Abstract
The single-valued projection (sv) is a relation between scattering amplitudes of gauge bosons in heterotic and open superstring theories. Recently we have studied sv from the aspect of nonlinear sigma models [1], where the gauge physics of open string sigma model is under the Wilson loop representation but the gauge physics of heterotic string sigma model is under the fermionic representation since the Wilson loop representation is absent in the heterotic case. There we showed that the sv comes from a sum of six radial orderings of heterotic vertices on the complex plane. In this paper, we propose a Wilson loop representation for the heterotic case and using the Wilson loop representation to show that sv comes from a sum of two opposite-directed contours of the heterotic sigma model. We firstly prove that the Wilson loop is the exact propagator of the fermion field that carry the gauge physics of the heterotic string in the fermionic representation. Then we construct the action of the heterotic string sigma model in terms of the Wilson loop, by exploring the geometry of the Wilson loop and by generalizing the nonabelian Stokes's theorem [2–4] to the fermionic case. After that, we compute some three loop and four loop diagrams as an example, to show how the sv for ζ2 and ζ3 arises from a sum of the contours of the Wilson loop. Finally we conjecture that this sum of contours of the Wilson loop is the mechanism behind the sv for general cases.
Highlights
For tree-level string amplitudes, the single-valued projection [5] is a map between gluon amplitudes of the open superstring and gluon amplitudes of heterotic string [6, 7, 8]
In the previous paper [1], we showed that the sv-map comes gauge physics open string stringy description Chan-Paton factor fermionic representation nonlinear sigma model exact propagator
We show that the sv-map comes from a sum of two opposite-directed integral contours, when the gauge physics of both the open and the heterotic string sigma models are under the Wilson loop representation
Summary
For tree-level string amplitudes, the single-valued projection (sv) [5] is a map between gluon amplitudes of the open superstring and gluon amplitudes of heterotic string [6, 7, 8]. After the gauge physics of both the open and the heterotic string sigma model are put under the Wilson loop representation, the sv-map [1] between them turns out to have a very simple geometric origin: the sv-map comes from the sum of two pathordered integrals of opposite directions. In the previous paper [1], we showed that the sv-map comes gauge physics open string stringy description Chan-Paton factor fermionic representation nonlinear sigma model exact propagator. We get the background field expansion of the Wilson loop for the heterotic string case, which corresponds consistently (in the sense of exact propagators) to the background field expansion of the fermionic representation obtained using reorganized perturbation in our previous paper [1]. We will compute several diagrams of three loop and four loop as an example and give the conjecture for the general case
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