Abstract
In this chapter recursion relations for intersection numbers are introduced. The starting point is a review of the fibre bundle structure of the moduli space and the behaviour of twisted cohomologies under such fibration. We introduce special classes of twisted forms having good decomposition properties, called fibration bases, which serve as bases for the expansion of more general forms. This effectively allows one to formulate recursion relations for scattering amplitudes on the moduli space by reducing the problem to a computation of intersection numbers on individual punctured spheres and gluing the results together. The matrices needed for this computation, called braid matrices, form representations of the pure braid group. The final expression involves only matrix inversion and multiplication, as well as taking one-dimensional residues.
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Notes
- 1.
We thank D. Fuchs, A. Schwarz, and E. Witten for suggesting this to us.
- 2.
It is known that one can construct a rank-(n−2)! Gauss–Manin connection in an auxiliary variable \(z_0 \in \mathbb {C}\mathbb {P}^1 {\setminus } \{0,1,\infty \}\), whose horizontal sections compute certain finite open-string integrals for n punctures in the asymptotic limit z 0→0, and for n+1 punctures at z 0→1 [234, 235]. The two limits are connected by an associator [236, 237], thus providing a recursion for these integrals in n. Using this setup Terasoma proved that coefficients of the Taylor expansion of such open-string integrals around α′ = 0 are \(\mathbb {Q}\)-linear combinations of multiple zeta values (MZVs) [234]. The conjecture [238] that coefficients of closed-string integrals fall into a special class of single-valued MZVs [239] was recently proven by Brown and Dupont [240], see also [241, 242].
- 3.
The manipulations in Lemma 3.1 require any solution \(\psi _{+,a}^r\) of the Pfaffian system of differential equations \(\boldsymbol \nabla _{p}^+ \psi _{+,a}^r = \sigma _{+,a}\). However, the homogeneous solution of this system, given simply by \(\psi _{+}^r = \mathbf {c}\, \mathcal {P}\exp ( - \int _{z_r}^{z_p} \boldsymbol {\omega }_p^+ )\) for a constant vector c, is multi-valued in the neighbourhood of each z p = z r and hence does not have a holomorphic expansion. This is why we only consider inhomogeneous solutions.
- 4.
In the cases when orthonormal bases are not known, i.e., \(\langle f_{-,a}^\vee | f_{+,b}\rangle _{\omega _p} =: {\mathbf {C}}_{ab} \neq \delta _{ab}\), following the above steps gives \(\sigma _{-,a} = \sum _b \langle \varphi _{-} | f_{+,b} \rangle _{\omega _p} {\mathbf {C}}^{-1}_{ba}\), \(\sigma _{+,a} = \sum _b {\mathbf {C}}^{-1}_{ab} \langle f_{-,b}^\vee | \varphi _{+} \rangle _{\omega _p}\), and \((\boldsymbol \omega _{p}^+)_{ab} = \sum _c {\mathbf {C}}^{-1}_{bc} \langle f_{-,c}^\vee | (d_{\Sigma _{p}} \!{+} \omega _{p-1} {\wedge }) f_{+,a} \rangle _{\omega _p}\), \((\boldsymbol \omega _{p}^-)_{ab} = -\sum _c \langle (d_{\Sigma _{p}} \!{-} \omega _{p-1} {\wedge }) f_{-,a}^\vee | f_{+,c} \rangle _{\omega _p} {\mathbf {C}}_{cb}^{-1}\). The recursion relations keep their form; the functions \(\psi _{\pm ,a}^r\) are solutions of \(\boldsymbol \nabla _p^\pm \psi _{\pm ,a}^r = \sigma _{\pm ,a}\) with σ ±,a and \(\boldsymbol \omega _p^{\pm }\) given above.
- 5.
Defining \({\mathbf {r}}_{ij} := \int \boldsymbol {\Omega }^{ij}_p d\log (z_i {-} z_j)\) it is straightforward to check that (3.38) and (2.4) imply
$$\displaystyle \begin{aligned}{}[ {\mathbf{r}}_{ij}, {\mathbf{r}}_{ik} ] + [ {\mathbf{r}}_{ij}, {\mathbf{r}}_{jk} ] + [ {\mathbf{r}}_{ik}, {\mathbf{r}}_{jk} ] = 0, \end{aligned} $$(3.39)which are the classical Yang–Baxter relations, see, e.g., [243]. They can be understood as the linearized version of the quantum Yang–Baxter relations
$$\displaystyle \begin{aligned} {\mathbf{R}}_{ij} {\mathbf{R}}_{ik} {\mathbf{R}}_{jk} = {\mathbf{R}}_{jk} {\mathbf{R}}_{ik} {\mathbf{R}}_{ij} \end{aligned} $$(3.40)for \({\mathbf {R}}_{ij} := \mathbb {I} + \alpha ' {\mathbf {r}}_{ij} + \ldots \) in the limit α′→ 0.
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Mizera, S. (2020). Recursion Relations from Braid Matrices. In: Aspects of Scattering Amplitudes and Moduli Space Localization. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-53010-5_3
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