Bootstrapping mixed correlators in $$ \mathcal{N} $$ = 4 super Yang-Mills

Abstract

We perform a numerical bootstrap study of the mixed correlator system containing the half-BPS operators of dimension two and three in $$ \mathcal{N} $$ N = 4 Super Yang-Mills. This setup improves on previous works in the literature that only considered single correlators of one or the other operator. We obtain upper bounds on the leading twist in a given representation of the R-symmetry by imposing gaps on the twist of all operators rather than the dimension of a single one. As a result we find a tension between the large N supergravity predictions and the numerical finite N results already at N∼ 100. A few possible solutions are discussed: the extremal spectrum suggests that at large but finite N, in addition to the double trace operators, there exists a second tower of states with smaller dimension. We also obtain new bounds on the dimension of operators which were not accessible with a single correlator setup. Finally we consider bounds on the OPE coefficients of various operators. The results obtained for the OPE coefficient of the lightest scalar singlet show evidences of a two dimensional conformal manifold.

Modular invariant holographic correlators for $$ \mathcal{N} $$ = 4 SYM with general gauge group

“…Unfortunately, we find that our large N finite τ predictions for CFT data do not saturate the bootstrap bounds at large N for either case, even at the self dual point τ = e iπ/3 that was conjectured to saturate the bounds in [43]. Furthermore, we computed bounds using both the single correlator setup in [42,43] as well as the mixed correlator setup in [44] (which included p = 2, 3 half-BPS operators), and found that the upper bounds were almost indistinguishable at large N , even though the mixed correlator setup does not apply to the orbifold theories (since they do not contain odd p half-BPS operators), so we would have expected SU(N ) to saturate the mixed bounds and the other cases to saturate the single correlator bounds.…”

“…The SO(2N ) value is also larger than the SU(N ) value for large c, so if any theory saturates the single correlator bounds of [42,43], it should be the SO(2N ) theory at τ = e iπ/3 . A numerical bootstrap study was also done in [44] using mixed correlators between the stress tensor multiplet and the next lowest half-BPS multiplet with ∆ = 3. Since this multiplet is absent for SO(2N ), only the SU(N ) theory could appear in these bounds.…”

“…Since this multiplet is absent for SO(2N ), only the SU(N ) theory could appear in these bounds. In figure 1 we show a comparison between the single and mixed correlator bounds which we recomputed at very high bootstrap precision for low spins, and the analytic estimates at large c. 12 For completeness we give a full description of our mixed correlator bootstrap setup (independent of [44]) in appendix E. None of the analytic estimates for ∆ 0 and ∆ 2 are close to saturating the bounds, 13 and the single correlator and mixed correlator bounds are in fact almost indistinguishable in this regime, which suggests that no physical theory is saturating these bounds.…”

Two Applications of the Analytic Conformal Bootstrap: A Quick Tour Guide

“…However, there are still undetermined data hidden in H and one would like to find a way to study the dimensions and the squared OPE coefficients a ∆, appearing there. Various approaches have been pursued in the years, from numerical bootstrap techniques [45,46,[51][52][53] to the more analytic ones [54][55][56][57][58][59][60][61][62][63][64][65][66], involving the use of modern tools such as the Lorentzian inversion formula [9,67,68], large spin perturbation theory [5] and unitarity methods [8,69]. These studies have shed some light on the spectrum of long operators in N = 4 and they have provided non-trivial tests of the AdS/CFT correspondence, especially in the large N (or equivalently large c = N 2 −1 4 ) limit and at infinite 't Hooft coupling λ.…”

Bootstrapping (D, D) Conformal Matter

“…Using the associativity of the operator product expansion (OPE), the ethos of the bootstrap is to harness the power of unitarity to impose strict bounds on the values that the conformal data of a(n S)CFT can take, with minimal assumptions on the spectrum of the theory. The superconformal bootstrap has been applied to SCFTs with at least eight supercharges; in three [18][19][20][21][22][23] and four [24][25][26][27][28][29][30][31][32][33][34][35][36][37] dimensions there is a significant body of literature, however five [38] and six [39][40][41] dimensional analyses have been carried out comparatively less. In general, the bounds that are imposed by crossing symmetry and associativity of the OPE appear to be saturated by SCFTs that have a construction via string theory.…”