We introduce a class of four dimensional field theories constructed by quotienting ordinary N = 4 U(N ) SYM by particular combinations of R-symmetry and SL(2, Z) automorphisms. These theories appear naturally on the worldvolume of D3 branes probing terminal singularities in F-theory, where they can be thought of as non-perturbative generalizations of the O3 plane. We focus on cases preserving only 12 supercharges, where the quotient gives rise to theories with coupling fixed at a value of order one. These constructions possess an unconventional large N limit described by a non-trivial F-theory fibration with base AdS 5 × (S 5 /Z k ). Upon reduction on a circle the N = 3 theories flow to well-known N = 6 ABJM theories.

Anomalies can be elegantly analyzed by means of the Dai-Freed theorem. In this framework it is natural to consider a refinement of traditional anomaly cancellation conditions, which sometimes leads to nontrivial extra constraints in the fermion spectrum. We analyze these more refined anomaly cancellation conditions in a variety of theories of physical interest, including the Standard Model and the SU (5) and Spin(10) GUTs, which we find to be anomaly free. Turning to discrete symmetries, we find that baryon triality has a Z 9 anomaly that only cancels if the number of generations is a multiple of 3. Assuming the existence of certain anomaly-free Z 4 symmetry we relate the fact that there are 16 fermions per generation of the Standard model -including right-handed neutrinos -to anomalies under time-reversal of boundary states in four-dimensional topological superconductors. A similar relation exists for the MSSM, only this time involving the number of gauginos and Higgsinos, and it is non-trivially, and remarkably, satisfied for the SU (3) × SU (2) × U (1) gauge group with two Higgs doublets. We relate the constraints we find to the well-known Ibañez-Ross ones, and discuss the dependence on UV data of the construction. Finally, we comment on the (non-)existence of K-theoretic θ angles in four dimensions. A On reduced bordism groups 63 B Tables of bordism groups of a point 64 C Bordism groups for Z k 65 D 3d currents 71 E Alternate generators for Ω Spin 5 (BZ n ) 73 -1 -An important technical point is that (2.4) and (2.5) require regularization as usual in quantum field theory. If a regularization preserving the symmetry G for an arbitrary background gauge field can be found, then (2.4) is not anomalous. In particular, this always happens whenever there is a G-invariant mass term m d d xψψ (2.8) for the fermions. In this case, Pauli-Villars regularization is available [5], which is manifestly gauge invariant.

We construct configurations of NS-, D4-, and D6-branes in type IIA string theory, realizing the recently discussed non-supersymmetric meta-stable minimum of 4d N = 1 SU(N c ) super-Yang-Mills theories with massive flavors. We discuss their lift to M-theory and the mechanism of pseudo-moduli stabilization. We extend the construction to many other examples of meta-stable minima, including the SO/Sp theories, SU(N c ) with matter in two-index tensor representations, and to a chiral gauge theory.4 Meta-stable non-supersymmetric vacua had been discussed in the supersymmetry model building literature, see e.g. [2].5 Gauge sectors of the kind described in [1] have been embedded also in heterotic compactifications in [8,9].However, the existence of local meta-stable minima in these constructions, where gravity is not decoupled, remains an open question.

We discuss the geometric origin of discrete higher form symmetries of quantum field theories in terms of defect groups from geometric engineering in M-theory. The flux non-commutativity in M-theory gives rise to (mixed) ’t Hooft anomalies for the defect group which constrains the corresponding global structures of the associated quantum fields. We analyze the example of 4d $$ \mathcal{N} $$ N = 1 SYM gauge theory in four dimensions, and we reproduce the well-known classification of global structures from reading between its lines. We extend this analysis to the case of 7d $$ \mathcal{N} $$ N = 1 SYM theory, where we recover it from a mixed ’t Hooft anomaly among the electric 1-form center symmetry and the magnetic 4-form center symmetry in the defect group. The case of five-dimensional SCFTs from M-theory on toric singularities is discussed in detail. In that context we determine the corresponding 1-form and 2-form defect groups and we explain how to determine the corresponding mixed ’t Hooft anomalies from flux non-commutativity. Several predictions for non-conventional 5d SCFTs are obtained. The matching of discrete higher-form symmetries and their anomalies provides an interesting consistency check for 5d dualities.