Superconformal Bootstrap Research Articles

Superconformal two-point functions of the Nahm pole defect in [formula omitted] super-Yang-Mills theory

We derive a manifestly superconformal expression for the leading-order two-point functions of all single trace chiral primary operators in 4d N=4 super-Yang-Mills theory with a co-dimension one Nahm pole defect. Notably, our result involves only a finite number of superblocks that correspond to 1/2-BPS representations in the defect channel. The derivation builds on a non-trivial exact result for traces of fuzzy spherical harmonics in combination with powers of su(2) generators, which leads to a remarkably compact expression for the bulk-to-defect couplings. Apart from being interesting in its own right as a closed-form expression at finite N, the final result is an essential prerequisite for applying the defect superconformal bootstrap programme to this model.

6D (2,0) bootstrap with the soft-actor-critic algorithm

We study numerically the 6D (2,0) superconformal bootstrap using the soft-Actor-Critic (SAC) algorithm as a stochastic optimizer. We focus on the four-point functions of scalar superconformal primaries in the energy-momentum multiplet. Starting from the supergravity limit, we perform searches for adiabatically varied central charges and derive two curves for a collection of 80 CFT data (70 of these data correspond to unprotected long multiplets and 10 to protected short multiplets). We conjecture that the two curves capture the A- and D-series (2,0) theories. Our results are competitive when compared to the existing bounds coming from standard numerical bootstrap methods, and data obtained using the OPE inversion formula. With this paper we are also releasing our Python implementation of the SAC algorithm, BootSTOP. The paper discusses the main functionality features of this package.

2d small N=4 Long-multiplet superconformal block

We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study. We derive the superconformal block by utilizing the global part of the super Virasoro algebra and set up the crossing equations for the non-BPS long-multiplet 4-point function. Along the way, we build global N=4 superconformal short and long multiplets and compute all possible 2,3-point functions of long-multiplets that are needed to construct the superconformal blocks and the crossing equations. Since we consider a long-multiplet 4-point function, the number of crossing equations is huge, and we expect it to give a strong constraint than the usual superconformal bootstrap analysis, which relies on BPS 4-point functions. In addition, we present an alternative way to derive crossing equations using N=4 superspace and comment on a puzzle.

Long multiplet bootstrap

Applications of the bootstrap program to superconformal field theories promise unique new insights into their landscape and could even lead to the discovery of new models. Most existing results of the superconformal bootstrap were obtained form correlation functions of very special fields in short (BPS) representations of the superconformal algebra. Our main goal is to initiate a superconformal bootstrap for long multiplets, one that exploits all constraints from superprimaries and their descendants. To this end, we work out the Casimir equations for four-point correlators of long multiplets of the two-dimensional global mathcal{N}=2 superconformal algebra. After constructing the full set of conformal blocks we discuss two different applications. The first one concerns two-dimensional (2,0) theories. The numerical bootstrap analysis we perform serves a twofold purpose, as a feasibility study of our long multiplet bootstrap and also as an exploration of (2,0) theories. A second line of applications is directed towards four-dimensional mathcal{N}=3 SCFTs. In this context, our results imply a new bound cge frac{13}{24} for the central charge of such models, which we argue cannot be saturated by an interacting SCFT.

Carving out the end of the world or (superconformal bootstrap in six dimensions)

We bootstrap mathcal{N}=left(1,0right) superconformal field theories in six dimensions, by analyzing the four-point function of flavor current multiplets. Assuming E8 flavor group, we present universal bounds on the central charge CT and the flavor central charge CJ. Based on the numerical data, we conjecture that the rank-one E-string theory saturates the universal lower bound on CJ, and numerically determine the spectrum of long multiplets in the rank-one E-string theory. We comment on the possibility of solving the higher-rank E-string theories by bootstrap and thereby probing M-theory on {mathrm{AdS}}_7times {mathrm{S}}^4/{mathrm{mathbb{Z}}}_2 .

(2, 2) superconformal bootstrap in two dimensions

We find a simple relation between two-dimensional BPS mathcal{N}=2 superconformal blocks and bosonic Virasoro conformal blocks, which allows us to analyze the crossing equations for BPS 4-point functions in unitary (2, 2) superconformal theories numerically with semidefinite programming. We constrain gaps in the non-BPS spectrum through the operator product expansion of BPS operators, in ways that depend on the moduli of exactly marginal deformations through chiral ring coefficients. In some cases, our bounds on the spectral gaps are observed to be saturated by free theories, by mathcal{N}=2 Liouville theory, and by certain Landau-Ginzburg models.

Mathcal{N} = 4 superconformal bootstrap of the K3 CFT

We study two-dimensional (4, 4) superconformal field theories of central charge c = 6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap. This is made possible through a surprising relation between the BPS mathcal{N} = 4 superconformal blocks with c = 6 and bosonic Virasoro conformal blocks with c = 28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly from the CFT perspective the signature of a continuous spectrum above a gap at the singular moduli, and find numerically an upper bound on this gap that is saturated by the A1 mathcal{N} = 4 cigar CFT. We also derive an analytic upper bound on the first nonzero eigenvalue of the scalar Laplacian on K3 in the large volume regime, that depends on the K3 moduli data. As two byproducts, we find an exact equivalence between a class of BPS mathcal{N} = 2 superconformal blocks and Virasoro conformal blocks in two dimensions, and an upper bound on the four-point functions of operators of sufficiently low scaling dimension in three and four dimensional CFTs.

S-duality invariant perturbation theory improved by holography

We study anomalous dimensions of unprotected low twist operators in the four-dimensional SU(N ) mathcal{N} = 4 supersymmetric Yang-Mills theory. We construct a class of interpolating functions to approximate the dimensions of the leading twist operators for arbitrary gauge coupling τ . The interpolating functions are consistent with previous results on the perturbation theory, holographic computation and full S-duality. We use our interpolating functions to test a recent conjecture by the mathcal{N} = 4 superconformal bootstrap that upper bounds on the dimensions are saturated at one of the duality-invariant points τ = i and τ = eiπ/3. It turns out that our interpolating functions have maximum at τ =eiπ/3, which are close to the conjectural values by the conformal bootstrap. In terms of the interpolating functions, we draw the image of conformal manifold in the space of the dimensions. We find that the image is almost a line despite the conformal manifold is two-dimensional. We also construct interpolating functions for the subleading twist operator and study level crossing phenomenon between the leading and subleading twist operators. Finally we study the dimension of the Konishi operator in the planar limit. We find that our interpolating functions match with numerical result obtained by Thermodynamic Bethe Ansatz very well. It turns out that analytic properties of the interpolating functions reflect can expectation on a radius of convergence of the perturbation theory.

Lessons from crossing symmetry at large N

We consider the four-point correlator of the stress tensor multiplet in $$ \mathcal{N}=4 $$ SYM. We construct all solutions consistent with crossing symmetry in the limit of large central charge c ∼ N 2 and large g 2 N . While we find an infinite tower of solutions, we argue most of them are suppressed by an extra scale Δgap and are consistent with the upper bounds for the scaling dimension of unprotected operators observed in the numerical superconformal bootstrap at large central charge. These solutions organize as a double expansion in 1/c and 1/Δgap. Our solutions are valid to leading order in 1/c and to all orders in 1/Δgap and reproduce, in particular, instanton corrections previously found. Furthermore, we find a connection between such upper bounds and positivity constraints arising from causality in flat space. Finally, we show that certain relations derived from causality constraints for scattering in AdS follow from crossing symmetry.

Exact correlators of BPS Operators from the 3d superconformal bootstrap

We use the superconformal bootstrap to derive exact relations between OPE coefficients in three-dimensional superconformal field theories with ${\cal N} \geq 4$ supersymmetry. These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In ${\cal N} = 4$ SCFTs, the non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we find are powerful enough to allow us to determine an infinite number of OPE coefficients in the interacting SCFT ($U(2)_2 \times U(1)_{-2}$ ABJ theory) that constitutes the IR limit of $O(3)$ ${\cal N} = 8$ super-Yang-Mills theory. More generally, in ${\cal N} = 8$ SCFTs with a unique stress tensor, we are led to conjecture that many superconformal multiplets allowed by group theory must actually be absent from the spectrum, and we test this conjecture in known ${\cal N} = 8$ SCFTs using the superconformal index. For generic ${\cal N} = 8$ SCFTs, we also improve on numerical bootstrap bounds on OPE coefficients of short and semi-short multiplets and discuss their relation to the exact relations between OPE coefficients we derived. In particular, we show that the kink previously observed in these bounds arises from the disappearance of a certain quarter-BPS multiplet, and that the location of the kink is likely tied to the existence of the $U(2)_2 \times U(1)_{-2}$ ABJ theory, which can be argued to not possess this multiplet.