It is well-known that if we gauge a Z n symmetry in two dimensions, a dual Z n symmetry appears, such that re-gauging this dual Z n symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category. 2 Recently in [5], Gaiotto, Kapustin, Komargodski and Seiberg performed an impressive study of the phase structure of thermal 4d su(2) Yang-Mills theory. One important step in the analysis is the symmetry structure of the thermal system, which is essentially three-dimensional. As a dimensional reduction from 4d, the system has a Z 2 × Z 2 0-form symmetry and a Z 2 1-form symmetry, with a mixed anomaly. Then the authors gauged the Z 2 1-form symmetry, and found that the total 0-form symmetry is now D 8 . This D 8 was then used very effectively to study the phase diagram, but that part of their paper does not directly concern us here. Their analysis of turning an anomalous Abelian symmetry by gauging a non-anomalous subgroup into a non-Abelian symmetry is a 3d analogue of what we explain in 2d. See their Sec. 4.2, Appendix B and Appendix C. Clearly an important direction to pursue is to generalize their and our constructions to arbitrary combinations of possibly-higher-form symmetries in arbitrary spacetime dimensions, but that is outside of the scope of this paper.

We delineate a procedure to classify 6d N = (1, 0) gauge theories composed, in part, of a semi-simple gauge group and hypermultiplets. We classify these theories by requiring that satisfy some consistency conditions. The primary consistency condition is that the gauge anomaly can be cancelled by adding tensor multiplets which couple to the gauge fields by acting as sources of instanton strings. Based on the number of tensor multiplets required to cancel the anomaly, we conjecture that the UV completion of these consistent gauge theories (if it exists) should be either a 6d N = (1, 0) SCFT or a 6d N = (1, 0) little string theory.

Little string theories (LSTs) are UV complete non-local 6D theories decoupled from gravity in which there is an intrinsic string scale. In this paper we present a systematic approach to the construction of supersymmetric LSTs via the geometric phases of F-theory. Our central result is that all LSTs with more than one tensor multiplet are obtained by a mild extension of 6D superconformal field theories (SCFTs) in which the theory is supplemented by an additional, non-dynamical tensor multiplet, analogous to adding an affine node to an ADE quiver, resulting in a negative semidefinite Dirac pairing. We also show that all 6D SCFTs naturally embed in an LST. Motivated by physical considerations, we show that in geometries where we can verify the presence of two elliptic fibrations, exchanging the roles of these fibrations amounts to T-duality in the 6D theory compactified on a circle. *

It is possible to describe fermionic phases of matter and spin-topological field theories in 2+1d in terms of bosonic "shadow" theories, which are obtained from the original theory by "gauging fermionic parity". The fermionic/spin theories are recovered from their shadow by a process of fermionic anyon condensation: gauging a one-form symmetry generated by quasi-particles with fermionic statistics. We apply the formalism to theories which admit gapped boundary conditions. We obtain Turaev-Viro-like and Levin-Wen-like constructions of fermionic phases of matter. We describe the group structure of fermionic SPT phases protected by Z f 2 × G. The quaternion group makes a surprise appearance.

Following a recent proposal, we delineate a general procedure to classify 5d SCFTs via compactifications of 6d SCFTs on a circle (possibly with a twist by a discrete global symmetry). The path from 6d SCFTs to 5d SCFTs can be divided into two steps. The first step involves computing the Coulomb branch data of the 5d KK theory obtained by compactifying a 6d SCFT on a circle of finite radius. The second step involves computing the limit of the KK theory when the inverse radius along with some other mass parameters is sent to infinity. Under this RG flow, the KK theory reduces to a 5d SCFT. We illustrate these ideas in the case of untwisted compactifications of rank one 6d SCFTs that can be constructed in F-theory without frozen singularities. The data of the corresponding KK theory can be packaged in the geometry of a Calabi-Yau threefold that we explicitly compute for every case. The RG flows correspond to flopping a collection of curves in the threefold and we formulate a concrete set of criteria which can be used to determine which collection of curves can induce the relevant RG flows, and, in principle, to determine the Calabi-Yau geometries describing the endpoints of these flows. We also comment on how to generalize our results to arbitrary rank.

We study 6d superconformal field theories (SCFTs) compactified on a circle with arbitrary twists. The theories obtained after compactification, often referred to as 5d Kaluza-Klein (KK) theories, can be viewed as starting points for RG flows to 5d SCFTs. According to a conjecture, all 5d SCFTs can be obtained in this fashion. We compute the Coulomb branch prepotential for all 5d KK theories obtainable in this manner and associate to these theories a smooth local genus one fibered Calabi-Yau threefold in which is encoded information about all possible RG flows to 5d SCFTs. These Calabi-Yau threefolds provide hitherto unknown M-theory duals of F-theory configurations compactified on a circle with twists. For certain exceptional KK theories that do not admit a standard geometric description we propose an algebraic description that appears to retain the properties of the local Calabi-Yau threefolds necessary to determine RG flows to 5d SCFTs, along with other relevant physical data.

According to a conjecture, all 5d SCFTs should be obtainable by rankpreserving RG flows of 6d SCFTs compactified on a circle possibly twisted by a background for the discrete global symmetries around the circle. For a 6d SCFT admitting an F-theory construction, its untwisted compactification admits a dual M-theory description in terms of a "parent" Calabi-Yau threefold which captures the Coulomb branch of the compactified 6d SCFT. The RG flows to 5d SCFTs can then be identified with a sequence of flop transitions and blowdowns of the parent Calabi-Yau leading to "descendant" Calabi-Yau threefolds which describe the Coulomb branches of the resulting 5d SCFTs. An explicit description of parent Calabi-Yaus is known for untwisted compactifications of rank one 6d SCFTs. In this paper, we provide a description of parent Calabi-Yaus for untwisted compactifications of arbitrary rank 6d SCFTs. Since 6d SCFTs of arbitrary rank can be viewed as being constructed out of rank one SCFTs, we accomplish the extension to arbitrary rank by identifying a prescription for gluing together Calabi-Yaus associated to rank one 6d SCFTs. 1 lbhardwaj@fas.harvard.edu 2 patrickjefferson@fas.harvard.edu 42 4.8 e 6 , e 7 , e 8 and f 4 48 4.9 su(3) on −3 50 -i -5 Gluing rules 50 5.1 Gluing of su(m), m ≥ 1 or m =6 and su(n), n ≥ 1 50 5.2 Gluing of sp(m), m ≥ 1 and su(n), n ≥ 1 52 5.3 Gluing of sp(m), m ≥ 1 and so(2r), r ≥ 4 52 5.4 Gluing of sp(m), m ≥ 1 and so(2r + 1), r ≥ 4 53 5.5 Gluing of sp(m), m ≥ 1 and so(7) 53 5.6 Gluing of sp(m), m ≥ 1 and g 2 54 5.7 Gluings of sp(0) = E-string 54 5.7.1 Simply laced 55 5.7.2 Non-simply laced 66 6 Future work 76 A User's guide for Mathematica notebook Pushforward.nb 77 • The set of compact holomorphic surfaces S i .

We describe general methods for determining higher-form symmetry groups of known 5d and 6d superconformal field theories (SCFTs), and 6d little string theories (LSTs). The 6d theories can be described as supersymmetric gauge theories in 6d which include both ordinary non-abelian 1-form gauge fields and also abelian 2-form gauge fields. Similarly, the 5d theories can also be often described as supersymmetric non-abelian gauge theories in 5d. Naively, the 1-form symmetry of these 6d and 5d theories is captured by those elements of the center of ordinary gauge group which leave the matter content of the gauge theory invariant. However, an interesting subtlety is presented by the fact that some massive BPS excitations, which includes the BPS instantons, are charged under the center of the gauge group, thus resulting in a further reduction of the 1-form symmetry. We use the geometric construction of these theories in M/F-theory to determine the charges of these BPS excitations under the center. We also provide an independent algorithm for the determination of 1-form symmetry for 5d theories that admit a generalized toric construction (i.e. a 5-brane web construction). The 2-form symmetry group of 6d theories, on the other hand, is captured by those elements of the center of the abelian 2-form gauge group that leave all the massive BPS string excitations invariant, which is much more straightforward to compute as it is encoded in the Green-Schwarz coupling associated to the 6d theory.