The free σCFTs

Abstract

We introduce the conformal field theories that describe the shadows of the lowest dimension composites made out of massless free scalars and fermions in d dimensions. We argue that these theories can be consistently defined as free CFTs for even d ≥ 4. We use OPE techniques to study their spectrum and show that for d → ∞ it matches that of free bosonic CFTs in d = 6 and d = 4 dimensions. For these σCFTs we calculate c T in d = 6, 8, 10 and 12 dimensions using the OPE and also a direct construction of their higher-derivative energy momentum tensors. Our results agree with the general proposal of arXiv:1601.07198.Free conformal field theories (CFTs) can be intriguing. This is one of the main lessons higher-spin holography 1 has taught us. For all their apparent simplicity, free CFTs emerge holographically as the result of a complicated symmetry that makes possible the consistent interaction of infinite towers of higher-spin gauge fields. And even though symmetry breaking may be the most physically relevant property of a theory, the exploitation of its symmetric phase usually yields interesting knowledge.In this work we initiate the study of two peculiar free CFTs that are universally connected to free massless scalars φ(x) and Dirac fermions ψ(x) in d dimensions. For those free theories, the simplest composite fields one can construct are 2 φ 2 (x) andψ(x)ψ(x) and parametrize the usual mass deformations.Considering then x-dependent mass terms such as d d xσ 2 (x)φ 2 (x) and d d xσ 1 (x)ψ(x)ψ(x) one elevates σ 2 (x) and σ 1 (x) to genuine scalar fields that determine the effective action of the massive phase. Moreover, since the canonical scaling dimensions of φ(x) and ψ(x) are ∆ φ = d 2 − 1 and ∆ ψ = d−1 2 , the fields σ 2 (x) and σ 1 (x) exhibit universal scaling behaviour in any d, namely ∆ σ 2 = 2 and ∆ σ 1 = 1. In the CFT language σ 2 (x) and σ 1 (x) are the shadows 3 of the composites φ 2 (x) andψ(x)ψ(x) respectively.The fields σ 2 (x) and σ 1 (x) are not part of the spectrum of the corresponding free CFTs for any d.They show up, however, in the spectrum of non-trivial bosonic and fermionic CFTs in 2 < d < 6 [4][5][6][7] where they generically acquire anomalous dimensions. Remarkably, taking the strict N → ∞ limit of the non-trivial theories one is left with theories where all fields have canonical dimensions and yet σ 2 (x) and σ 1 (x) are still in the spectrum replacing φ 2 (x) andψ(x)ψ(x) respectively. Nevertheless, at least for the above range of dimensions, it has been noticed that all the non-trivial CFT models do become free field theories for d = 2, 4, 6 and hence it is natural to ask about the fate of the σ-fields in these cases. For the bosonic O(N ) vector model in 2 < d < 4 this has been partially discussed in [8]. It was there shown that if one starts with both of the fields φ 2 (x) and σ 2 (x) in the theory, the second one decouples in the free field theory limit and the first one becomes redundant at the non-trivial fixed point. This mechanism was connected to the O(N ) → O(N − 1) global symmetr...

C T for higher derivative conformal fields and anomalies of (1, 0) superconformal 6d theories

“…The above values of C T in (3.18) and (3.21) are in agreement with the general expression for the 4-derivative conformal scalar in dimension d found in [30,55] …”

“…It would be interesting to reproduce (5.26) in alternative flat-space approaches like the ones used in the higher-derivative scalar cases in [55] and [30].…”

C T for non-unitary CFTs in higher dimensions

“…Note added: for higher-derivative scalar theories calculations of C T have also been carried out by Guerrieri et al [41], who further considered the theory with eight derivatives. Prompted by their discussion there is a quick derivation of C T,ϕ,2p , agreeing with (2.13) for any p, based on known results for conformal partial-wave expansions of four-point functions as a sum over conformal primaries.…”

Generalized Wilson-Fisher Critical Points from the Conformal Operator Product Expansion

“…All other correlation functions, either of ϕ f or of its composites, are given by Wick contractions. We are particularly interested in the cases in which Δ ϕ f ¼ ðd=2Þ − k. They correspond to GFCFTs that have a Lagrangian description as massless free theories with a ∂ 2k kinetic term that can be coupled to the stress energy tensor [12][13][14][15]. The k ¼ 1 case corresponds to the free canonical theories.…”

“…The k ¼ 1 case corresponds to the free canonical theories. Some GFCFTs with k > 1 play an important role in the study of 1=N expansions in Gross-Neveu and OðNÞ vector models in high dimensions [12][13][14]. Applying the Wick contractions to the product ϕ …”