Gluon scattering in AdS at finite string coupling from localization

TL;DR

This work computes open-string (gluon) scattering on D7 branes in AdS$_5\times S^5/\mathbb{Z}_2$ by leveraging an analytic bootstrap together with exact localization of the mass-deformed sphere free energy $F(\mu_i)$. The leading holographic corrections at large $N$ and finite $\tau$ are fixed, with the $F^4$ term expressed in terms of Jacobi theta functions and the flat-space limit linking to the Veneziano amplitude; the protected $D^2F^4$ term is governed by the non-holomorphic Eisenstein series $E_{3/2}(\tau)$. At nonzero string coupling, a novel $Z_{\text{extra}}$ instanton contribution controls the $\tau$-dependence, and the results respect $SL(2,\mathbb{Z})$ duality and $SO(8)$ triality. Overall, the paper demonstrates AdS/CFT at finite coupling for open-string sectors and reveals precise modular structures tying localization data to string amplitudes.

Abstract

We consider gluons scattering in Type IIB string theory on AdS$_5\times S^5/\mathbb{Z}_2$ in the presence of D7 branes, which is dual to the flavor multiplet correlator in a certain 4d $\mathcal{N}=2$ $USp(2N)$ gauge theory with $SO(8)$ flavor symmetry and complexified coupling $τ$. We compute this holographic correlator in the large $N$ and finite $τ$ expansion using constraints from derivatives of the mass deformed sphere free energy, which we compute to all orders in $1/N$ and finite $τ$ using supersymmetric localization. In particular, we fix the $F^4$ higher derivative correction to gluon scattering on AdS at finite string coupling $τ_s=τ$ in terms of Jacobi theta functions, which feature the expected relations between the $SL(2,\mathbb{Z})$ duality and the $SO(8)$ triality of the CFT, and match it to the known flat space term. We also use the flat space limit to compute $D^2F^4$ corrections of the correlator at finite $τ$ in terms of a non-holomorphic Eisenstein series. At weak string coupling, we find that the AdS correlator takes a form which is remarkably similar to that of the flat space Veneziano amplitude.

Figures (1)

• Figure 1: The localization inputs for $\mathcal{F}_{\bf v}$, $\mathcal{F}_{\bf c}$, and $\mathcal{F}_{\bf s}$ in the $USp(4)$ theory as a function of $i/\tau$ from the free point $\tau=i\infty$ until the $S$-duality invariant point $\tau=i$ with $\mathop{\rm Re}\nolimits(\tau)=0$ (left), and as a function of $\tau$ along the $S$-duality invariant arc with $|\tau|=1$ from $\tau=i$ to the $T$-duality invariant point $\tau=e^{i \pi /3}$ (right). These functions were computed up to 2-instanton order using \ref{['ZextraFinal']}, and show some small errors (e.g. $\mathcal{F}_{\bf v}\neq \mathcal{F}_{\bf c}$ at $\tau=i$ in the right hand plot). The analogous plots for $USp(2)$ are shown in Chester:2022sqb, where we could go to much higher instanton order due to the equivalence to $SU(2)$ SQCD.