Carving out the Space of Open-String S-matrix

TL;DR

The paper investigates open string four-point amplitudes as dual objects: a space-time S-matrix constrained by factorization and unitarity, and a worldsheet-derived 1D CFT correlator governed by OPE consistency. It shows that space-time unitarity enforces an expansion of the disk-integrand function f(z) in SL(2,R) blocks, with Virasoro symmetry emerging at the boundary of allowed factorization, i.e., at a kink in the solution space. Through a detailed analysis of monodromy relations, it defines a monodromy plane in EFT-coupling space and demonstrates that the intersection with the EFThedron sharply confines the allowed couplings, with higher-derivative data driving the region toward the exact string values; numerical evidence suggests the four-point open superstring amplitude is geometrically fixed by this intersection. The authors also show how KLT relates open and closed string EFTs, mapping the constrained open-string data to a compatible closed-string EFT. Together, these results propose a principled, geometry-driven route to reconstructing string amplitudes from consistency conditions alone.

Abstract

In this paper, we explore the open string amplitude's dual role as a space-time S-matrix and a 1D CFT correlation function. We pursue this correspondence in two directions. First, beginning with a general disk integrand dressed with a Koba-Nielsen factor, we demonstrate that exchange symmetry for the factorization residue of the amplitude forces the integrand to be expandable on SL(2,R) conformal blocks. Furthermore, positivity constraints associated with unitarity imply the SL(2,R) blocks must come in linear combinations for which the Virasoro block emerges at the "kink" in the space of solutions. In other words, Virasoro symmetry arises at the boundary of consistent factorization. Next, we consider the low energy EFT description, where unitarity manifests as the EFThedron in which the couplings must live. The existence of a worldsheet description implies, through the Koba-Nielsen factor, monodromy relations which impose algebraic identities amongst the EFT couplings. We demonstrate at finite derivative order that the intersection of the "monodromy plane" and the EFThedron carves out a tiny island for the couplings, which continues to shrink as the derivative order is increased. At the eighth derivative order, on a three-dimensional monodromy plane, the intersection fixes the width of this island to around 1.5$\%$ (of $ζ(3)$) and 0.2$\%$ (of $ζ(5)$) with respect to the super string answer. This leads us to conjecture that the four-point open superstring amplitude can be completely determined by the geometry of the intersection of the monodromy plane and the EFThedron.

Figures (13)

• Figure 1: The plot for the allowed solution for an ansatz for $\chi_4=\frac{(i+a_4 i^2) (j+a_4 j^2)}{a_1 d^2 +a_2 d+a_3}$ under (ii) and (iii). For illustrative purposes we have set $a_4=5, a_1=9880$, the Virasoro values. The lines denote the boundaries carved out by (iii) as a function of the space-time dimension. The point in the figure is the Virasoro value for $a_2$ and $a_3$.
• Figure 2: The gray region represents the three-dimensional intersection of the monodromy plane and EFThedron at eighth derivative order. The red region represents the region that can be projected from the four-dimensional geometry that would appear at ninth derivative order.
• Figure 3: The scalar coefficient $\mathcal{C}_0^{(2)}$ for the conformal block $(p,h)$. We see that there are regions of $(p,h)$ where the coefficient becomes negative, thus violating unitarity.
• Figure 4: Solution space for $(a_1,a_2,a_3)$. This diagram is carved out by inequalities corresponding to different dimensions, with $d=3,24,25$ shown here. In each dimension, the saturated inequality is a two-dimensional plane. The Virasoro point lives on the intersection of these planes.
• Figure 5: $a_1=9880$ solution space. The point in figure is the Virasoro coefficient.