Bootstrapping $\mathcal{N}=4$ super-Yang-Mills on the conformal manifold

Abstract

We combine supersymmetric localization results with numerical bootstrap techniques to compute upper bounds on the low-lying CFT data of N = 4 super-Yang-Mills theory as a function of the complexified gauge coupling τ . In particular, from the stress tensor multiplet four-point function, we extract the scaling dimension of the lowest-lying unprotected scalar operator and its OPE coefficient. While our method can be applied in principle to any gauge group G, we focus on G = SU (2) and SU (3) for simplicity. At weak coupling, the upper bounds we find are very close to the corresponding four-loop results. We also give preliminary evidence that these upper bounds become small islands under reasonable assumptions.

Snowmass White Paper: Bootstrapping String Theory

“…The numerical bootstrap effort uses the tools being developed in the broader conformal bootstrap program for CFTs in general spacetime dimensions (for reviews and references, see [54][55][56][57][58]), in particular testing for the unitarity and crossing symmetry of four-point functions using semi-definite programming, numerically implemented, for instance, with the SDPB semi-definite program solver [59]. Numerical studies in the holographic theories mentioned above have been performed for 4d N = 4 SYM in [60][61][62][63], for 3d ABJ(M) theory in [53,[64][65][66][67], and for the 6d (2, 0) theory in [68], and they include information obtained using supersymmetric localization [50,[69][70][71][72][73][74]. Of the results obtained thus far, we would like to highlight the precise islands in OPE coefficient space in 3d N = 8 ABJM theory (see Figure 1), which represent a first step in determining the low-lying CFT data of ABJM theory numerically.…”

The Hybrid Bootstrap

“…, and α even 0 , β even 0 are defined in (4.41) of [13]. Now we can compute the individual contribution of { , T } using (6) and sum up the double twist contribution using (11) to get an approximate generating function. To further improve it, we may iterate the inversion formula a few times, i.e.…”

“…To solve (9), we use GFF value as the initial value for ∆ and iterate (9) for 3 times. In first step, we take { , T } as individual contributions using (6) and sum up double twist contribution from 0 = 4 using (11). To do the double twist summation, we take the identity contribution for λ 2 σσ[σσ] 0 /λ 2 GFF and the { , T } contributions for δh( h) in (10).…”

“…See[1] for a thorough review. See[2][3][4][5][6] for some recent computations, where rigorous error bars are determined in various CFTs.2 See also[17] for a review of the analytic bootstrap 3. Alternative approaches might be to follow[19] or[20] 4.…”

“…For Section 2.2, the block and OPE coefficient is in the Convention 6, because this convention is more convenient for the analytic computation. When using the result of Section 2.2 in the numerics, one has to first convert the convention accordingly 6. The data from our numerics has a small difference between f σσT /∆ σ , f T /∆ .…”