The large $ \mathcal{N} $ = 4 superconformal $ \mathcal{W} $ ∞ algebra

Abstract

The most general large N = 4 superconformal W ∞ algebra, containing in addition to the superconformal algebra one supermultiplet for each integer spin, is analysed in detail. It is found that the W ∞ algebra is uniquely determined by the levels of the two su(2) algebras, a conclusion that holds both for the linear and the non-linear case. We also perform various cross-checks of our analysis, and exhibit two different types of truncations in some detail.

The operator product expansion between the 16 lowest higher spin currents in the $$\mathcal{N}=4$$ N = 4 superspace

“…The above 16 higher spin currents can be represented by four 2 So far, the field contents on the right-hand side of the OPEs are the same as the ones in [12] in the N = 2 superspace. For the lowest N = 4 (higher spin) multiplet, the exact relations between the 16 higher spin currents and the ones in this paper (or the ones in [24]) are known explicitly. For the next lowest N = 4 higher spin multiplet, they will depend on (N , k) in a complicated way.…”

“…For s = 1, this is true because the lowest higher spin-1 current commutes with the spin-1 current U (z 1 ) (and four spin-1 2 currents); this is the so-called Goddard-Schwimmer mechanism [28]. For arbitrary s, the description of [24] implies that this is also true. In this calculation, we can easily see that there is no contribution from the second term of …”

“…As in previous sections, we can also compare expression (3.1) with the corresponding quantity in [24] by following the previous procedure for the 16 currents. Let us express the answer as follows:…”

Even spin N = 4 $$ \mathcal{N}=4 $$ holography

“…It corresponds to an even-spin version of the 1-parameter family of large N = 4 W-algebras denoted W N =4 ∞ [λ], constructed in [40], at λ = 0. For generic λ the structure constants of this algebra were shown in [40] to be completely fixed by two parameters k ± , corresponding to the levels of the affine su(2) ± subalgebras of the large N = 4 superconformal algebra. These can be exchanged by the central charge c and the parameter λ as c = 6k…”

“…In the quantum case, i.e. for finite N and k (and therefore finite central charge c), the model is not expected to be a mere truncation of the original [40]. In [38] it was found that there are two natural ways in which the free parameter γ of the quantum bosonic even spin W ∞ -algebra can be identified with λ at finite c. These two ways agree in the classical limit c → ∞, and they correspond to two different quantisations of the classical DS reduction of the even spin bosonic algebra hs e [λ].…”