We use the S-matrix bootstrap to carve out the space of unitary, crossing symmetric and supersymmetric graviton scattering amplitudes in ten dimensions. We focus on the leading Wilson coefficient $alpha$ controlling the leading correction to maximal
supergravity. The negative region $alpha<0$ is excluded by a simple dual argument based on linearized unitarity (the desert). A whole semi-infinite region $alpha gtrsim 0.14$ is allowed by the primal bootstrap (the garden). A finite intermediate region is excluded by non-perturbative unitarity (the swamp). Remarkably, string theory seems to cover all (or at least almost all) the garden from very large positive $alpha$ -- at weak coupling -- to the swamp boundary -- at strong coupling.
We use the numerical S-matrix bootstrap method to obtain bounds on the two leading Wilson coefficients of the chiral lagrangian controlling the low-energy dynamics of massless pions thus providing a proof of concept that the numerical S-matrix bootst
rap can be used to derive non-perturbative bounds on EFTs in more than two spacetime dimensions.
We initiate an exploration of the conformal bootstrap for $n>4$ point correlation functions. Here we bootstrap correlation functions of the lightest scalar gauge invariant operators in planar non-abelian conformal gauge theories as their locations ap
proach the cusps of a null polygon. For that we consider consistency of the OPE in the so-called snowflake channel with respect to cyclicity transformations which leave the null configuration invariant. For general non-abelian gauge theories this allows us to strongly constrain the OPE structure constants of up to three large spin $J_j$ operators (and large polarization quantum number $l_{j}$) to all loop orders. In $ mathcal{N}=4$ we fix them completely through the duality to null polygonal Wilson loops and the recent origin limit of the hexagon explored by Basso, Dixon and Papathanasiou.
Using duality in optimization theory we formulate a dual approach to the S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a consequence of unitarity, crossing symmetry and analyticity of the scattering matrix. We then explain
how to optimize such bounds numerically, and prove that they provide the same bounds obtained from the usual primal formulation of the S-matrix Bootstrap, at least once convergence is attained from both perspectives. These techniques are then applied to the study of a gapped system with two stable particles of different masses, which serves as a toy model for bootstrapping popular physical systems.
We explore a new way of probing scattering of closed strings in $AdS_5times S^5$, which we call `the large $p$ limit. It consists of studying four-point correlators of single-particle operators in $mathcal{N}=4$ SYM at large $N$ and large t Hooft cou
pling $lambda$, by looking at the regime in which the dual KK modes become short massive strings. In this regime the charge of the single-particle operators is order $lambda^{1/4}$ and the dual KK modes are in between fields and strings. Starting from SUGRA we compute the large $p$ limit of the correlators by introducing an improved $AdS_5times S^5$ Mellin space amplitude, and we show that the correlator is dominated by a saddle point. Our results are consistent with the picture of four geodesics shooting from the boundary of $AdS_5times S^5$ towards a common bulk point, where they scatter as if they were in flat space. The Mandelstam invariants are put in correspondence with the Mellin variables and in turn with certain combinations of cross ratios. At the saddle point the dynamics of the correlator is directly related to the bulk Mellin amplitude, which in the process of taking large $p$ becomes the flat space ten-dimensional S-matrix. We thus learn how to embed the full type IIB S-matrix in the $AdS_5times S^5$ Mellin amplitude, and how to stratify the latter in a large $p$ expansion. We compute the large $p$ limit of all genus zero data currently available, pointing out additional hidden simplicity of known results. We then show that the genus zero resummation at large $p$ naturally leads to the Gross-Mende phase for the minimal area surface around the bulk point. At one-loop, we first uncover a novel and finite Mellin amplitude, and then we show that the large $p$ limit beautifully asymptotes the gravitational S-matrix.
The octagon function is the fundamental building block yielding correlation functions of four large BPS operators in N=4 super Yang-Mills theory at any value of the t Hooft coupling and at any genus order. Here we compute the octagon at strong coupli
ng, and discuss various interesting limits and implications, both at the planar and non-planar level.
We explain how the t Hooft expansion of correlators of half-BPS operators can be resummed in a large-charge limit in N=4 super Yang-Mills theory. The full correlator in the limit is given by a non-trivial function of two variables: One variable is th
e charge of the BPS operators divided by the square root of the number Nc of colors; the other variable is the octagon that contains all the t Hooft coupling and spacetime dependence. At each genus g in the large Nc expansion, this function is a polynomial of degree 2g+2 in the octagon. We find several dual matrix model representations of the correlators in the large-charge limit. Amusingly, the number of colors in these matrix models is formally taken to zero in the relevant limit.
We analyze the pentagon transitions involving arbitrarily many flux-tube gluonic excitations and bound states thereof in planar N=4 Super-Yang-Mills theory. We derive all-loop expressions for all these transitions by factorization and fusion of the e
lementary transitions for the lightest gluonic excitations conjectured in a previous paper. We apply the proposals so obtained to the computation of MHV and NMHV scattering amplitudes at any loop order and find perfect agreement with available perturbative data up to four loops.
We study three-point correlation functions of local operators in planar $mathcal{N}=4$ SYM at weak coupling using integrability. We consider correlation functions involving two scalar BPS operators and an operator with spin, in the so called SL(2) se
ctor. At tree level we derive the corresponding structure constant for any such operator. We also conjecture its one loop correction. To check our proposals we analyze the conformal partial wave decomposition of known four-point correlation functions of BPS operators. In perturbation theory, we extract from this decomposition sums of structure constants involving all primaries of a given spin and twist. On the other hand, in our integrable setup these sum rules are computed by summing over all solutions to the Bethe equations. A perfect match is found between the two approaches.
We compute structure constants in N=4 SYM at one loop using Integrability. This requires having full control over the two loop eigenvectors of the dilatation operator for operators of arbitrary size. To achieve this, we develop an algebraic descripti
on called the Theta-morphism. In this approach we introduce impurities at each spin chain site, act with particular differential operators on the standard algebraic Bethe ansatz vectors and generate in this way higher loop eigenvectors. The final results for the structure constants take a surprisingly simple form. For some quantities we conjecture all loop generalizations. These are based on the tree level and one loop patterns together and also on some higher loop experiments involving simple operators.
