Minimal Basis for Gauge Theory Amplitudes

Identities based on monodromy for integrations in string theory are used to derive relations between different color-ordered tree-level amplitudes in both bosonic and supersymmetric string theory. These relations imply that the color-ordered tree-level $n$-point gauge theory amplitudes can be expanded in a minimal basis of $(n-3)!$ amplitudes. This result holds for any choice of polarizations of the external states and in any number of dimensions.

Figure 1

Contour of integration for the amplitude $\mathcal{A}(1,3,2,4)$.

Figure 2

Flipped contour for the amplitude $\mathcal{A}(1,3,2,4)$.

Figure 3

Contour for the amplitude $\mathcal{A}({\beta }_1,\dots ,{\beta }_r,1,{\alpha }_1,\dots ,{\alpha }_s,n)$.

Figure 4

Flipped contour of Fig. 3.

Figure 5

Integrals associated with the contours of Fig. 4.