We discuss in detail level/rank duality in three-dimensional Chern-Simons theories and various related dualities in three-dimensional Chern-Simons-matter theories. We couple the dual Lagrangians to appropriate background fields (including gauge fields, spin c connections and the metric). The non-trivial maps between the currents and the line operators in the dual theories is accounted for by mixing of these fields. In order for the duality to be valid we must add finite counterterms depending on these background fields. This analysis allows us to resolve a number of puzzles with these dualities, to provide derivations of some of them, and to find new consistency conditions and relations between them. In addition, we find new level/rank dualities of topological Chern-Simons theories and new dualities of Chern-Simons-matter theories, including new boson/boson and fermion/fermion dualities.August 2016 1. The two-dimensional central charges (and correspondingly the three-dimensional framing anomaly) do not match between the two sides of these dualities. What should we add to them in order to fix it? 2. The chiral algebra of a standard two-dimensional rational conformal field theory includes only integral-dimension operators. This is true in SU (N ) K for all K, U (N ) K,K for even K, and U (N ) K,K±N for odd K (see section 2). Correspondingly, the threedimensional Chern-Simons descriptions of these theories do not depend on the choice of spin structure. This is not true in the other theories in (1.3). They are spin-Chern-Simons theories and depend on the choice of spin structure. 1 How can a non-spin Chern-Simons theory be dual to a spin Chern-Simons theory?3. Consider the cases where the two sides of the duality (1.3) do not depend on the choice of spin structure, i.e. the first duality with even N and the second duality with odd N and K. Even here, the two-dimensional conformal weights of the representations and correspondingly the three-dimensional spins of the line observables do not quite match between the two sides of the duality. They match only modulo one-half. This does not occur in (1.2). But it is puzzling in (1.3).Below we will resolve these puzzles. First, we will add to all the non-spin theories in (1.3) an almost trivial, purely gravitational sector consisting of a single non-trivial line observable of spin one-half. This sector was discussed in [14] and will be reviewed in section 2.3. (In the condensed matter literature this sector is often denoted as {1, f }.) We can also try to add this sector to the spin theories in (1.3), but since they already include such 1 Ordinary, non-spin Chern-Simons theories are independent of a choice of spin structure and the spin theories depend on that choice. The spin theories include a line with half-integral spin (hence it is nontrivial), which is transparent, i.e. it has no braiding with all the lines in the theory. This line does not decouple from the rest of the theory, because it can be obtained by fusing other lines. We can consider such a transparent line als...

In general quantum field theories (QFTs), ordinary (0-form) global symmetries and 1-form symmetries can combine into 2-group global symmetries. We describe this phenomenon in detail using the language of symmetry defects. We exhibit a simple procedure to determine the (possible) 2-group global symmetry of a given QFT, and provide a classification of the related 't Hooft anomalies (for symmetries not acting on spacetime). We also describe how QFTs can be coupled to extrinsic backgrounds for symmetry groups that differ from the intrinsic symmetry acting faithfully on the theory. Finally, we provide a variety of examples, ranging from TQFTs (gapped systems) to gapless QFTs. Along the way, we stress that the "obstruction to symmetry fractionalization" discussed in some condensed matter literature is really an instance of 2-group global symmetry.

We study 3d and 4d systems with a one-form global symmetry, explore their consequences, and analyze their gauging. For simplicity, we focus on Z N one-form symmetries. A 3d topological quantum field theory (TQFT) T with such a symmetry has N special lines that generate it. The braiding of these lines and their spins are characterized by a single integer p modulo 2N . Surprisingly, if gcd(N, p) = 1 the TQFT factorizes T = T ⊗ A N,p . Here T is a decoupled TQFT, whose lines are neutral under the global symmetry and A N,p is a minimal TQFT with the Z N one-form symmetry of label p. The parameter p labels the obstruction to gauging the Z N one-form symmetry; i.e. it characterizes the 't Hooft anomaly of the global symmetry. When p = 0 mod 2N , the symmetry can be gauged. Otherwise, it cannot be gauged unless we couple the system to a 4d bulk with gauge fields extended to the bulk. This understanding allows us to consider SU (N ) and P SU (N ) 4d gauge theories. Their dynamics is gapped and it is associated with confinement and oblique confinement -probe quarks are confined. In the P SU (N ) theory the low-energy theory can include a discrete gauge theory. We will study the behavior of the theory with a spacedependent θ-parameter, which leads to interfaces. Typically, the theory on the interface is not confining. Furthermore, the liberated probe quarks are anyons on the interface. The P SU (N ) theory is obtained by gauging the Z N one-form symmetry of the SU (N ) theory. Our understanding of the symmetries in 3d TQFTs allows us to describe the interface in the P SU (N ) theory. December, 2018a

Abstract:In the last few years several dualities were found between the low-energy behaviors of Chern-Simons-matter theories with unitary gauge groups coupled to scalars, and similar theories coupled to fermions. In this paper we generalize those dualities to orthogonal and symplectic gauge groups. In particular, we conjecture dualities between SO(N ) k Chern-Simons theories coupled to N f real scalars in the fundamental representation, and SO(k) −N +N f /2 theories coupled to N f real (Majorana) fermions in the fundamental. For N f = 0 these are just level-rank dualities of pure Chern-Simons theories, whose precise form we clarify. They lead us to propose new gapped boundary states of topological insulators and superconductors. For k = 1 we get an interesting low-energy duality between N f free Majorana fermions and an SO(N ) 1 Chern-Simons theory coupled to N f scalar fields (with N f ≤ N − 2).