Stringy and Quantum Corrections Beyond Supergravity

A supergravity prediction is not one approximation with one error bar. It is the baseline term in several expansions whose failures have different physical signatures: local higher derivatives, closed-string topology and bulk loops, Kaluza–Klein towers, Lorentzian brane states, and Euclidean brane saddles. The safe procedure is therefore observable-specific: name the state and kinematics, compute every relevant control parameter, apply selection rules, and give a separate stop condition for each omitted sector.

This page develops that procedure in type-IIB string theory on AdS$_5\times S^5$ and applies it to a normalized stress-tensor-multiplet four-point function. The powers of $N$ and $\lambda$ below belong to that theory, normalization, and order of limits; they are not a universal AdS/CFT formula.

Required background. Corrections and thresholds supplies the large-$N$ loop and local-contact distinction. Choosing a ten- or five-dimensional description supplies oscillator and compactification thresholds. What a consistent truncation licenses distinguishes nonlinear closure from low-energy decoupling.

Helpful background. The four-derivative curvature basis supplies field-redefinition discipline. Metric and ghost loops and nonanalytic long-distance terms explain why a quantum loop is not merely another local operator.

The route is to translate bulk controls, identify correction signatures, organize one convention-fixed asymptotic Mellin benchmark, and then force the approximation to fail at a pole.

Five correction axes answer different questions

Start with a specified two-derivative supergravity observable $\mathcal A_{\mathrm{sugra}}$. Five questions must then be asked independently.

1. Are the background and the process soft in string units? Background curvature is tested by $\alpha' \lvert\mathrm{Riemann}\rvert$; a local process with characteristic invariant $Q^2$ is tested by $\alpha'Q^2$. Passing the first test does not pass the second.
2. Are quantum topologies suppressed? In a fixed weakly coupled type-IIB frame, another closed-string handle is relatively weighted by $g_s^2$. After holographic normalization, connected correlators instead expose a fixed-$\lambda$ expansion in $1/N^2$. These are related bookkeeping systems, not interchangeable labels for every term.
3. Can an omitted compactification mode be resolved? A Wilsonian expansion requires $Q$ below the first coupled Kaluza–Klein mass. A consistent truncation can close classically without such a mass hierarchy, but it does not delete generic KK poles or remove the tower from quantum loops.
4. Does the charge sector contain extended brane states? A Lorentzian wrapped brane is a state, defect, or charged object. Its mass, stability, probe limit, and backreaction must be checked; it is not automatically an exponentially small correction to the vacuum amplitude.
5. Are there allowed Euclidean brane saddles? Their weights are exponential in an action, but charge conservation, fermion zero modes, determinants, and Stokes data decide whether a given observable receives them at all.

This separation is more than terminology. Mellin space is a transform of a conformal correlator in which local contact vertices become polynomials and exchanged states produce pole families; the Gamma-factor convention must be specified before interpreting either feature. A local $R^4$ vertex produces a Mellin polynomial, an exchanged KK or string state produces poles, a loop produces discontinuities and logarithms together with scheme-dependent local subtraction polynomials, and an instanton changes selected coefficient functions through exponentially weighted Fourier sectors.

Translate AdS5 and S5 controls to boundary variables

Use the site convention

$$g_{\mathrm{YM}}^2=4\pi g_s, \qquad \lambda=g_{\mathrm{YM}}^2N, \qquad L^4=4\pi g_sN\alpha'^2=\lambda\alpha'^2.$$

It follows that

$$\epsilon_{\mathrm{bg}} \equiv\frac{\alpha'}{L^2} =\lambda^{-1/2}, \qquad \epsilon_Q \equiv\alpha'Q^2 =\frac{(QL)^2}{\sqrt{\lambda}},$$

and

$$g_s=\frac{\lambda}{4\pi N}, \qquad \frac{G_5}{L^3}=\frac{\pi}{2N^2}, \qquad m_sL=\frac{L}{\sqrt{\alpha'}}=\lambda^{1/4}.$$

These relations follow from D3-brane flux quantization and the five-sphere reduction; the full normalization derivation is given on the D3-brane parameter and evidence page. At fixed AdS-scale kinematics, $QL=O(1)$, the first allowed type-IIB local interaction $\alpha'^3R^4$ is relatively of order $\epsilon_Q^3\sim\lambda^{-3/2}$ when its matrix element is nonzero. If $QL$ grows to $\lambda^{1/4}$, then $\epsilon_Q=O(1)$ even though the background remains weakly curved.

For unit-normalized single-trace operators, the connected correlator has the fixed-$\lambda$ genus organization

$$\mathcal G_{\mathrm{conn}} =N^{-2}\mathcal G_0(\lambda) +N^{-4}\mathcal G_1(\lambda)+\cdots.$$

The raw ten-dimensional handle weight is

$$g_s^2=\frac{\lambda^2}{16\pi^2N^2}.$$

Consequently, every $\mathcal G_h(\lambda)$ has its own strong-coupling expansion. One must not replace all $N^{-4}$ terms by a $\lambda$-independent “one-loop supergravity” function. Use uppercase $S,T,U$ for the dimensionful ten-dimensional Mandelstam invariants, with $S+T+U=0$, and reserve lowercase $s,t,u$ below for dimensionless Mellin variables. With its tensor structure suppressed, the protected $R^4$ contribution to the normalized flat-space four-supergraviton amplitude contains

$$\frac{\alpha'^3STU}{64} \left[ 2\zeta(3)+\frac{2\pi^2}{3}g_s^2 +\text{D-instanton Fourier modes} \right].$$

Thus the local genus-one $R^4$ contribution scales parametrically as $N^{-4}\lambda^{1/2}$ in the normalized four-point function, whereas the ordinary one-loop supergravity term scales as $N^{-4}$ and has nonlocal loop structure. The genus-one $R^4$ term divided by its genus-zero $R^4$ partner is governed by $g_s^2$; the supergravity loop/tree ratio also contains the dimensional conversion between the string and AdS scales. This protected completion and its order-of-limits dependence are derived in Binder et al. 2019, §5, eqs. (5.3)–(5.5), PDF and tested further at finite complex coupling in Alday, Chester, and Hansen 2021, §§1, 3–4, PDF.

A sufficient parametric corner for weakly curved, weakly coupled type-IIB supergravity is therefore

$$1\ll\lambda\ll N,$$

up to convention-dependent numerical factors and observable-specific qualifications. It does not produce a generic five-dimensional Wilsonian gap: on AdS$_5\times S^5$, $m_{\mathrm{KK}}L=O(1)$.

Correction signatures

The following comparison is a diagnostic, not an ordering prescription. A row may vanish by a selection rule, and a nominally small row may dominate near its own pole.

Five independent correction sectors and their first failure signals

Sector	Control in the stated IIB frame	Characteristic signature	What can suppress it	Stop condition
Tree-level higher derivatives	α′/L2 and α′Q2; the first allowed local term here is α′3R4	Local contact terms; polynomial Mellin dependence order by order	Supersymmetry or an observable-specific zero can remove a coefficient	α′Q2 ≳ 1 or approach to a massive-string pole
Closed-string genera and bulk loops	gs2 between matching topologies; 1/N2 at fixed λ after boundary normalization	Unitarity cuts, logarithms, multi-trace OPE corrections, plus local counterterm ambiguities	Large N, weak gs, supersymmetry, or a protected observable	A loop term, species sum, or local genus correction becomes comparable to the retained order
Kaluza–Klein tower	Q/mKK, harmonic quantum numbers, and couplings	Extra exchange poles and compact-space tensor structures	Harmonic selection rules or exact classical closure of a retained sector	A coupled channel approaches an omitted KK pole; classical closure alone does not license loop omission
Lorentzian wrapped Dp states	Q/MDp, charge, cycle volume, probe/backreaction ratio	Heavy charged states, defects, new thresholds, or brane backreaction	No allowed stable cycle, charge mismatch, or a parametrically large mass	The state is kinematically accessible, macroscopically occupied, or no longer a probe
Euclidean D-branes and D-instantons	exp(−SE) together with determinants and zero-mode saturation	Exponentials and axionic Fourier phases invisible at every perturbative genus order	Charge or zero-mode selection rules can make the contribution vanish	The action is not large, the asymptotic saddle decomposition changes across a Stokes wall, or the chosen weak frame fails

Two rows deserve explicit formulas.

For a ten-dimensional massless scalar reduced on the round five-sphere,

$$-\nabla_{S^5}^2Y_\ell =\frac{\ell(\ell+4)}{L^2}Y_\ell, \qquad m_\ell^2L^2=\ell(\ell+4), \qquad \Delta_\ell=\ell+4.$$

The first nonconstant harmonic is therefore at the AdS scale, not parametrically above it. Whether it appears in a particular amplitude depends on integrals of harmonics and the $SO(6)$ representation channel Kim, Romans, and van Nieuwenhuizen 1985, Tables III–VI, pp. 397–399.

For an unwarped D$p$-brane wrapping a spatial $p$-cycle $\Sigma_p$ in string frame,

$$T_{D p} =\frac{1}{(2\pi)^p g_s\alpha'^{(p+1)/2}}, \qquad M_{D p}=T_{D p}\operatorname{Vol}(\Sigma_p).$$

Warping, worldvolume flux, Wess–Zumino couplings, BPS minimal-volume conditions, and global charge-cancellation constraints can change the physical answer. The formula describes a Lorentzian state, not an instanton weight. As a clean check, a D3 giant graviton wrapping an $S^3\subset S^5$ obeys $E=J/L$, $r^2=(J/N)L^2$, and $J\le N$ McGreevy, Susskind, and Toumbas 2000, §3.2, eqs. (3.40)–(3.47), PDF.

By contrast, a charge-$k$ D-instanton contributes, in the same weak IIB frame,

$$e^{-S_k} =\exp\left[ -\frac{2\pi\lvert k\rvert}{g_s} +2\pi i kC_0 \right] =\exp\left[ -\frac{8\pi^2\lvert k\rvert N}{\lambda} +ik\theta \right],$$

where $C_0=\theta/(2\pi)$ in this convention. Taking $k$ over signed nonzero integers includes the conjugate anti-instanton sectors. Determinant and zero-mode prefactors remain implicit. An instanton corrects permitted coefficient functions; it is not a universal additive term with a fixed power of $N$ Green and Gutperle 1997, eq. (2) and §§5–6, PDF.

First application: an AdS5 four-point expansion

Consider $SU(N)$ $\mathcal N=4$ super-Yang–Mills theory and the half-BPS stress-tensor-multiplet superprimary $\mathcal O_2(x,Y)$, where $Y$ is a null $SO(6)$ polarization. Fix

$$\left\langle \mathcal O_2(x_1,Y_1)\mathcal O_2(x_2,Y_2) \right\rangle =\frac{(Y_1\cdot Y_2)^2}{\lvert x_{12}\rvert^4}.$$

Following Binder et al., strip the protected/free contribution and the fixed spacetime and $SO(6)$ tensor factors, then call the Mellin transform of the remaining connected dynamical function $\mathcal M_2(s,t)$. This is their reduced dynamical convention, with the universal Gamma-function measure kept outside $\mathcal M_2$; the identical-scalar Mellin convention explains why moving that measure changes apparent poles. Set

$$c=\frac{N^2-1}{4}, \qquad u=4-s-t.$$

Take the ‘t Hooft limit first—$N\to\infty$ at fixed $\lambda$—then expand at large $\lambda$ while holding the Mellin variables $s,t$ fixed. The reduced connected Mellin amplitude is

$$\begin{aligned} \mathcal M_2(s,t) =\frac{8}{c}\Bigg[ &\frac{1}{(s-2)(t-2)(u-2)} +\frac{15\zeta(3)}{\lambda^{3/2}} \\ &+\frac{315\zeta(5)}{4\lambda^{5/2}} \left(s^2+t^2+u^2-3\right) +\cdots \Bigg] +O(c^{-2}). \end{aligned}$$

This is an asymptotic hierarchy, not a numerical error bound. The rational term is the complete tree-level supergravity contribution in this reduced convention. The constant term is the planar $R^4$ contact interaction. The quadratic polynomial is the planar $D^4R^4$ interaction. There is no planar $\lambda^{-2}$ contact term in this amplitude because the $D^2R^4$ coefficient vanishes. The normalization, flat-space matching, and localization constraint are worked out in Binder et al. 2019, §2.2, eqs. (2.11)–(2.16), and §4, eqs. (4.1)–(4.13), PDF. The D3-brane evidence comparison and the classification of this four-point statement explain what this benchmark establishes without repeating the derivation here.

The omitted terms are best named by sector:

• Higher planar string terms. Further local interactions and eventually the massive-string pole tower continue the genus-zero answer. The displayed powers assume fixed Mellin variables.
• Order $c^{-2}$. One-loop supergravity contributes nonlocal discontinuities and scheme-dependent local subtraction polynomials needed for renormalization. Genus-one string corrections add their own strong-coupling series, including the protected local $R^4$ term discussed above Alday, Bissi, and Perlmutter 2019, §§1–3, PDF; Alday 2021, §§1–3, PDF.
• Compactification sectors. The consistent five-dimensional truncation justifies the selected classical supergravity calculation. It does not prove that all KK states are absent from quantum loops or from other external channels.
• Nonperturbative sectors. D-instanton Fourier modes multiply allowed interactions with weights of order $e^{-8\pi^2\lvert k\rvert N/\lambda+ik\theta}$ times prefactors. Their ordering changes if $\lambda$ scales with $N$.

Two independent checks constrain the displayed local coefficients: the large-Mellin flat-space limit reproduces the ten-dimensional type-IIB amplitude Penedones 2011, §5, PDF, while integrated correlator constraints from supersymmetric localization fix combinations of the same Mellin data Binder et al. 2019, §§3–4, PDF. Agreement of two projections is strong evidence for those coefficients; it does not control every omitted sector.

Build an observable-specific error budget

For any proposed correction to a supergravity result, use the following seven-step method.

1. Fix the theory and frame. State the compactification, flux integers, asymptotic dilaton and axion, Einstein or string frame, and the coupling convention.
2. Fix the observable. Give the external operators or states, normalization, charge channel, Lorentzian or Euclidean prescription, and kinematic range.
3. Define the baseline. Say which two-derivative fields and diagrams are included. If a lower-dimensional action is used, identify whether it follows from a consistent truncation or only from a low-energy integration-out argument.
4. Compute independent controls. At minimum report $\alpha'/L^2$, every relevant $\alpha'Q_i^2$, $g_s$, the gravitational loop parameter, $Q/m_{\mathrm{KK}}$, wrapped-brane masses or actions, and the instanton action.
5. Apply selection rules before ranking terms. Check supersymmetry, internal harmonics, charges, zero modes, and whether the proposed local operator has a nonzero matrix element.
6. Match analytic signatures and use an independent check. Contact polynomials cannot replace poles; loops require discontinuities and renormalization; instantons require the correct Fourier phase. Compare with a flat-space limit, localization identity, unitarity cut, uplift, duality, or another calculation that tests different assumptions.
7. State the remainder and stop rule. Name omitted sectors separately and say exactly when each approximation is withdrawn. A list of small parameters is not yet an error estimate unless their coefficients and uniformity over the stated kinematics are controlled.

The output should be a conditional statement such as: “At fixed $s,t$, through planar $D^4R^4$ order and away from omitted poles, the displayed Mellin amplitude has an asymptotic planar remainder beginning at higher string order; all $c^{-2}$, KK-loop, and nonperturbative sectors remain outside the claimed accuracy.” That sentence is much stronger scientifically than “string corrections are small.”

Adversarial control: a pole defeats every finite local series

Let a coupled heavy state of mass $m$ contribute in a Lorentzian channel

$$\mathcal A(s)=\frac{g^2}{m^2-s}.$$

For $\lvert s\rvert<m^2$, integrating out the state gives

$$\mathcal A_K(s) =\frac{g^2}{m^2} \times \sum_{n=0}^{K}\left(\frac{s}{m^2}\right)^n,$$

with exact remainder

$$R_K(s) =\mathcal A(s)-\mathcal A_K(s) =\frac{g^2}{m^2} \times \frac{(s/m^2)^{K+1}}{1-s/m^2}.$$

The relative remainder is $\lvert R_K/\mathcal A\rvert=\lvert s/m^2\rvert^{K+1}$. It becomes nonuniform as $s\to m^2$, and no finite contact series reproduces the pole.

The same obstruction is visible in the tree-level ten-dimensional Virasoro–Shapiro factor. With $S+T+U=0$,

$$f(S,T) =-\frac{STU\,\alpha'^3}{64} \prod_{X=S,T,U} \frac{\Gamma(-\alpha'X/4)} {\Gamma(1+\alpha'X/4)} +O(g_s^2).$$

Its low-energy expansion begins $1+\zeta(3)\alpha'^3STU/32+\cdots$, but the first massive $S$-channel pole occurs at $\alpha'S/4=1$ Binder et al. 2019, §1, eqs. (1.2)–(1.5), PDF. In the Penedones–Borel flat-space scaling, the integral samples Mellin variables of order $L^2S\sim\sqrt{\lambda}$ at this ten-dimensional pole. That is not the first intrinsic AdS exchange pole: heavy single-trace string primaries have twists of order $\lambda^{1/4}$, so their Mellin pole families begin at that smaller parametric order Alday and Hansen 2023, eqs. (13)–(16) and (22)–(23), PDF. Both descriptions identify the boundary string threshold $EL\sim\lambda^{1/4}$. Near it, restore the string tower; adding finitely many $D^{2k}R^4$ vertices is not a controlled repair.

Current near-flat-space work can determine curvature corrections and infinitely many Wilson coefficients for external states with general KK charges, but it retains rather than removes the distinction between the low-energy contact series and the resolved string spectrum Wang, Wu, and Yuan 2025, main result and Supplemental Material, PDF.

Now make the KK failure independent of all the other controls. Choose

$$N=10^6, \qquad \lambda=10^4.$$

Then

$$\frac{\alpha'}{L^2}=10^{-2}, \qquad \lambda^{-3/2}=10^{-6}, \qquad g_s=\frac{1}{400\pi}\simeq7.96\times10^{-4}, \qquad m_sL=10.$$

The unit D-instanton magnitude is suppressed by $e^{-800\pi^2}$. Nevertheless, an $\ell=1$ scalar harmonic has $m_1L=\sqrt5$. If a chosen channel has nonzero coupling $g_1$, tuning to its pole defeats every local $R^4$-only approximation even though $\alpha'm_1^2=5/\sqrt\lambda=0.05$. This does not claim that every $\mathcal O_2$ channel couples to that harmonic; it demonstrates why the coupling and the pole location must be checked separately.

Strongest surviving claim. Away from all coupled thresholds, the local action remains a useful asymptotic or Wilsonian description to the stated order. Near a KK pole, restore that KK field. Near a massive-string pole, use the full string amplitude or an equivalent description containing the resolved tower.

Large N at fixed intermediate coupling

Take $N\to\infty$ at fixed $\lambda=O(1)$. Then $g_s\to0$, handles are suppressed, and the D-instanton magnitude is exponentially small. But

$$\frac{\alpha'}{L^2}=\lambda^{-1/2}=O(1),$$

so the surviving description is a genus-zero, full-string problem, not two-derivative supergravity. The worldsheet theory need not be semiclassical or tractable when $\lambda=O(1)$. This is the precise downgrade carried out in the D3-brane adversarial limit.

The opposite trajectory is also instructive. If $\lambda$ grows proportionally to $N$, then $g_s$ approaches a constant and $e^{-8\pi^2N/\lambda}$ need not vanish with $N$. Nonperturbative ordering is therefore trajectory-dependent.

Common pitfalls

“Large $N$ means supergravity.” Large $N$ suppresses the fixed-$\lambda$ genus expansion. Weak curvature additionally requires large $\lambda$, and weak string coupling requires $\lambda/N$ small.

“A consistent truncation makes KK modes heavy.” Consistency is a nonlinear closure statement for selected classical solutions. On AdS$_5\times S^5$, omitted KK masses are generally of order $1/L$, and the tower can matter in other channels or in loops.

“Every $N^{-4}$ term is a supergravity loop.” A fixed genus has its own $\lambda$ expansion. Local genus-one string vertices and nonlocal one-loop supergravity terms share an $N$ order but have different coupling powers and analytic structures.

“A wrapped brane is an instanton.” A spatially wrapped Lorentzian brane is a state or defect. An Euclidean brane is a saddle with an exponential weight; its existence and contribution require separate charge and zero-mode checks.

“If $R^4$ vanishes, supergravity is exact.” A zero matrix element removes one operator in one observable. Another higher-derivative term, a KK exchange, a loop, or a nonperturbative sector may be the leading correction.

Exercises

1. Find a correlated supergravity window

Let $\lambda=N^a$ as $N\to\infty$. For which $a$ do both background curvature and string topology become parametrically controlled?

Solution

The curvature control is

$$\frac{\alpha'}{L^2}=N^{-a/2},$$

which tends to zero for $a>0$. The string coupling is

$$g_s=\frac{N^{a-1}}{4\pi},$$

which tends to zero for $a<1$. Therefore the simultaneous parametric window is

$$0<a<1.$$

At $a=0$, curvature corrections do not vanish. At $a=1$, $g_s$ no longer tends to zero, even if its numerical constant happens to be smaller than one.

2. Classify four terms before comparing them

Classify tree supergravity, planar $R^4$, one-loop supergravity, and a charge-$k$ D-instanton by their $N$ and $\lambda$ behavior and analytic signature in the $\mathcal O_2$ four-point function.

Solution

With unit-normalized external operators, tree supergravity is $O(c^{-1})\sim O(N^{-2})$ and is meromorphic because of exchange poles. The planar $R^4$ term is $O(c^{-1}\lambda^{-3/2})$ at fixed Mellin variables and is a local contact polynomial.

One-loop supergravity begins at $O(c^{-2})\sim O(N^{-4})$ and carries discontinuities, logarithms, and local subtraction ambiguities. This must not be confused with the protected local genus-one $R^4$ contribution, which scales parametrically as $c^{-2}\lambda^{1/2}$.

A D-instanton corrects selected coefficient functions by

$$\exp\left[ -\frac{8\pi^2\lvert k\rvert N}{\lambda} +ik\theta \right]$$

times determinants and zero-mode prefactors. It has no universal power-law $N$ order because its behavior depends on the trajectory in $(N,\lambda)$ space and on selection rules.

3. Measure the failure of a local pole expansion

For the exchange amplitude above, truncate after $K=2$. Find the relative remainder at $x=s/m^2=0.2$ and $0.9$, and state what happens at $x=1$.

Solution

Because

$$\left\lvert\frac{R_2}{\mathcal A}\right\rvert=\lvert x\rvert^3,$$

the relative remainders are

$$0.2^3=0.008, \qquad 0.9^3=0.729.$$

The same finite expansion that is accurate below threshold is already poor near the pole. At $x=1$, the exact amplitude diverges and the local geometric series is outside its domain of convergence. The stop rule is to restore the exchanged state before using kinematics near that pole.

4. Apply a selection rule before power counting

Suppose the $R^4$ matrix element vanishes in a chosen charge channel, while exchange of an omitted KK mode is allowed. What is the strongest justified statement?

Solution

One may state only that the leading local $R^4$ correction vanishes for that observable and channel. It does not follow that the full order $\lambda^{-3/2}$ correction, every string correction, or every correction to supergravity vanishes without a complete operator and exchange analysis.

If the kinematics approach the allowed KK pole, that exchange is the leading omitted effect regardless of the $R^4$ zero. Away from the pole, its local expansion may be used only when $Q/m_{\mathrm{KK}}$ is small and the required coefficients have been matched.

Where to continue

• Loop diagrams and local subtraction in AdS develops the $c^{-2}$ renormalization problem.
• Mellin contact polynomials and exchange poles separates local vertices from resolved states.
• Finite-gap locality error budgets turns a spectral gap into a derivative-expansion claim.
• Curvature operators and field redefinitions supplies the general gravity-EFT handoff.
• Top-down, bottom-up, and ultraviolet-completion claims applies this correction discipline to model claims.

The coefficient formulas and their primary supporting literature were checked through 10 August 2026. The conclusions remain asymptotic and conditional on the stated theory, frame, observable, kinematics, and order of limits.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Alday, L. F. (2021), “On Genus-One String Amplitudes on AdS$_5\times S^5$,” Journal of High Energy Physics 2021(04), 005. DOI. Open PDF.
• Alday, L. F., Bissi, A., and Perlmutter, E. (2019), “Genus-One String Amplitudes from Conformal Field Theory,” Journal of High Energy Physics 2019(06), 010. DOI. Open PDF.
• Alday, L. F., Chester, S. M., and Hansen, T. (2021), “Modular Invariant Holographic Correlators for $\mathcal N=4$ SYM with General Gauge Group,” Journal of High Energy Physics 2021(12), 159. DOI. Open PDF.
• Alday, L. F., and Hansen, T. (2023), “The AdS Virasoro–Shapiro Amplitude,” Journal of High Energy Physics 2023(10), 023. DOI. Open PDF.
• Binder, D. J., Chester, S. M., Pufu, S. S., and Wang, Y. (2019), “$\mathcal N=4$ Super-Yang–Mills Correlators at Strong Coupling from String Theory and Localization,” Journal of High Energy Physics 2019(12), 119. DOI. Open PDF.
• Green, M. B., and Gutperle, M. (1997), “Effects of D-Instantons,” Nuclear Physics B 498, 195–227. DOI. Open PDF.
• Kim, H. J., Romans, L. J., and van Nieuwenhuizen, P. (1985), “Mass Spectrum of Chiral Ten-Dimensional $N=2$ Supergravity on $S^5$,” Physical Review D 32, 389–399. DOI.
• McGreevy, J., Susskind, L., and Toumbas, N. (2000), “Invasion of the Giant Gravitons from Anti-de Sitter Space,” Journal of High Energy Physics 2000(06), 008. DOI. Open PDF.
• Penedones, J. (2011), “Writing CFT Correlation Functions as AdS Scattering Amplitudes,” Journal of High Energy Physics 2011(03), 025. DOI. Open PDF.
• Wang, B., Wu, D., and Yuan, E. Y. (2025), “Kaluza–Klein AdS Virasoro–Shapiro Amplitude near Flat Space,” Physical Review Letters 135, 041603. DOI. Open PDF.