Lakshya Bhardwaj, Simone Giacomelli, Max Hübner, Sakura SchäferNameki
SciPost Phys. 13, 101 (2022) ·
published 26 October 2022

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A relative theory is a boundary condition of a higherdimensional topological quantum field theory (TQFT), and carries a nontrivial defect group formed by mutually nonlocal defects living in the relative theory. Prime examples are $6d$ $\mathcal{N}=(2,0)$ theories that are boundary conditions of $7d$ TQFTs, with the defect group arising from surface defects. In this paper, we study codimensiontwo defects in $6d$ $\mathcal{N}=(2,0)$ theories, and find that the line defects living inside these codimensiontwo defects are mutually nonlocal and hence also form a defect group. Thus, codimensiontwo defects in a $6d$ $\mathcal{N}=(2,0)$ theory are relative defects living inside a relative theory. These relative defects provide boundary conditions for topological defects of the $7d$ bulk TQFT. A codimensiontwo defect carrying a nontrivial defect group acts as an irregular puncture when used in the construction of $4d$ $\mathcal{N}=2$ Class S theories. The defect group associated to such an irregular puncture provides extra "trapped" contributions to the 1form symmetries of the resulting Class S theories. We determine the defect groups associated to large classes of both conformal and nonconformal irregular punctures. Along the way, we discover many new classes of irregular punctures. A key role in the analysis of defect groups is played by two different geometric descriptions of the punctures in Type IIB string theory: one provided by isolated hypersurface singularities in CalabiYau threefolds, and the other provided by ALE fibrations with monodromies.
Max Hübner, David R. Morrison, Sakura SchäferNameki, YiNan Wang
SciPost Phys. 13, 030 (2022) ·
published 19 August 2022

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We realize higherform symmetries in Ftheory compactifications on noncompact elliptically fibered CalabiYau manifolds. Central to this endeavour is the topology of the boundary of the noncompact elliptic fibration, as well as the explicit construction of relative 2cycles in terms of Lefschetz thimbles. We apply the analysis to a variety of elliptic fibrations, including geometries where the discriminant of the elliptic fibration intersects the boundary. We provide a concrete realization of the 1form symmetry group by constructing the associated charged line operator from the elliptic fibration. As an application we compute the symmetry topological field theories in the case of elliptic threefolds, which correspond to mixed anomalies in 5d and 6d theories.
Lakshya Bhardwaj, Max Hübner, Sakura SchäferNameki
SciPost Phys. 12, 040 (2022) ·
published 26 January 2022

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We study confinement in 4d N=1 theories obtained by deforming 4d N=2 theories of Class S. We argue that confinement in a vacuum of the N=1 theory is encoded in the 1cycles of the associated N=1 curve. This curve is the spectral cover associated to a generalized Hitchin system describing the profiles of two Higgs fields over the Riemann surface upon which the 6d (2,0) theory is compactified. Using our method, we reproduce the expected properties of confinement in various classic examples, such as 4d N=1 pure SuperYangMills theory and the CachazoSeibergWitten setup. More generally, this work can be viewed as providing tools for probing confinement in nonLagrangian N=1 theories, which we illustrate by constructing an infinite class of nonLagrangian N=1 theories that contain confining vacua. The simplest model in this class is an N=1 deformation of the N=2 theory obtained by gauging $SU(3)^3$ flavor symmetry of the $E_6$ MinahanNemeschansky theory.
Lakshya Bhardwaj, Max Hübner, Sakura SchäferNameki
SciPost Phys. 11, 096 (2021) ·
published 24 November 2021

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We determine the 1form symmetry group for any 4d N = 2 class S theory constructed by compactifying a 6d N=(2,0) SCFT on a Riemann surface with arbitrary regular untwisted and twisted punctures. The 6d theory has a group of mutually nonlocal dimension2 surface operators, modulo screening. Compactifying these surface operators leads to a group of mutually nonlocal line operators in 4d, modulo screening and flavor charges. Complete specification of a 4d theory arising from such a compactification requires a choice of a maximal subgroup of mutually local line operators, and the 1form symmetry group of the chosen 4d theory is identified as the Pontryagin dual of this maximal subgroup. We also comment on how to generalize our results to compactifications involving irregular punctures. Finally, to complement the analysis from 6d, we derive the 1form symmetry from a Type IIB realization of class S theories.
Dr Hubner: "We thank both referees for the..."
in Submissions  report on Generalized Symmetries in Ftheory and the Topology of Elliptic Fibrations