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

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. 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. 2.

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

  3. 3.

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

  4. 4.

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

  5. 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. 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. 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. 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.

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Correspondence to Saiei-Jaeyeong Matsubara-Heo.

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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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