Abstract
We explore a class of CFT’s with higher spin currents and charges. Away from the free or N = ∞ limit the non-conservation of currents is governed by operators built out of the currents themselves, which deforms the algebra of charges by, and together with, its action on the currents. This structure is encoded in a certain A∞/L∞-algebra. Under quite general assumptions we construct invariants of the deformed higher spin symmetry, which are candidate correlation functions. In particular, we show that there is a finite number of independent structures at the n-point level. The invariants are found to have a form reminiscent of a one-loop exact theory. In the case of Chern-Simons vector models the uniqueness of the invariants implies the three-dimensional bosonization duality in the large-N limit.
Highlights
The unbroken higher spin symmetry can be summarized by saying that (i) the higher spin currents Js, with s = 2 member being the stress-tensor, generate the higher spin charges Qs; (ii) Qs have to form a Lie algebra hs, the higher spin algebra, which is infinitedimensional and contains the conformal algebra so(d, 2); (iii) currents J form a module over hs:
A remarkable feature of the higher spin symmetry breaking in vector models is that the non-conservation of higher spin currents is still driven by the higher spin currents themselves, by the double-trace, [JJ], operators that are built out of J’s
The main result of the paper is remarkably simple and calls for a better explanation: slightly-broken higher spin symmetry admits a family of invariants WSBHS[J] that can compactly be written as WSBHS[J] = Tr log [1 − ν−1J] p.p. , Wfree[J] = Tr log [1 − ν−1J] p.p. , (5.1)
Summary
It is useful to start from the case of a free CFT in d ≥ 3 where higher spin symmetry is realized as an ordinary global symmetry. For the case of the 3d free boson/fermion CFT’s, the higher spin algebra is just the even subalgebra Ae2 of the Weyl algebra A2 [40,41,42] This is given by even functions f (a, a†) in two pairs of creation/annihilation operators [ai, a†j] = δji , i, j = 1, 2. This interpretation reduces the computation of correlators to simple Gaussian integrals [26,27,28,29], cf [54] It is vital for the three-dimensional bosonization to work, at least in the large-N limit, that the higher spin algebras of free scalar and fermion CFT’s are isomorphic to each other; they should lead to the same A∞- and L∞-algebras and to the same invariants thereof. The higher spin algebras are usually simple and rigid (= admit no deformations as associative algebras)
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