Little String Amplitudes (and the Unreasonable Effectiveness of 6D SYM)

TL;DR

<3-5 sentence high-level summary>This work connects double-scaled little string theory (DSLST) to six-dimensional ${ m (1,1)}$ super-Yang-Mills on its Coulomb branch by computing four-point amplitudes of massless states on the DSLST side and matching them to perturbative SYM amplitudes. The DSLST amplitudes are obtained from worldsheet correlators in the cigar coset ${SL(2)_k/U(1)}$ and the ${SU(2)_k/U(1)}$ minimal models, recast into Liouville conformal blocks and evaluated numerically in an $oldsymbol{ extalpha'}$ expansion. The authors demonstrate striking agreement with 1- and 2-loop gluon amplitudes for gauge groups ${SU(k)}$ at a ${ m Z}_k$-invariant point on the Coulomb branch, supporting a nontrivial UV structure control and suggesting a non-renormalization property for Cartan gluons. The results illuminate how the UV completion encoded in DSLST constrains higher-dimensional operators in 6D SYM and motivate further exploration of higher-genus DSLST and higher-loop SYM correspondences.

Abstract

We study tree level scattering amplitudes of four massless states in the double scaled little string theory, and compare them to perturbative loop amplitudes in six-dimensional super-Yang-Mills theory. The little string amplitudes are computed from correlators in the cigar coset CFT and in N=2 minimal models. The results are expressed in terms of integrals of conformal blocks and evaluated numerically in the alpha' expansion. We find striking agreements with up to 2-loop scattering amplitudes of massless gluons in 6D SU(k) SYM at a Z_k invariant point on the Coulomb branch. We comment on the issue of UV divergence at higher loop orders in the gauge theory and discuss the implication of our results.

Figures (5)

• Figure 1: The 1-loop scalar integral $I_4^{1-loop}(s_{12},s_{14})$.
• Figure 2: In (a), the planar 2-loop scalar integral. In (b), the non-planar 2-loop scalar integral.
• Figure 3: The four UV divergent scalar integrals for the 3-loop 4-gluon scattering amplitude in 6D SYM. The signs for the internal vertices denote the two index structures in the double line notation; plus for the left vertex and minus for the right vertex in Figure \ref{['fig:DL']}. We label the diagrams following the notation in Figure 2 of Bern:2007hh. The above sign assignments together with the other four obtained by flipping all the $+/-$ are the only eight diagrams that contribute to the UV divergence of the scattering amplitudes of four Cartan gluons.
• Figure 4: The two index structures for the 3-gluon vertex in the double line notation.
• Figure 5: The two UV-divergent scalar integrals in the 3-loop 4-point amplitude. Since we are only interested in the divergent part, we have set the external momenta to be zero.The number indicates the propagator should be raised to the corresponding power.