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.We then formulate twisted homology on such a cut Riemann surface keeping loop momenta µ I as well as surface moduli Ω IJ fixed. Therefore all statements we make are about loop integrands for open strings. The introduction of loop momenta also helps in the interpretation of our results in the field-theory limit.Key identities used to deduce relations between different color-ordered amplitudes at genus zero are the monodromy relations introduced by Plahte [1] and later studied in [2,4]. From the perspective of twisted homology, they become simply a statement of the vanishing of terms which are a total boundary under the twisted boundary operator [16,24], that is, contours that can be deformed to a point. Using this framework one can prove that the dimension of twisted homology of M 0,n is (n−3)! [25] (in the physics literature this result first appeared in [2,4]), which has a geometric interpretation in terms of the Euler characteristic of M 0,n .Our setup essentially trivializes finding monodromy relations for higher-genus surfaces.-3 -This allows us to write down explicit identities for all planar open-string integrands with particle inserted on any boundary to arbitrary genus g, 1 arbitrary number of punctures n, all orders in α , and any external string state. For the correct counting of the number of independent cycles it is important to realize that there are always two monodromy relations for each surface, obtained from contracting the contours C + and C − in Figure 1. These two relations explain heuristically the reduction from the naive (n − 1)! to (n − 3)! distinct tree-level amplitudes. If one builds up recursively the amplitude by adding punctures, at each step there are two fewer independent amplitudes resulting in the previous reduction. The story is sometimes presented differently in the literature, where complex conjugation, or real and imaginary parts, are invoked. While this is effectively identical to using these two relations we believe that the problem is better formulated in those terms. At low genus (g = 1, 2, 3), similar monodromy relations were recently written down in [30][31][32]46] and checked to the first couple of orders in α . These...

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