Monodromy relations from twisted homology

Abstract

We reformulate the monodromy relations of open-string scattering amplitudes as boundary terms of twisted homologies on the configuration spaces of Riemann surfaces of arbitrary genus. This allows us to write explicit linear relations involving loop integrands of open-string theories for any number of external particles and, for the first time, to arbitrary genus. In the non-planar sector, these relations contain seemingly unphysical contributions, which we argue clarify mismatches in previous literature. The text is mostly self-contained and presents a concise introduction to twisted homologies. As a result of this powerful formulation, we can propose estimates on the number of independent loop integrands based on Euler characteristics of the relevant configuration spaces, leading to a higher-genus generalization of the famous $(n-3)!$ result at genus zero.

Figures (12)

• Figure 1: Genus-$g$ Riemann surface sliced along the $A_I$-cycles (blue). Loop momenta $\ell_I^\mu$ flow through the two pairs of resulting disks for each $I=1,2,\ldots,g$ (the point denoted with a black box is identified). The resulting Riemann surface can be understood as two copies of open-string worldsheets glued together along $B_I$-cycles (red) and $D$ (dark green), on which punctures are placed. Monodromy relations are a consequence of the fact that the contours ${\cal C}_{\pm}$ (light green) can each be contracted to a point.
• Figure 2: Translation between multi-valued integrands on the covering space $\widehat{X}$ and paths with local coefficients on the base space $X$, which gives an illustration of what is meant by "picking up a phase when crossing a branch cut".
• Figure 3: Choice of branch cuts for the loading $T_0(z_1)$ with monodromies $\alpha_j=e^{-2\pi i \alpha'k_1\cdot k_j}$. Note that $\alpha_2 = \prod_{j=3}^{n} \alpha_j^{-1} = e^{-2\pi i \alpha' k_1 {\cdot} k_2}$ is not independent of the other ones.
• Figure 4: Examples of twisted chains and cycles.
• Figure 5: Contours producing boundary relations on the twisted homology. The left- and right-most ends of the horizontal contours are connected, since they are defined on $\mathbb{CP}^1$.