Exceptional moduli spaces for exceptional $\mathcal{N}=3$ theories

Abstract

It is expected on general grounds that the moduli space of 4d N = 3 theories is of the form C 3r /Γ, with r the rank and Γ a crystallographic complex reflection group (CCRG). As in the case of Lie algebras, the space of CCRGs consists of several infinite families, together with some exceptionals. To date, no 4d N = 3 theory with moduli space labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show that the 4d N = 3 theories proposed in [1], constructed via non-geometric quotients of type-e 6d (2,0) theories, realize nearly all such exceptional moduli spaces. In addition, we introduce an extension of this construction to allow for twists and quotients by outer automorphism symmetries. This gives new examples of 4d N = 3 theories going beyond simple S-folds. 8 in Type IIA. Exchanging the two circles S 1 9 ↔ S 1 8 and lifting back to M-theory gives the original configuration-see Figure 1.Given these symmetries, we can consider various (potentially non-geometric) orbifolds involving them. We will be interested in a quotient by

Non-invertible symmetries of $$ \mathcal{N} $$ = 4 SYM and twisted compactification

“…We then conclude that the resulting 3d theory should have the moduli space C 4n /G(k, 1, n), as expected from the invariant polynomials. This analysis can be done also for the other cases (see [40] for more explicit examples) and leads to the same results as the ones implied by JHEP08(2022)053 the invariant polynomials, so from now on we will only present the arguments following from the invariant polynomials.…”

“…Let us now consider twisted compactifications of N = 4 SYM with gauge algebra su(N ). The basic strategy is similar to the one developed in [40]-namely, we aim to study the resulting theories by understanding the structure of their moduli spaces. Recall that N = 4 SYM with gauge algebra g has moduli space C 3N /W(g), where N and W(g) are the rank and Weyl group of g, respectively.…”

“…transformations z → e 2πi k z with z one of the coordinates of C n . We shall not review aspects of complex reflection groups here, but rather refer the reader to the many expositions on this subject that are now available in the physics literature [40][41][42][43] as well as the resources available in the mathematics literature, e.g. [44].…”

“…Noting that the invariant polynomials of W(su(N )) are of dimension We can also arrive at this result from a direct analysis of the moduli space, as done in [40], which we shall now briefly review. The moduli space of the N = 4 theory is spanned by the vevs of the three complex scalar fields in the vector multiplet.…”

Dirac pairings, one-form symmetries and Seiberg-Witten geometries

“…For example, at rank 2 the ex-ceptional complex reflection group G 8 in the Shephard-Todd classification [31] (with Coulomb branch scaling dimensions 8 and 12) only admits integer symplectic forms with minimum invariant factors (1,2), so are never principal; see section 3.2 of [28]. Furthermore, this Coulomb branch arises as the moduli space of an N =3 SCFT with a known M-theory construction [32,33]. The counting of global structures of theories with non-principal Dirac pairing, reviewed below, implies that this theory has at least 3 distinct global structures.…”