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} = 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} = 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.
Highlights
In this paper we are interested in the study of discrete higher form symmetries for quantum field theories that arise by geometric engineering in M-theory
We stress that D is not the group of higher form symmetries of a quantum field theory yet, rather it is the group of higher form symmetries acting on the geometric engineering Hilbert space, which is the Hilbert space that the given string theory assigns to the given noncompact geometry
As a simpler warm-up example, in section 2 we discuss the case of 7d gauge theories with simple -laced lie groups G ∈ ADE; we find agreement with the global structure obtained by considering Wilson and ’t Hooft operators in the 7d side, and the global structure predicted by M-theory flux non-commutativity
Summary
In this paper we are interested in the study of discrete higher form symmetries for quantum field theories that arise by geometric engineering in M-theory. The operators that are measuring these charges are the flux operators in the string theory we are adopting for the engineering that are sourced by the corresponding kind of p brane These operators are the charge operators for the D(p−k+1) factor of the defect group. Our main aim is to explain how the discrete higher form symmetries arise in this context and what is their relation with the M-theory defect group It is well-known that geometric engineering in string theory gives an alternative formulation of field theories that often proves useful when studying models that cannot be realized perturbatively, which is the case for all SCFTs in dimension higher than four [14, 15]. A more detailed summary of our results can be found in section 1.2 below
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