Higher form symmetries and M-theory

Abstract

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.

Higher-form symmetries of 6d and 5d theories

“…This implies that the left-most and right-most black dots capture the instanton contribution. Indeed, this fact was already observed in [11], see also related observations in [28].…”

“…Indeed, we will see that 1-form symmetries 5d and 6d SCFTs are discrete. From a geometric engineering point of view, higher form symmetries were discussed using the M-theory realization of 5d SCFTs on Calabi-Yau threefolds, as well as other M-theory geometric engineering constructions such as G 2 -holonomy compactifications to 4d in [23,27,28]. Related works in Type IIB, for 4d SCFTs in particular Argyres-Douglas theories were obtained in [25,29,30].…”

“…This severely limits the non-compact complex surfaces that can be wrapped by M5-branes. See[27,28] for more discussion about this discrete data. It is this above mentioned choice of discrete data that gives rise to the 5d theories in the above mentioned class of 5d theories that we will be studying 10.…”

Discrete and higher-form symmetries in SCFTs from wrapped M5-branes

“…Moreover, anomalies provide an organizing principle in exploring the landscape of such theories. Discrete higherform symmetries for field theories engineered by M-theory on a singular local geometry have been recently studied in [4,5].…”

“…The Z2 action is (y 1 , y 2 , y 3 , y 4 , y5 ) → (−y 1 , −y 2 , −y 3 , −y 4 , y5 ) in terms the Cartesian coordinates y 1,2,3,4,5 of R 5 ⊃ S4 . This action has two fixed points on S 4 at y 5 = ±1.…”