Equivalence of twistor prescriptions for super Yang-Mills

TL;DR

This work proves that the connected-curve and maximally disconnected twistor prescriptions for tree-level amplitudes in ${\mathcal N}=4$ SYM are equivalent when the integration contours are chosen compatibly. The authors establish the equivalence explicitly for degree-2 curves by localizing to a common degeneration locus of intersecting lines and then extend the argument to general degree $d$ via iterative degenerations and the introduction of intermediate prescriptions. A unifying picture emerges: all prescriptions localize to the same subspace of moduli space and yield the same residues, with intermediate tree-graph prescriptions providing a diagrammatic and computational bridge. The results offer a flexible toolkit for computing amplitudes and pave the way for addressing contours, real versions, and loop extensions within twistor-string frameworks.

Abstract

There is evidence that one can compute tree level super Yang-Mills amplitudes using either connected or completely disconnected curves in twistor space. We argue that the two computations are equivalent, if the integration contours are chosen in a specific way, by showing that they can both be reduced to the same integral over a moduli space of singular curves. We also formulate a class of new ``intermediate'' prescriptions to calculate the same amplitudes.

Figures (9)

• Figure 1: An instanton contribution: (a) from a connected curve of degree 2; (b) from a pair of degree 1 curves. The dotted line represents a propagator in holomorphic Chern-Simons theory.
• Figure 2: A curve of degree 2 can degenerate into a pair of intersecting lines.
• Figure 3: A contribution to Yang-Mills amplitudes with $5$ positive and $5$ negative helicity gluons, represented (a) as four disconnected lines in twistor space, (b) as a graph $\Gamma$ with four MHV vertices.
• Figure 4: A different version of Figure \ref{['fig-mhv-tree']}, representing the same single-trace amplitude with the index line made manifest. The circles represent degree 1 curves in twistor space.
• Figure 5: A degenerate configuration of two intersecting lines in ${\mathbb C\mathbb P^{3|4}}$ can be deformed into a smooth connected curve of degree 2 or into two disconnected lines. The transition between the two branches of moduli space is reminiscent of a conifold transition.