The Geometry of Decoupling Fields

Abstract

We consider 4d field theories obtained by reducing the 6d (1,0) SCFT of N M5-branes probing a C 2 /Z k singularity on a Riemann surface with fluxes. We follow two different routes. On the one hand, we consider the integration of the anomaly polynomial of the parent 6d SCFT on the Riemann surface. On the other hand, we perform an anomaly inflow analysis directly from eleven dimensions, from a setup with M5-branes probing a resolved C 2 /Z k singularity fibered over the Riemann surface. By comparing the 4d anomaly polynomials, we provide a characterization of a class of modes that decouple along the RG flow from six to four dimensions, for generic N , k, and genus. These modes are identified with the flip fields encountered in the Lagrangian descriptions of these 4d models, when they are available. We show that such fields couple to operators originating from M2-branes wrapping the resolution cycles. This provides a geometric origin of flip fields. They interpolate between the 6d theory in the UV, where the M2-brane operators are projected out, and the 4d theory in the IR, where these M2-brane operators are part of the spectrum.

Compactification of 6d ${\cal N}=(1,0)$ quivers, 4d SCFTs and their holographic dual Massive IIA backgrounds

“…This was followed by many works studying compactifications (twisted or with fluxes) from a purely QFT point of view. In the particular case of compactifications of 6d to 4d systems (preserving minimal SUSY in both dimensions), we find the works [18]- [19]. For a very nice summary of these developments from a field theoretical perspective, see [20].…”

“…• More generally, bringing together the field theoretical techniques of [18]- [19] with the holographic results expressed in this paper or in [28], seems like a problem to study. • It should be interesting to exploit the recently discovered plethora of embeddings of 5d supergravity into Massive IIA [29].…”

“…• The presence of extra fields contributing to the central charges (as emphasised in Appendix C) is one of the most urgent topics suggested by this work. May be a treatment along the lines of [19] proves effective. In fact, involving the anomaly polynomials I 8 and I 6 for the SCFT 6 and SCFT 4 may show the need to add flip fields.…”

“…Using the projection (A. 19) and the fact that k = ±1 one can show that .22) Replacing this in the variation, yields Now we turn to (A.39). By replacing (A.56) and a (r) = 0, we get…”

A 5d perspective on the compactifications of 6d SCFTs to 4d $\mathcal{N}=1$ SCFTs

“…Moving down to 4d, many properties of 4d N = 2 Lagrangian as well as strongly coupled non-Lagrangian theories and relations to 6d (2, 0) SCFTs compactified on Riemann surfaces were found in the seminal work of Gaiotto [17], and a classification was suggested in [18]. In recent years a similar effort was focused on understanding properties of less understood 4d N = 1 models using relations to compactifications of 6d (1, 0) SCFTs on a Riemann surface with flux [19][20][21][22][23][24][25][26][27][28][29][30][31][32] (see also [33] for a recent review). 1 These efforts have revealed many dualities and symmetry enhancements of the underlying 4d N = 1 theories understood from geometry, yet there are many models that are not well understood including many strongly coupled theories and SCFTs.…”