AdS Virasoro-Shapiro Amplitude

• AdS Virasoro-Shapiro amplitude is a generalization of the flat-space Virasoro-Shapiro function, incorporating AdS curvature corrections through a genus-zero worldsheet integral with single-valued multiple polylogarithms.
• It encodes strong-coupling planar-limit CFT data for four-point correlators in N=4 SYM by extracting anomalous dimensions and OPE coefficients of single-trace operators with and without R-charge.
• The amplitude framework is validated via integrability, localization, and effective action analyses, ensuring consistency with flat-space limits while revealing detailed AdS curvature corrections.

The AdS Virasoro-Shapiro amplitude generalizes the celebrated flat-space Virasoro-Shapiro function to string scattering in an Anti-de Sitter (AdS) background, with particular emphasis on AdS₅×S⁵ and its holographic duals in $\mathcal{N}=4$ SYM. Its structure encodes the strong-coupling, planar-limit @@@@2@@@@ data for four-point correlators of half-BPS operators, including full towers of anomalous dimensions and OPE coefficients of single-trace operators, both at zero and nonzero $R$-charge (Kaluza-Klein, KK charges). Unlike flat space, the AdS amplitude incorporates AdS curvature corrections in a manifestly single-valued genus zero worldsheet integral over the Riemann sphere, where the integrand includes insertions of single-valued multiple polylogarithms of weight three and higher. The AdS Virasoro-Shapiro paradigm serves as a key testing ground for dispersive sum rules, integrability, localisation, and bootstrap approaches.

1. Dual Conformal Correlators and the Genus-Zero String Amplitude

In planar $\mathcal{N}=4$ SYM, single-trace half-BPS operators $\mathcal{O}_p(x,y) = (y \cdot \phi)^p + \dots$ (dimension $p$, transforming in the $[0,p,0]$ of SU(4)$_R$) generate four-point correlators whose connected component decomposes as

$$\frac{y_{12}^2 y_{34}^p}{(x_{12}^2)^2 (x_{34}^2)^p} \mathcal{G}(U,V;\alpha,\bar\alpha) = \text{(free)} + \frac{(z-\alpha)\cdots(\bar{z}-\bar{\alpha})}{(z\bar{z})^2 (\alpha\bar{\alpha})^2} \mathcal{T}(U,V)$$

where $U, V$ are spacetime cross-ratios and $\alpha, \bar\alpha$ are R-symmetry cross-ratios.

Holographically, this correlator is dual to a genus-zero worldsheet CFT computation on AdS₅×S⁵. In the flat-space limit, the string amplitude reduces to

$$A^{(0)}(S,T) = \int_{\mathbb{C}} d^2 z\, |z|^{-2S-2} |1-z|^{-2T-2} \frac{1}{(S+T)^2}$$

where $(S,T,U)$ are Mandelstam invariants in AdS units, related to Mellin variables ($s,t$) via a Borel transform. This reproduces the familiar gamma-function structure of the flat-space Virasoro-Shapiro amplitude (with $S+T+U=-4$ for canonical normalization).

2. Worldsheet Integral Representation and Curvature Corrections

To first order in the AdS curvature expansion ($\alpha'/R^2 \sim 1/\sqrt{\lambda}$), the full amplitude admits the structure

$A(S,T) = \sum_{i=1,2} \sum_{\text{perm}} B_i(S,T)$

with

$$B_i(S,T) = \int_{\mathbb{C}} d^2 z\, |z|^{-2S-2} |1-z|^{-2T-2} G_i^{(1)}(S,T;z)$$

The sum over “perm” captures channel crossing symmetry (interchanging $T$ and $U$). The measure is the standard Lebesgue measure on the Riemann sphere.

The amplitude organizes as a series in the small parameter $1/\sqrt{\lambda}$: $$A(S,T) = \sum_{k=0}^\infty \lambda^{-3/2 - k/2} A^{(k)}(S,T)$$ with $A^{(0)}$ from flat space and $A^{(1)}$ the first nontrivial AdS curvature correction.

3. Single-Valued Multiple Polylogarithms in the Integrand

The novel feature of AdS curvature corrections is the appearance of single-valued multiple polylogarithms (SVMPLs) of weight three in the worldsheet integrand. Specifically, the ansatz for $G_i^{(1)}(S,T;z)$ is as a linear combination of SVMPLs: $$\left\{\mathcal{L}^s_{000}(z),\,\mathcal{L}^s_{001}(z),\,\mathcal{L}^s_{010}(z),\,\zeta(3),\,\mathcal{L}^a_{000}(z),\,\mathcal{L}^a_{001}(z),\,\mathcal{L}^a_{010}(z)\right\}$$ Here, for any word $w \in \{0, 1\}^3$,

$$\mathcal{L}^s_w(z) = \mathcal{L}_w(z) + \mathcal{L}_w(1-z) + \mathcal{L}_w(\bar{z}) + \mathcal{L}_w(1-\bar{z})$$

$$\mathcal{L}^a_w(z) = \mathcal{L}_w(z) - \mathcal{L}_w(1-z) + \mathcal{L}_w(\bar{z}) - \mathcal{L}_w(1-\bar{z})$$

Such combinations are real-analytic and single-valued on the sphere, preventing any branch cuts within the integration domain. For example, $$\mathcal{L}_{001}(z) = \mathrm{Li}_3(z) - \ln|z| \mathrm{Li}_2(z) + \frac12 \ln^2|z| \ln(1-z)$$.

The weight-3 SVMPLs ensure that the Wilson coefficients in the low-energy expansion are manifestly elements of the single-valued multiple zeta value (MZV) ring.

4. Flat-Space Limit and Consistency

In the strong-coupling (large-$\lambda$ or $R\to\infty$) limit, the SVMPL insertions decouple, and one recovers the flat-space amplitude: $$A(S,T) \longrightarrow \int_{\mathbb{C}} d^2 z\, |z|^{-2S-2} |1-z|^{-2T-2} \frac{1}{(S+T)^2} = A^{(0)}(S,T)$$ This is a crucial check, demonstrating that the AdS Virasoro-Shapiro amplitude is a deformation of the flat-space result, with curvature corrections vanishing asymptotically.

5. Extraction of OPE Data with Kaluza-Klein Charge and Odd Spin

The amplitude encodes the CFT data via its pole structure. Near $S = \tau$ (where $\tau$ is the twist of an exchanged operator), one may expand: $$A(S,T)\sim\sum_{\tau,\ell} \frac{C_{\tau,\ell}(p)\, Q_\ell(1+\frac{2T}{S})}{S-\tau} + \dots$$ Matching this expansion to the worldsheet integral allows extraction of the strong-coupling expansion of operator twists $\tau = \Delta - \ell$ and squared OPE coefficients $C_{\tau,\ell}(p)$, including those for operators with nonzero $R$-charge and odd spin.

The leading strong-coupling normalization for odd spin $\ell$ and averaged over degeneracy $\delta$ is

$$\langle f_{t,2}\rangle_{\delta,\ell=2\delta-3} = -\frac{r_{1/2}(\delta)}{\delta}(\delta-1)(p^2-4),\qquad r_{1/2}(\delta) = \frac{4^{2-2\delta}\, \delta^{2\delta-2}(2\delta-2)}{\Gamma(\delta)\, \Gamma(\delta-\tfrac12)}$$

More generally, determinant-type relations (from dispersive sum rules with Mack-polynomial projectors) or direct matching to integral residues provide the OPE data for all $\delta, \ell$.

6. Cross-Checks: Integrability, Localization, and 10D Effective Action

The construction passes stringent checks:

Integrability: For the leading Regge trajectory ($\ell=2\delta-2$), one finds

$$\tau = 2\sqrt{\delta} \lambda^{1/4} - \ell - 2 + O(\lambda^{-1/4})$$

This precisely matches the quasi-classical string spectra and Quantum Spectral Curve data, e.g., for $\tau^{[0,p-2,0]}(1,0)=13/4, 5, 29/4, \dots$.

Localization: The “supergravity” Wilson coefficients $\xi_{0,0}^{(1)}(p)=0$, $\xi_{1,0}^{(1)}(p)=\frac14 p(p-2) \zeta(5)$ agree with supersymmetric localization results via integrated correlators.

10D Effective Action: Structures such as $\zeta(3)^2, \zeta(7)$, in the low-energy expansion

$$A^{(1)}(S,T) = p(p-2)S\,\zeta(5) + \frac{\zeta(3)^2}{3}\big[(p^2-6p-36)S^2 + 2(p^2+18)TU\big] + \dots$$

reproduce known higher-derivative couplings after Kaluza-Klein reduction on $S^5$, as in IIB effective theory.

7. Impact and Generalization

The AdS Virasoro-Shapiro amplitude with KK modes provides a rigorous framework to decode strong-coupling CFT data for $\mathcal{N}=4$ SYM, including operators of arbitrary $R$-charge and spin. The explicit integral structure, in terms of SVMPLs, reveals an underlying worldsheet conformal field theory with universal analytic properties and single-valuedness constraints. This ensures compatibility with flat-space scattering physics, while enabling computation of AdS curvature corrections.

The approach generalizes to arbitrary combinations of KK modes and is robust under consistency checks from integrability, dispersive sum rules, and effective action analysis. It predicts an infinite tower of anomalous dimensions and OPE coefficients for single-trace operators at strong coupling, tightly interwoven with emergent hidden conformal symmetries and AdS/CFT duality (Fardelli et al., 2023).