Fermion conformal bootstrap in 4d

Crossing symmetry provides a powerful tool to access the non-perturbative dynamics of conformal and superconformal field theories. Here we develop the mathematical formalism that allows to construct the crossing equations for arbitrary four-point functions in theories with superconformal symmetry of type I, including all superconformal field the- ories in d = 4 dimensions. Our advance relies on a supergroup theoretic construction of tensor structures that generalizes an approach which was put forward in [1] for bosonic theories. When combined with our recent construction of the relevant superblocks, we are able to derive the crossing symmetry constraint in particular for four-point functions of arbitrary long multiplets in all 4-dimensional superconformal field theories.

Section: Discussionmentioning

confidence: 99%

“…Many authors have studied tensor structures for spinning four-point functions in conformal field theories, see e.g. [6,[41][42][43][44][45][46]. The tensor factor Ω(x i ) is restricted but not determined by conformal symmetry.…”

Section: Jhep10(2020)147mentioning

confidence: 99%

The superconformal equation

Burić

Schomerus

Sobko

2020

“…We emphasize that the OPE relations are essential to meaningfully impose this constraint: they prevent the appearance of vector operators of dimensions very close to d that would numerically be indistinguishable from V (1) . Furthermore, because of the fake primary effect [46,47] the block for V (1) can be mimicked in our numerical analysis by a spin 2 operator of dimension 3, and therefore the constraint that ∆ τ > 3 (strictly) is also essential to ensure that it is really absent. This latter argument relies on the observation that, for a spin 2 operator whose dimension ∆ → 3, the corresponding combination of blocks that enters in the crossing equation (4.7) is:…”

Section: Jhep12(2020)182mentioning

confidence: 99%

Bootstrapping boundary-localized interactions

Behan

Pietro

Lauria

et al. 2020

We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

“…The most important one is to compute superconformal blocks for spinning superprimaries, both protected and long. A place to start from could be the case of spin 1/2 [36] where the conformal blocks are already available and the extra work needed to include supersymmetry is minimal. The aim of the Mathematica package that we introduced is to render all these tasks relatively straightforward and systematic.…”

Section: Discussionmentioning

confidence: 99%

Differential operators for superconformal correlation functions

Manenti

2020

We present a systematic method to expand in components four dimensional superconformal multiplets. The results cover all possible N = 1 multiplets and some cases of interest for N = 2. As an application of the formalism we prove that certain N = 2 spinning chiral operators (also known as "exotic" chiral primaries) do not admit a consistent three-point function with the stress tensor and therefore cannot be present in any local SCFT. This extends a previous proof in the literature which only applies to certain classes of theories. To each superdescendant we associate a superconformally covariant differential operator, which can then be applied to any correlator in superspace. In the case of threepoint functions, we introduce a convenient representation of the differential operators that considerably simplifies their action. As a consequence it is possible to efficiently obtain the linear relations between the OPE coefficients of the operators in the same superconformal multiplet and in turn streamline the computation of superconformal blocks. We also introduce a Mathematica package to work with four dimensional superspace. 23 7. Conclusions25 Appendix A. Details on notation and conventions 26 -1 -Appendix B. Acting on different points 28 Appendix C. Superspace expansion 29 Appendix D. Some identities for the superspace derivatives 30 References 32 In[11]:= chi = Compare[ChiralDp[tOφφb, η2, θ3, x3], variables → Array[C,4]] 20 In the package the derivatives ChiralD represent the derivatives D. When acting on the t use ChiralD instead (D =[esc]scD[esc]). Also note that, when acting on the t, the operators D are sent to D and D to D. See (4.8). 21 In more complicated applications it is better to separate the various orders in θ3 andθ3 and act on each piece only with the operators inside ChiralD (like ∂η∂ θ , ∂η∂xθ etc. . .) that do not give zero when the θ's are suppressed. In the documentation of the package there is a worked out example. 22 See (2.11). 23 This instruction will generate an error but it can be ignored. It can be avoided by replacing the C[i]'s with variables consisting in a single Symbol. It comes up because there is an automatic routine that sets ConstQ to true for the variables in variables, which works only if they are Symbols. This is not an issue because ConstQ[C[i]] is true by default after SUSY3pf is called.