VOAs labelled by complex reflection groups and 4d SCFTs

Abstract

We define and study a class of N = 2 vertex operator algebras W G labelled by complex reflection groups. They are extensions of the N = 2 super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group G. If G is a Coxeter group, the N = 2 super Virasoro algebra enhances to the (small) N = 4 superconformal algebra. With the exception of G = Z 2 , which corresponds to just the N = 4 algebra, these are non-deformable VOAs that exist only for a specific negative value of the central charge. We describe a free-field realization of W G in terms of rank(G) βγbc ghost systems, generalizing a construction of Adamovic for the N = 4 algebra at c = −9. If G is a Weyl group, W G is believed to coincide with the N = 4 VOA that arises from the four-dimensional super Yang-Mills theory whose gauge algebra has Weyl group G. More generally, if G is a crystallographic complex reflection group, W G is conjecturally associated to an N = 3 4d superconformal field theory. The free-field realization allows to determine the elusive "R-filtration" of W G , and thus to recover the full Macdonald index of the parent 4d theory.

Free field realizations from the Higgs branch

“…A remark is in order. This free field realization is almost identical to one that was presented in [21] and recently generalized in [12]. 28 The version given here has several advantages.…”

“…We have seen that the free field realizations arising from the Higgs branch EFT of a four-dimensional SCFT come equipped with natural R-filtrations that we propose (as in [12] for a different class of theories) to identify with the physical R-filtration inherited from four dimensions. Having access to this extra structure is of vital importance in promoting VOA spectral data to four dimensional SCFT data.…”

“…Namely, given an abstract presentation of the VOA, there is no known general recipe to define the R-filtration. A remarkable and important feature of the free field realizations described in this paper (as well as those in [12]) is that they come equipped with a natural (and intrinsic) filtration with the right properties to be identified with the R-filtration inherited from four-dimensions. Indeed, by anticipating this application, the form of our free field constructions can be partly motivated by the requirement that they give rise to the right commutative vertex algebra structure at the associated-graded level.…”

“…The expression (6.5) is written in terms of four-dimensional quantum numbers, and the match with the vacuum character of the associated VOA is obtained by recalling that the conformal weight in the chiral algebra is given by h = E − R. The explicit expression for the Macdonald index in all the examples studied in this paper is known. 35 The index for the a 0 and a 1 entries of the DC series and the examples of section 5 can be found in [11]; the a 2 entry coincides with χ[AD (A1,D4) ] and its Macdonald index can be found in [34]; the d 4 , e 6 , e 7 , and e 8 entries are class S theories with regular punctures, so the indices can be obtained using the methods introduced in [32]; finally the case of N = 4 SYM is reviewed in [12]. The specific case of interest, V − 4 3 (sl (2)), is the associated VOA of the four dimensional Argyres-Douglas theory of type (A 1 , D 3 ), and the Macdonald index can be found in [11].…”

On the correspondence between surface operators in Argyres-Douglas theories and modules of chiral algebra

“…The Macdonald limit of the superconformal index [4] with surface operator inserted is also an interesting object to study, as it gives a refined character of the space of the same set of local operators. As discussed in [15,59,30,60], the refinement is realized by a new grading for the number of "basic" operators used to produce a descendant state, which in Virasoro minimal models counts the number of L − ∀ n 's in each state. At least for the vacuum module, a clear way to add this new grading in the computation of the character can be found in the POSET approach to minimal models [61,62], whose generalization to higer-rank cases and non-vacuum modules is worth studying.…”