Extremal cubic couplings in AdS relate bulk fields such that ∆ + ∆ = ∆ . Such couplings lead to divergent 3-point Witten diagrams, and do not occur in theories with maximal supersymmetry. We consider the simplest theories where such coupling are non-zero, which is type IIB string theory with $\textit{N}$ D3 branes probing various configurations of sevenbranes, which are dual to certain 4d $$\mathcal{N}=2$$ SCFTs. At large $\textit{N}$, the low energy effective theory is supergravity on $\textit{AdS}$ × $\textit{S}$ with a singularity, whose fixed point locus is $\textit{AdS}$ × $\textit{S}$ . These theories have infinite towers of graviton modes, as well as gluon modes on the sevenbranes. We compute the nonzero coupling between these modes, which is in general (super)-extremal. We use these couplings to compute the graviton exchange term in the holographic correlator of gluon KK modes $$\langle 22pp\rangle $$ , which appears at the same order 1/$\textit{N}$ as 1-loop gluon exchange, and receives contributions from a whole tower of graviton modes. We use this graviton exchange term to compute the unmixing of the single trace graviton modes with double traces of gluon modes, which explains the divergent 3-point diagrams. Finally, for $$\langle 2222\rangle $$ for the simplest 4d $$\mathcal{N}=2$$ gauge theory, we use supersymmetric localization and the new graviton exchange term to completely fix the correlator at order 1/$\textit{N}$ .

United Kingdom

Shai Chester

Rishi Mouland

Jesse Muiden

Extremal couplings, graviton exchange, and gluon scattering in AdS

We consider mixed gluon-graviton scattering in Type IIB string theory on AdS × $\textit{S}$ /$\textit{ℤ}$ in the presence of D7 branes, which is dual to a mixed correlator of the SO(8) and SU(2) flavor multiplets of a certain 4d $$ \mathcal{N} $$ = 2 USp(2$\textit{N}$) gauge theory with complexified coupling $\textit{τ}$. We compute this holographic correlator in the large $\textit{N}$ and finite $\textit{τ}$ expansion using constraints from derivatives of the mass deformed sphere free energy, which we compute using supersymmetric localization at large $\textit{N}$ and finite $\textit{τ}$ in terms of modular invariant non-holomorphic Eisenstein series. In particular, we combine this constraint with the known flat space limit to fix the $\textit{R}$ $\textit{F}$ higher derivative correction to the correlator in terms of the weight one non-holomorphic Eisenstein series, and also to fix the logarithmic threshold. We also compute the one-loop correction to the correlator, and match it to the expected flat space limit result.

United States

Pietro Ferrero

Daniele Pavarini

Modular invariant gluon-graviton scattering in AdS at one loop

We study higher-derivative corrections to the graviton scattering amplitude in M-theory, via the stress tensor correlator of 3d $$ \mathcal{N} $$ = 8 U($\textit{N}$) × U($\textit{N}$) ABJM theory (dual to graviton scattering in M-theory on AdS × $\textit{S}$ /$\textit{ℤ}$ ). We use the conformal bootstrap combined with an integral constraint derived from supersymmetric localization in order to constrain semishort OPE coefficients appearing in the stress tensor correlator. We obtain islands that are significantly more precise than those in previous studies that did not use the integral constraint. Using these islands, we can estimate the powers and coefficients in a large central charge expansion. This allows us to accurately read off the N LO contribution, from the protected $\textit{D}$ $\textit{R}$ correction, and also estimate the N LO contribution, from the unprotected $\textit{D}$ $\textit{R}$ correction.

Ross Dempsey

Silviu Pufu

Higher-derivative corrections in M-theory from precision numerical bootstrap

We consider tree level scattering of gravitons in type IIA string theory on $\textit{AdS}$ × $$ {\mathbbm{CP}}^3 $$ to all orders in $\textit{α}$ , which is dual to the stress tensor correlator in U($\textit{N}$) × U($\textit{N}$) ABJM theory in the planar large $\textit{N}$ limit and to all orders in large $\textit{λ}$ ~ $\textit{N}$/$\textit{k}$. The small curvature expansion of this correlator, defined via a Borel transform, is given by the flat space Virasoro-Shapiro amplitude plus AdS curvature corrections. We fix curvature corrections by demanding that their resonances are consistent with the superconformal block expansion of the correlator and with a worldsheet ansatz in terms of single-valued multiple polylogarithms. The first correction is fully fixed in this way, and matches independent results from integrability, as well as the $\textit{R}$ correction at finite AdS curvature that was previously fixed using supersymmetric localization. We are also able to fix the second curvature correction by using a few additional assumptions, and find that it also satisfies various non-trivial consistency checks. We use our results to fix the tree level $\textit{D}$ $\textit{R}$ correction at finite AdS curvature, and to give many predictions for future integrability studies.

Tobias Hansen

De-liang Zhong

The type IIA Virasoro-Shapiro amplitude in AdS

We consider type IIB string theory with N D3 branes and various configurations of sevenbranes, such that the string coupling g  s  is fixed to a constant finite value. These are the simplest realizations of F-theory, and are holographically dual to rank N Argyres-Douglas conformal field theories (CFTs) with SU(2) and SU(3) flavor groups, and Minahan-Nemeschansky CFTs with E 6, E 7 and E 8 flavor groups. We use the Seiberg-Witten curves of these theories to compute the mass deformed sphere free energy F (m) at large N in terms of novel matrix models with non-polynomial potentials. We show how F (m) can be used along with the analytic bootstrap to fix the large N expansion of flavor multiplet correlators in these CFTs, which are dual to scattering of gluons on AdS 5 × S 3, and in the flat space limit determine the effective theory of sevenbranes in F-theory. As a first step in this program, we use the matrix models to compute the log N term in F (m) and thereby fix the logarithmic threshold in the AdS 5 × S 3 holographic correlator, which matches the flat space prediction.

Brazil

Connor Behan

Towards bootstrapping F-theory

We combine supersymmetric localization with the numerical conformal bootstrap to bound the scaling dimension and OPE coefficient of the lowest-dimension unprotected operator in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SU(N) super-Yang-Mills theory for a wide range of N and Yang-Mills couplings g YM. We find that our bounds are approximately saturated by weak coupling results at small g YM. Furthermore, at large N our bounds interpolate between integrability results for the Konishi operator at small g YM and strong-coupling results, including the first few stringy corrections, for the lowest-dimension double-trace operator at large g YM. In particular, our scaling dimension bounds describe the level splitting between the single- and double-trace operators at intermediate coupling.

Level repulsion in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Yang-Mills via integrability, holography, and the bootstrap

We analyze correlation functions of SU(k) × SU(2) F  flavor currents in a family of three-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 superconformal field theories, combining analytic bootstrap methods with input from supersymmetric localization. Via holographic duality, we extract gluon and graviton scattering amplitudes of M-theory on AdS4 × S 7/ℤ  k  which contains a ℂ2/ℤ k  orbifold singularity. From these results, we derive aspects of the effective description of M-theory on the orbifold singularity beyond its leading low energy limit. We also determine a threshold correction to the holographic correlator from the combined contribution of two-loop gluon and tree-level bulk graviton exchange.

Yifan Wang

Xi Yin

Bootstrapping M-theory orbifolds

We compute the AdS Veneziano amplitude for type IIB gluon scattering in AdS 5 × S 3 to all orders in α  ′  in a small curvature expansion. This is achieved by combining a dispersion relation in the dual 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 SCFT with an ansatz for the amplitude as a worldsheet integral in terms of multiple polylogarithms. The first curvature correction is fully fixed in this way and satisfies consistency checks in the high energy limit, the low energy expansion as previously fixed using supersymmetric localisation, and for the energy of massive string operators, which we independently compute using a semiclassical expansion. We also combine localisation with this first curvature correction to fix the unprotected D 4 F 4 correction to the amplitude at finite curvature.

Luis Alday

The AdS Veneziano amplitude at small curvature

Integrated correlation functions in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 supersymmetric Yang-Mills theory with gauge group SU(N) can be expressed in terms of the localised S 4 partition function, Z  N , deformed by a mass m. Two such cases are   <math> <msub> <mi>C</mi> <mi>N</mi> </msub> <mo>=</mo> <msup> <mfenced> <mrow> <mo>Im</mo> <mi>τ</mi> </mrow> </mfenced> <mn>2</mn> </msup> <msub> <mi>∂</mi> <mi>τ</mi> </msub> <msub> <mi>∂</mi> <mover> <mi>τ</mi> <mo>¯</mo> </mover> </msub> <msubsup> <mi>∂</mi> <mi>m</mi> <mn>2</mn> </msubsup> <mo>log</mo> <msub> <mfenced> <msub> <mi>Z</mi> <mi>N</mi> </msub> </mfenced> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </math>  $$ {\mathcal{C}}_N={\left(\operatorname{Im}\tau \right)}^2{\partial}_{\tau }{\partial}_{\overline{\tau}}{\partial}_m^2\log {\left.{Z}_N\right|}_{m=0} $$  and   <math> <msub> <mi>H</mi> <mi>N</mi> </msub> <mo>=</mo> <msubsup> <mi>∂</mi> <mi>m</mi> <mn>4</mn> </msubsup> <mo>log</mo> <msub> <mfenced> <msub> <mi>Z</mi> <mi>N</mi> </msub> </mfenced> <mrow> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> </math>  $$ {\mathcal{H}}_N={\partial}_m^4\log {\left.{Z}_N\right|}_{m=0} $$ , which are modular invariant functions of the complex coupling τ. While   <math> <msub> <mi>C</mi> <mi>N</mi> </msub> </math>  $$ {\mathcal{C}}_N $$  was recently written in terms of a two-dimensional lattice sum for any N and τ,   <math> <msub> <mi>H</mi> <mi>N</mi> </msub> </math>  $$ {\mathcal{H}}_N $$  has only been evaluated up to order 1/N 3 in a large-N expansion in terms of modular invariant functions with no known lattice sum realisation. Here we develop methods for evaluating   <math> <msub> <mi>H</mi> <mi>N</mi> </msub> </math>  $$ {\mathcal{H}}_N $$  to any desired order in 1/N and finite τ. We use this new data to constrain higher loop corrections to the stress tensor correlator, and give evidence for several intriguing relations between   <math> <msub> <mi>H</mi> <mi>N</mi> </msub> </math>  $$ {\mathcal{H}}_N $$  and   <math> <msub> <mi>C</mi> <mi>N</mi> </msub> </math>  $$ {\mathcal{C}}_N $$  to all orders in 1/N. We also give evidence that the coefficients of the 1/N expansion of   <math> <msub> <mi>H</mi> <mi>N</mi> </msub> </math>  $$ {\mathcal{H}}_N $$  can be written as lattice sums to all orders. Lastly, these large N and finite τ results are used to accurately estimate the integrated correlators at finite N and finite τ.

Daniele Dorigoni

Michael Green

Congkao Wen

Relations between integrated correlators in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 supersymmetric Yang-Mills theory

We consider gluons scattering in Type IIB string theory on AdS5 × S 5 /ℤ2 in the presence of D7 branes, which is dual to the flavor multiplet correlator in a certain 4d   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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, 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 2 F 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.