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A bstract We revisit the symmetries of massless two-dimensional adjoint QCD with gauge group SU( N ). The dynamics is not sufficiently constrained by the ordinary symmetries and anomalies. Here we show that the theory in fact admits ∼ 2 2 N non-invertible symmetries which severely constrain the possible infrared phases and massive excitations. We prove that for all N these new symmetries enforce deconfinement of the fundamental quark. When the adjoint quark has a small mass, m ≪ g YM , the theory confines and the non-invertible symmetries are softly broken. We use them to compute analytically the k -string tension for N ≤ 5. Our results suggest that the k -string tension, T k , is T k ∼ | m | sin( πk/N ) for all N . We also consider the dynamics of adjoint QCD deformed by symmetric quartic fermion interactions. These operators are not generated by the RG flow due to the non-invertible symmetries, thus violating the ordinary notion of naturalness. We conjecture partial confinement for the deformed theory by these four-fermion interactions, and prove it for SU( N ≤ 5) gauge theory. Comparing the topological phases at zero and large mass, we find that a massless particle ought to appear on the string for some intermediate nonzero mass, consistent with an emergent supersymmetry at nonzero mass. We also study the possible infrared phases of adjoint QCD allowed by the non-invertible symmetries, which we are able to do exhaustively for small values of N . The paper contains detailed reviews of ideas from fusion category theory that are essential for the results we prove.

A bstract We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived, and crossing symmetry is proven. The key ingredients are open-to-closed maps and a boundary crossing relation, by which we show that a diagonal basis exists in the defect Hilbert spaces. We then introduce regular TFTs, provide their explicit constructions for the Fibonacci, Ising and Haagerup ℋ 3 fusion categories, and match our formulae with previous bootstrap results. We end by explaining how non-regular TFTs are obtained from regular TFTs via generalized gauging.

A bstract We explore a situation where a global symmetry of the ultraviolet (UV) theory does not act faithfully on the local infrared (IR) degrees of freedom, but instead acts effectively as a higher-form symmetry. We refer to this phenomenon as symmetry transmutation , where the UV symmetry is “transmuted” into a higher-form symmetry in the IR. Notably, unlike emergent (accidental) symmetries, which are approximate, these symmetries are exact. We illustrate the ubiquity of this phenomenon in various continuum and lattice systems and provide examples where the ’t Hooft anomalies of the UV symmetry are matched by those of the new higher-form symmetry in the IR. We also show that in certain phases and for certain energies, the UV baryon-number symmetry of one-flavor QCD is transmuted into a discrete one-form global symmetry. Finally, we compare our symmetry transmutation to the well-known phenomenon of symmetry fractionalization.

A bstract It is known that the asymptotic density of states of a 2d CFT in an irreducible representation ρ of a finite symmetry group G is proportional to (dim ρ ) 2 . We show how this statement can be generalized when the symmetry can be non-invertible and is described by a fusion category C . Along the way, we explain what plays the role of a representation of a group in the case of a fusion category symmetry; the answer to this question is already available in the broader mathematical physics literature but not yet widely known in hep-th. This understanding immediately implies a selection rule on the correlation functions, and also allows us to derive the asymptotic density.

A bstract We study four-dimensional gauge theories on oriented and non-spin spacetime manifolds. On such manifolds, each line operator arises only either as a boson or a fermion. Based on physical arguments, a method of systematically assigning spin labels to line operators is proposed, and several consistency checks are performed. This is used to classify all possible sets of allowed line operators — including their spins — for gauge theories with simple Lie algebras. The Lagrangian descriptions of the theories with these sets of allowed line operators are given. Finally, the one-form symmetries of these theories are studied by coupling to background gauge fields, and their ’t Hooft anomalies are computed.

A bstract We test recently proposed IR dualities and supersymmetry enhancement by studying the supersymmetry on domain walls. In the SU(3) Wess-Zumino model studied in [ 1 , 2 ], we show that domain walls exhibit supersymmetry enhancement. This model was conjectured to be dual to an N = 2 abelian gauge theory. We show that domain walls on the gauge theory side are consistent with the proposed duality, as they are described by the same effective theory on the wall. In [ 3 ], a third model was conjectured to be dual to the same IR theory. We study the phases and domain walls of this model and we show that they also agree. We then consider the analogous SU(5) Wess-Zumino model, and study its mass deformations and phases. We argue that even though one might expect supersymmetry enhancement in this model as well, the analysis of its domain walls shows that there is none. Finally, we study the N = 2 model in [ 4 ] which was conjectured to have N = 4 supersymmetry in the IR. In this case we don't see the supersymmetry enhancement on the domain wall; however, we argue that half-BPS domain walls of the N = 2 algebra are quarter-BPS of the N = 4 algebra. This is then in agreement with the conjectured enhancement, even though it does not show that it takes place.

We discuss invertible and non-invertible topological condensation defects arising from gauging a discrete higher-form symmetry on a higher codimensional manifold in spacetime, which we define as higher gauging. A q -form symmetry is called p -gaugeable if it can be gauged on a codimension- p manifold in spacetime. We focus on 1-gaugeable 1-form symmetries in general 2+1d QFT, and gauge them on a surface in spacetime. The universal fusion rules of the resulting invertible and non-invertible condensation surfaces are determined. In the special case of 2+1d TQFT, every (invertible and non-invertible) 0-form global symmetry, including the Z 2 electromagnetic symmetry of the Z 2 gauge theory, is realized from higher gauging. We further compute the fusion rules between the surfaces, the bulk lines, and lines that only live on the surfaces, determining some of the most basic data for the underlying fusion 2-category. We emphasize that the fusion “coefficients” in these non-invertible fusion rules are generally not numbers, but rather 1+1d TQFTs. Finally, we discuss examples of non-invertible symmetries in non-topological 2+1d QFTs such as the free U (1) Maxwell theory and QED.

We explore topological defects and boundaries of quantum field theories in various dimensions. In particular, we study non-invertible topological defects that generate non-invertible symmetries. These generalized symmetries lead to powerful dynamical constraints that we explore in this dissertation.In the first part, we study two-dimensional Quantum Chromodynamics (QCD), which serves as a toy model for the four-dimensional QCD that describes the strong nuclear force in nature. In particular, we study two-dimensional QCD with a fermion in the adjoint representation of the gauge group. We discover an exponential number of non-invertible symmetries in this theory. Using these generalized symmetries, we solve the confinement problem for massless and massive adjoint fermion and compute the tension of confining strings analytically among the other results we obtain.Next, we study boundaries of topological quantum field theories (TQFTs) in 2+1-dimensions. TQFTs characterize the low energy behavior of gapped quantum systems with only topological degrees of freedom. Such theories, despite being gapped in the bulk, sometimes carry interesting gapless \"edge modes\" on their boundaries. We find new obstructions to having a gapped boundary, which necessitates gapless edge modes, in 2+1-dimensions known as higher central charges. We obtain these new invariants using non-invertible symmetries and their generalized gauging.Finally, we construct non-invertible symmetries in arbitrary spacetime dimensions known as condensation defects. The construction is by gauging a higher-form symmetry on a higher-codimension submanifold in spacetime, which we denote as higher gauging. We find the fusion rules of condensation surface defects and emphasize that the fusion \"coefficients\" in these non-invertible fusion rules are generally not numbers but rather 1+1d TQFTs.

We study 't Hooft anomalies of global symmetries in 1+1d lattice Hamiltonian systems. We consider anomalies in internal and lattice translation symmetries. We derive a microscopic formula for the \"anomaly cocycle\" using topological defects implementing twisted boundary conditions. The anomaly takes value in the cohomology group \\(H^3(G,U(1)) H^2(G,U(1))\\). The first factor captures the anomaly in the internal symmetry group \\(G\\), and the second factor corresponds to a generalized Lieb-Schultz-Mattis anomaly involving \\(G\\) and lattice translation. We present a systematic procedure to gauge internal symmetries (that may not act on-site) on the lattice. We show that the anomaly cocycle is the obstruction to gauging the internal symmetry while preserving the lattice translation symmetry. As an application, we construct anomaly-free chiral lattice gauge theories. We demonstrate a one-to-one correspondence between (locality-preserving) symmetry operators and topological defects, which is essential for the results we prove. We also discuss the generalization to fermionic theories. Finally, we construct non-invertible lattice translation symmetries by gauging internal symmetries with a Lieb-Schultz-Mattis anomaly.

Using the modified Villain lattice Hamiltonian formulation of the 1+1d compact boson theory, we construct exactly solvable abelian chiral lattice gauge theories in two spacetime dimensions. As a concrete example, we derive an explicit quadratic lattice Hamiltonian for the \"34-50\" chiral gauge theory. We further show that \\(N\\) copies of the modified Villain theory realize the \\(O(N,N;Z)\\) T-duality transformations, which we then use to solve and analyze these lattice gauge theories.