) should be denoted by \(m_1(\alpha |\beta )\) [28, 29].

  4. One can also regard the hourglass as a bubble [8].

  5. The physical intuition of this condition is that when the parameters \(s_{ij}\) become real valued, this corresponds to physical states in string theory going “on-shell.” When this occurs, the Riemann surface degenerates and the dimension of the homology group drops. The condition (\(*\)) is meant to prevent this and stay at a generic point in the space of kinematics, the space of Mandelstam invariants “\(s_{ij}\)”.

  6. If \(K(\alpha )\cap K(\beta )=\varnothing \) in \(\overline{\mathcal {M}}_{0,n}(\mathbb {R})\), one has \(\langle reg[C^+(\alpha )],[C^-(\beta )]\rangle _h=0\) by the definition of the twisted homology intersection number.

  7. This is a formula of a homology intersection number on a complement of a hyperplane arrangement in a projective space. However, since the computation of intersection is a local problem, we can apply the formula even after blowing-up the projective space. The signature effect of blowing-up must be taken into account.

  8. To be more precise, the linear operator \(l_e:H^0(\Delta (\alpha );\mathcal {L})\rightarrow H^0(\Delta (\alpha );\mathcal {L})\) does not depend on the choice of the base point t but depends on \(\alpha \). In order to simplify the notation, we simply use the symbol \(l_e\) in which the dependency on \(\alpha \) is obscure.

References

  1. Aomoto, K., Kita, M.: Theory of hypergeometric functions. With an appendix by Toshitake Kohno. Translated from the Japanese by Kenji Iohara. Springer Monographs in Mathematics. Springer, Tokyo (2011)

  2. Armstrong, S.M., Carr, M., Devadoss, S.L., Engler, E., Leininger, A., Manapat, M.: Particle configurations and Coxeter operads. J. Homotopy Relat. Struct. 4(1), 83–109 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and iIts Applications, vol. 71. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  4. Bjerrum-Bohr, N.E.J., Damgaard, P.H., Søndergaard, T., Vanhove, P.: The momentum kernel of gauge and gravity theories. J. High Energy Phys. 1, 1–18 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  5. Cohen, D.C., Dimca, A., Orlik, P.: Nonresonance conditions for arrangements (English, French summary). Ann. Inst. Fourier (Grenoble) 53(6), 1883–1896 (2003)

    Article  MathSciNet  Google Scholar 

  6. Cho, K., Matsumoto, K.: Intersection theory for twisted cohomologies and twisted Riemann’s period relations. I. Nagoya Math. J. 139, 67–86 (1995)

    Article  MathSciNet  Google Scholar 

  7. Cachazo, F., He, S., Yuan, E.Y.: Scattering of massless particles: scalars, gluons and gravitons. J. High Energy Phys. 2014, 33 (2014)

    Article  Google Scholar 

  8. Devadoss, S.L.: Tessellations of moduli spaces and the mosaic operad. In: Homotopy Invariant Algebraic Structures. Contemporary Mathematics, vol. 239, pp. 91–114. Baltimore (1998). J. M. Boardman, J.-P. Meyer, J. Morava, W. S. Wilson (editors) American Mathematical Society, Providence (1999)

  9. Devadoss, S.L., Morava, J.: Navigation in tree spaces. Adv. Appl. Math. 67, 75–95 (2015)

    Article  MathSciNet  Google Scholar 

  10. Goto, Yoshiaki: Twisted cycles and twisted period relations for Lauricella’s hypergeometric function FC. Int. J. Math. 24(12), 1350094 (2013)

    Article  Google Scholar 

  11. Goto, Y., Matsubara-Heo, S.-J.: Homology and cohomology intersection numbers of GKZ systems. arXiv:2006.07848

  12. Hanamura, M., Yoshida, M.: Hodge structure on twisted cohomologies and twisted Riemann inequalities. I. (English summary). Nagoya Math. J. 154, 123–139 (1999)

    Article  MathSciNet  Google Scholar 

  13. Haraoka, Y.: Multiplicative middle convolution for KZ equations. Math. Z. 294(3–4), 1787–1839 (2020)

    Article  MathSciNet  Google Scholar 

  14. Huang, Y., Siegel, W., Yuan, E.Y.: Factorization of chiral string amplitudes. J. High Energy Phys. 9, 101 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  15. Kapranov, Mikhail M.: The permutoassociahedron, Mac Lane’s coherence theorem and asymptotic zones for the KZ equation. (English summary). J. Pure Appl. Algebra 85(2), 119–142 (1993)

    Article  MathSciNet  Google Scholar 

  16. Kapranov, M. M., Chow quotients of Grassmannians. I. In: I. M. Gel’fand Seminar. Advances in Soviet Mathematics, vol. 16, , pp. 29–110, Part 2. American Mathematical Society, Providence (1993)

  17. Kawai, H., Lewellen, D.C., Tye, S.-H.H.: A relation between tree amplitudes of closed and open strings. Nucl. Phys. B 269(1), 1–23 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  18. Kita, M., Yoshida, M.: Intersection theory for twisted cycles. II. Degenerate arrangements. Math. Nachr. 168, 171–190 (1994)

    Article  MathSciNet  Google Scholar 

  19. Knudsen, F.F.: The projectivity of the moduli space of stable curves. II. The stacks Mg, n. Math. Scand. 52(2), 161–199 (1983)

    Article  MathSciNet  Google Scholar 

  20. Kohno, T.: Homology of a local system on the complement of hyperplanes. Proc. Jpn. Acad. Ser. A Math. Sci. 62(4), 144–147 (1986)

    Article  MathSciNet  Google Scholar 

  21. Mimachi, K., Ohara, K., Yoshida, M.: Intersection numbers for loaded cycles associated with Selberg-type integrals. Tohoku Math. J. (2) 56(4), 531–551 (2004)

    Article  MathSciNet  Google Scholar 

  22. Mano, T., Watanabe, H.: Twisted cohomology and homology groups associated to the Riemann–Wirtinger integral. Proc. Am. Math. Soc. 140(11), 3867–3881 (2012)

    Article  MathSciNet  Google Scholar 

  23. Matsubara-Heo, S.-J.: Euler and Laplace integral representations of GKZ hypergeometric functions. arXiv: 1904.00565

  24. Matsubara-Heo, S.-J.: Computing cohomology intersection numbers of GKZ hypergeometric systems, to appear in Proceedings of Science, MathemAmplitudes 2019: Intersection Theory & Feynman Integrals (MA2019), 18–20 Dec 2019, Padova, Italy. arXiv:2008.03176

  25. Matsumoto, K., Yoshida, M.: Monodromy of Lauricella’s hypergeometric \(F_A\)-system. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13(2), 551–577 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Mimachi, K.: Intersection numbers for twisted cycles and the connection problem associated with the generalized hypergeometric function \({}_{n+1}F_n\). Int. Math. Res. Not. IMRN 8, 1757–1781 (2011)

    MATH  Google Scholar 

  27. Mimachi, K., Yoshida, M.: Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory. Commun. Math. Phys. 250(1), 23–45 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  28. Mizera, S.: Combinatorics and topology of Kawai–Lewellen–Tye relations. J. High Energy Phys. 8, 097 (2017)

    Article  MathSciNet  ADS  Google Scholar 

  29. Mizera, S.: Inverse of the string theory KLT kernel. J. High Energy Phys. 6, 084 (2017)

    Article  MathSciNet  ADS  Google Scholar 

  30. Ohara, K., Sugiki, Y., Takayama, N.: Quadratic relations for generalized hypergeometric functions \({}_pF_{p-1}\). Funkc. Ekvac. 46(2), 213–251 (2003)

    Article  Google Scholar 

  31. Petkovšek, M., Wilf, H.S., Zeilberger, D.: A=B. With a foreword by Donald E. Knuth. With a separately available computer disk. A K Peters, Ltd., Wellesley, MA (1996)

  32. Selberg, A.: Bemerkninger om et multipelt integral (Norwegian). Norsk Mat. Tidsskr. 26, 71–78 (1944)

    MathSciNet  Google Scholar 

  33. Stasheff, J.D.: Homotopy associativity of H-spaces. I. Trans. Am. Math. Soc. 108, 275–292 (1963)

    MathSciNet  MATH  Google Scholar 

  34. Whittaker E. T.; Watson G. N., A course of modern analysis. An introduction to the general theory of infinite processes and of analytic functions: with an account of the principal transcendental functions. Reprint of the fourth: edition, p. 1996. Cambridge University Press, Cambridge, Cambridge Mathematical Library (1927)

  35. Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math. 108(3), 575–633 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  36. Yoshida, M.: The democratic compactification of configuration spaces of point sets on the real projective line. Kyushu J. Math. 50(2), 493–512 (1996)

    Article  MathSciNet  Google Scholar 

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  1. Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe, 657-8501, Japan

    Saiei-Jaeyeong Matsubara-Heo

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Communicated by Ruben Minasian.

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Matsubara-Heo, SJ. A Tree Expansion Formula of a Homology Intersection Number on the Configuration Space \(\mathcal {M}_{0,n}\). Ann. Henri Poincaré 22, 2831–2852 (2021). https://doi.org/10.1007/s00023-021-01041-4

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